Hybrid B-Spline Collocation Method for Solving the ...

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Research Article Hybrid B-Spline Collocation Method for Solving the Generalized Burgers-Fisher and Burgers-Huxley Equations Imtiaz Wasim, 1 Muhammad Abbas , 1 and Muhammad Amin 2 1 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 2 Department of Mathematics, National College of Business Administration and Economics, Lahore 54000, Pakistan Correspondence should be addressed to Muhammad Abbas; [email protected] Received 17 August 2017; Revised 5 November 2017; Accepted 13 December 2017; Published 18 January 2018 Academic Editor: Chaudry M. Khalique Copyright © 2018 Imtiaz Wasim et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxley equations using hybrid B-spline collocation method. is technique is based on usual finite difference scheme and Crank-Nicolson method which are used to discretize the time derivative and spatial derivatives, respectively. Furthermore, hybrid B-spline function is utilized as interpolating functions in spatial dimension. e scheme is verified unconditionally stable using the Von Neumann (Fourier) method. Several test problems are considered to check the accuracy of the proposed scheme. e numerical results are in good agreement with known exact solutions and the existing schemes in literature. 1. Introduction Nonlinear partial differential equations (PDEs) play a signif- icant role in different fields of science and engineering. Many physical problems are designed in mathematical form using nonlinear PDEs [1–3]. e generalized Burgers-Fisher (GBF) and generalized Burgers-Huxley (GBH) equations describe many physical phenomena. Numerical treatment of these two equations have become a dominant tool due to complexities in finding their solutions. Consider one-dimensional nonlinear partial differential equation of the following form: + − = () , ≤ ≤ , 0 ≤ ≤ (1) subject to the initial condition, (, 0) = () , ≤ ≤ , (2) and the boundary conditions, (, ) = 1 () , (, ) = 2 () , 0≤≤, (3) where = (, ) and () are some nonlinear expressions in terms of (, ) provided that () = (1 − ) for GBF equation and () = (1− )( −) for GBH equation while , , , , are constants such that , ≥ 0, >0, ∈ (0, 1), ∈ (0, 1]. e nonlinear diffusion models generated from (1) have a significant role in nonlinear physics and of great practical interest. 1.1. Model I. e GBF equation has a lot of applications in the fields such as fluid mechanics [4], gas dynamics, plasma physics [4], number theory, elasticity [5], and heat conduction [6]. is equation becomes the classical Fisher equation when =0, =1 which is one of the significant structures in population biology and is given by − = (1 − ) . (4) Kolmogorov et al. [7] wrote down the same equation for the description of dynamic spread of a combustion front. It arises in several phenomena involving perturbation spreads in excitable mediums, spreading of bacterial colonies [8], spread of reaction fronts in chemically bistable systems [9], and switching in nonlinear optics [10]. Hindawi Mathematical Problems in Engineering Volume 2018, Article ID 6143934, 18 pages https://doi.org/10.1155/2018/6143934

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Research ArticleHybrid B-Spline Collocation Method for Solving the GeneralizedBurgers-Fisher and Burgers-Huxley Equations

ImtiazWasim1 Muhammad Abbas 1 andMuhammad Amin2

1Department of Mathematics University of Sargodha Sargodha 40100 Pakistan2Department of Mathematics National College of Business Administration and Economics Lahore 54000 Pakistan

Correspondence should be addressed to Muhammad Abbas mabbasuosedupk

Received 17 August 2017 Revised 5 November 2017 Accepted 13 December 2017 Published 18 January 2018

Academic Editor Chaudry M Khalique

Copyright copy 2018 Imtiaz Wasim et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study we introduce a new numerical technique for solving nonlinear generalized Burgers-Fisher and Burgers-Huxleyequations using hybrid B-spline collocationmethodThis technique is based on usual finite difference scheme and Crank-Nicolsonmethod which are used to discretize the time derivative and spatial derivatives respectively Furthermore hybrid B-spline functionis utilized as interpolating functions in spatial dimension The scheme is verified unconditionally stable using the Von Neumann(Fourier) method Several test problems are considered to check the accuracy of the proposed schemeThe numerical results are ingood agreement with known exact solutions and the existing schemes in literature

1 Introduction

Nonlinear partial differential equations (PDEs) play a signif-icant role in different fields of science and engineering Manyphysical problems are designed in mathematical form usingnonlinear PDEs [1ndash3] The generalized Burgers-Fisher (GBF)and generalized Burgers-Huxley (GBH) equations describemany physical phenomenaNumerical treatment of these twoequations have become a dominant tool due to complexitiesin finding their solutions

Consider one-dimensional nonlinear partial differentialequation of the following form

119906119905 + 120572119906120575119906119909 minus 120598119906119909119909 = 120573119891 (119906) 119886 le 119909 le 119887 0 le 119905 le 119879 (1)subject to the initial condition119906 (119909 0) = 119892 (119909) 119886 le 119909 le 119887 (2)and the boundary conditions119906 (119886 119905) = 1198921 (119905)

119906 (119887 119905) = 1198922 (119905) 0 le 119905 le 119879

(3)

where 119906 = 119906(119909 119905) and 119891(119906) are some nonlinear expressionsin terms of 119906(119909 119905) provided that 119891(119906) = 119906(1 minus 119906120575) for GBFequation and119891(119906) = 119906(1minus119906120575)(119906120575minus120574) for GBH equationwhile120572 120573 120574 120575 120598 are constants such that 120572 120573 ge 0 120575 gt 0 120574 isin (0 1)120598 isin (0 1] The nonlinear diffusion models generated from(1) have a significant role in nonlinear physics and of greatpractical interest

11 Model I The GBF equation has a lot of applicationsin the fields such as fluid mechanics [4] gas dynamicsplasma physics [4] number theory elasticity [5] and heatconduction [6] This equation becomes the classical Fisherequation when 120572 = 0 120575 = 1 which is one of the significantstructures in population biology and is given by

119906119905 minus 120598119906119909119909 = 120573119906 (1 minus 119906) (4)

Kolmogorov et al [7] wrote down the same equation forthe description of dynamic spread of a combustion front Itarises in several phenomena involving perturbation spreadsin excitable mediums spreading of bacterial colonies [8]spread of reaction fronts in chemically bistable systems [9]and switching in nonlinear optics [10]

HindawiMathematical Problems in EngineeringVolume 2018 Article ID 6143934 18 pageshttpsdoiorg10115520186143934

2 Mathematical Problems in Engineering

The exact solution of GBF is

119906 (119909 119905) = (1205742 + 1205742 tanh (1198891 (119909 minus 1198892119905)))1120575 (5)

subject to the initial condition

119906 (119909 0) = (1205742 + 1205742 tanh (1198891119909))1120575 (6)

and the boundary conditions

119906 (119886 119905) = (1205742 + 1205742 tanh (1198891 (119886 minus 1198892119905)))1120575119906 (119887 119905) = (1205742 + 1205742 tanh (1198891 (119887 minus 1198892119905)))1120575

(7)

where

1198891 = minus120572 + radic1205722 + 4120573 (1 + 120575)4 (1 + 120575) 1205741205751198892 = 120574120572(1 + 120575) minus

(1 + 120575 minus 120574) (minus120572 + radic1205722 + 4120573 (1 + 120575))2 (1 + 120575)

(8)

Wang et al [11] investigated the exact solution of GBHequation with the help of nonlinear transformations

12 Model II Satsuma et al [12] investigated the GBH equa-tion in 1987 This equation reduces to the Huxley equation[11] when 120572 = 0 120575 = 1 120598 = 1 which describes nerve pulsepropagation in nerve fibres and wall motion in liquid crystals[13 14] It can be expressed as follows

119906119905 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 120574) (9)

By considering a well-known experiment in liquid crystals asimilarity between the motion of a wall in liquid crystals andnerve propagation was discussed in [14] These models havebeen studied widely in the last decades due to their impor-tance in neurobiology Hodgkin and Huxley [15] suggestedtheir famous Hodgkin-Huxley model for nerve propagationin 1952 It takes the form of Burgers equation by considering120573 = 0 120575 = 1 120598 = 1 In nonlinear dissipative systems[16] it describes the far field of wave propagation and can beexpressed as follows

119906119905 + 120572119906119906119909 minus 119906119909119909 = 0 (10)

It becomes a FitzHugh-Nagumo (FN) equation when 120572 = 0120573 = 1 120575 = 1 120598 = 1 are chosen Basically it is reactiondiffusion equation utilized in circuit theory and biology [17]and its mathematical form is

119906119905 minus 119906119909119909 = 119906 (1 minus 119906) (119906 minus 120574) (11)

When 120572 = 0 120573 = 0 120575 = 1 this equation also reducesto prototype model named as Burgers-Huxley equationIt describes the interaction between diffusion transportsconvection and reaction mechanisms [18] and is given by

119906119905 + 120572119906119906119909 minus 120598119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 120574) (12)

The exact solution of GBH equation can be written as follows

119906 (119909 119905) = (05 + 05 tanh (1205901 (119909 minus 1205902119905)))1120575 (13)

subject to the initial condition

119906 (119909 0) = (05 + 05 tanh (1205901119909))1120575 (14)

and the boundary conditions

119906 (119886 119905) = (05 + 05 tanh (1205901 (119886 minus 1205902119905)))1120575 119906 (119887 119905) = (05 + 05 tanh (1205901 (119887 minus 1205902119905)))1120575 (15)

where

1205901 = minus1205721205752 (1 + 120575) 1205902 = 120572(1 + 120575) + 120573 (1 + 120575)120572 (16)

This solution was investigated by Xinyi and Yuekai [19] whichis the generalization of the preceding results

Several numerical techniques have been developed to findthe numerical solution of GBF and GBH equations Javidipresented the numerical solution of GBH equation usingspectral collocation method [20] and pseudospectral andpreconditioning [21] and Chebyshev polynomials to developa new domain decomposition algorithm [22] Golbabai andJavidi [23] applied a spectral domain decomposition tech-nique for the numerical solution of GBF equation Darvishi etal [24] investigated the numerical solution of GBH equationby adopting a spectral collocation method and Darvishi etalrsquos preconditioning Sari et al [25] presented the numericalsolution of GBF equation by applying a compact finitedifference scheme Hammad and El-Azab [26] computed thenumerical solution of two types of equations namely GBFand GBH using 2119873 order compact finite difference schemeA computationalmeshlessmethodwas developed by Khattak[27] for solving the GBH equation

Sari and Gurarslan [28] obtained the numerical solu-tion of the GBH equation using a polynomial differentialquadrature method Malik et al [29] developed a heuristicscheme for the numerical solution of the GBF equation basedon the hybridization of Exp-function method with natureinspired algorithm The problem was converted into a non-linear ordinary differential equation (ODE) by substitutionThe travelling wave solution was approximated by the Exp-function method with unknown parameters Dehghan etal [30] developed two numerical methods based on theinterpolating scaling functions and mixed collocation finitedifference schemes for the numerical solution of the GBHequation

Zhang et al [31] developed a new kind of exact finitedifference scheme for solving Burgers equation and Burgers-Fisher equation using the solitary wave solution Biazar andMuhammadi [32] solved GBH equation using differentialtransform method (DTM) Bratsos [33 34] solved GBHequation using a modified predictor-corrector method based

Mathematical Problems in Engineering 3

on a second- and fourth-order time finite difference schemeZarebnia and Aliniya [35] used a mesh-free collocationmethod based on sinc functions for solving the Burgers-Huxley equation Batiha et al [36] applied Hersquos variationaliteration method (VIM) without any discretization to solvethe GBH equation Morufu [37] developed an improvedalgorithm for solving GBF equation based on a Maple codeHashim et al [38] applied Adomian decomposition method(ADM) to get rapidly convergent analytical series solution ofthe GBH equation

Zhao et al [39] approximated the GBF equation using thepseudospectral method based on Crank-Nicolsonleapfrogscheme The approximate solutions were obtained for theGBH and GBF equations using the Adomian and discreteAdomian decompositionmethods [40 41] Inan and Bahadir[42] obtained a numerical solution of the GBH equationusing implicit exponential finite difference method Celik[43] proposed aChebyshevwavelet collocationmethod basedon truncated Chebyshev wavelet series for the solution ofGBH equation Moreover the numerical solution of pro-posed GBH equation was obtained using several numericalmethods named Galerkin method [44] implicit and fullyimplicit exponential finite difference methods [45] Haarwavelet method [46] conditionally bounded and symmetry-preserving method [47] linearly implicit compact scheme[48] positive and bounded finite element method [49]explicit solution scheme [50] exponential time differencingscheme [51] and higher order finite difference schemes [52]

The B-spline collocation scheme is a well-known inter-polating or approximating scheme which provides a goodapproximation rate computationally fast numerically con-sistent and has ability to reproduce the shape of the datawith second order of continuity as compared to polynomialsRecently several numerical schemes based on different typesof B-spline functions were applied to find the numericalsolutions of the differential equations Mittal and Tripathi[53] proposed a numerical scheme based on modified cubicB-spline functions to get the approximate solutions ofGBF and GBH equations Mittal and Jain [54] obtained anumerical solution of nonlinear Burgers equation using amodified cubic B-spline collocation method Singh et al [55]developed a numerical scheme for solving the GBH equationusingmodified cubic B-spline differential quadraturemethod(MCB-DQM) and numerical results can be obtained usingSSP-RK43 scheme Reza [56] implemented the cubic B-splinecollocation scheme based on the finite difference scheme forsolving the GBH equation Reza [57] developed a numericalmethod based on exponential B-spline with finite differenceapproximations to solve the GBF equation Recently Bukhariet al [58] applied local radial basis functions differentialcollocation (LRBDQ) method to compute the numericalsolution of GBH equation

13 Motivation of the Study The finite difference schemeis not the only tool for computing approximations to thesolution of boundary value problems There are variousapproximation techniques which have been examined bymany researchers Spline interpolation method is one ofthe most effective approximation methods on account of its

= 2

= 1

= 0

= 15

= minus102

04

06

08

10

12

xj+2 xj+3 xj+4xj+1

Figure 1 Hybrid B-spline functions with parameter 120583 = 2 15 1 0and minus1simplicity and practicality The main advantage of using thismethod is that it is able to approximate the analytical curveup to certain smoothness Therefore the spline method hasthe flexibility to get the approximation at any point in thedomain with more accurate results compared to the usualfinite difference method This thus provides the motivationfor this study on examining the accuracy of hybrid B-splineon solving nonlinear partial differential equation Howeverone of the limitations of classical B-spline interpolation isthat it does not possess any free parameter for the curvemodification Therefore the shape of the curve is incapableof being altered once the control points are determined Onthe other hand spline interpolation is a global interpolationany changes of the data point will require solving all the linearsystems again The advantage of using hybrid B-spline is thatit possesses a free parameter 120583 to control the global shape ofcurve An appropriate choice of the parameter rises the orderof accuracy of the scheme Hybrid B-spline basis functionreduces to cubic trigonometric B-spline and cubic B-splinefunctionwhen120583 = 0 and 1 respectivelyThis research focuseson the value of 120583 gt 1 Figure 1 depicts the graph of cubictrigonometric B-spline when 120583 = 0 cubic B-spline functionwhen 120583 = 1 and the effect of parameter 120583 for proposedhybrid B-spline function Therefore the superiority of thisspline interpolation method on proposed problem is to beexamined

Although a finite difference scheme is only able to give theapproximations at selected points this method is relativelysimple and very much easy to implement Hence an idea ofcombining finite difference approach with hybrid B-splineinterpolation method for solving proposed problem alsonaturally arose Here hybrid B-spline is used to model thesolution curve at each level of time Thus it is applied tointerpolate the solutions at time 119905 while finite differencescheme is used to discretize the time derivativeThe obtainedresults are more accurate than some available methods inthe literature Stability analysis of the proposed method ispresented and shown to be unconditionally stable withoutany restriction on the choice of step sizes ℎ andΔ119905 An advan-tage of the proposed hybrid B-spline collocation method(HBSCM) outlined in this study is that it produces a splinefunction on each new time line which can be used to obtainthe solution at any intermediate point in the spatial direction

4 Mathematical Problems in Engineering

whereas the finite difference approach yields the solution onlyat the selected points

This article is organized as follows In Section 2 hybridB-spline collocation method a combination of cubic B-spline function and cubic trigonometric B-spline functionwith one free parameter 120583 is constructed and applied toobtain the numerical solutions of the proposed equationsIn Section 3 the method is proved unconditionally stableby Von Neumann approach In Section 4 several numericalcases of GBF and GBH equations are considered to show thefeasibility of the proposed method Finally the conclusion ofthis study is provided

2 Materials and Methods

This section introduces the hybrid B-spline basis functionand derivation of proposed HBSCM for solving the GBF andGBH equations

21 Hybrid B-Spline Basis Function For the discretization ofthe grid region [119886 119887] times [0 119879] an equally divided meshΩ withgrid points (119909119895 119905119896) is considered Here 119909119895 = 119886 + 119895ℎ 119905119896 = 119896Δ119905where 119895 = 0 1 119899 and 119896 = 0 1 119873 while ℎ and Δ119905are spatial size and time step respectively Hybrid B-splinecollocation basis function can be written as follows

1198674119895 (119909)

=

1205836ℎ3 (119909 minus 119909119895)3 + (1 minus 120583)120577 1199033 (119909119895) 119909 isin [119909119895 119909119895+1]1205836ℎ3 (ℎ3 + 3ℎ2 (119909 minus 119909119895+1) + 3ℎ (119909 minus 119909119895+1)2 minus 3 (119909 minus 119909119895+1)3) + (1 minus 120583)120577 (119903 (119909119895) (119903 (119909119895) 119904 (119909119895+2) + 119904 (119909119895+3) 119903 (119909119895+1)) + 119904 (119909119895+4) 1199032 (119909119895+1)) 119909 isin [119909119895+1 119909119895+2]1205836ℎ3 (ℎ3 + 3ℎ2 (119909119895+3 minus 119909) + 3ℎ (119909119895+3 minus 119909)2 minus 3 (119909119895+3 minus 119909)3) + (1 minus 120583)120577 (119904 (119909119895+4) (119903 (119909119895+1) 119904 (119909119895+3) + 119904 (119909119895+4) 119903 (119909119895+2)) + 119903 (119909119895) 1199042 (119909119895+3)) 119909 isin [119909119895+2 119909119895+3]1205836ℎ3 (119909119895+4 minus 119909)3 + (1 minus 120583)120577 1199043 (119909119895+4) 119909 isin [119909119895+3 119909119895+4]

0 otherwise

(17)

where 119903(119909119895) = sin ((119909 minus 119909119895)2) 119904(119909119895) = sin ((119909119895 minus 119909)2) 120577 =sin (ℎ2) sin (ℎ) sin (3ℎ2) and 120583 isin 119877

The approximate solution 119906(119909119895 119905119896) to the exact solution119906exc(119909 119905) can be expressed as follows [59ndash64]

119906119896119895 (119909 119905) = 119895+1sum119898=119895minus1

119863119896119898 (119905)1198674119898 (119909) (18)

where 119863119896119898(119905) are time-dependent unknowns to be deter-mined

The values of1198674119895 (119909) and its derivatives at node 119909 = 119909119895 aregiven by

1198674119898 (119909119895) =

1198861 = 1205836 + (1 minus 120583) sin2 (ℎ2) cosec (ℎ) cosec(3ℎ2 ) 119898 = 119895 plusmn 11198862 = 21205833 + (1 minus 120583) 21 + 2 cos (ℎ) 119898 = 1198950 otherwise

1198891198891199091198674119898 (119909119895) =

1198863 = minus( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 minus 11198864 = ( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 + 10 otherwise

119889211988911990921198674119898 (119909119895) =

1198865 = 120583ℎ2 + (1 minus 120583) 3 (1 + 3 cos (ℎ)) cosec2 (ℎ2)16 (2 cos (ℎ2) + cos (3ℎ2)) 119898 = 119895 plusmn 11198866 = minus(2120583ℎ2 + (1 minus 120583) 3cos2 (ℎ2) cosec2 (ℎ2)(2 + 4 cos (ℎ)) ) 119898 = 1198950 otherwise

(19)

Mathematical Problems in Engineering 5

From (17)ndash(19) the values of 119906119896119895 (119909 119905) and their derivatives atthe knots are calculated in terms of time parameters119863119896119895 (119905) asfollows

119906119896119895 = 1198861119863119896119895minus1 + 1198862119863119896119895 + 1198861119863119896119895+1(119906119909)119896119895 = 1198863119863119896119895minus1 + 0119863119896119895 + 1198864119863119896119895+1(119906119909119909)119896119895 = 1198865119863119896119895minus1 + 1198866119863119896119895 + 1198865119863119896119895+1

(20)

Equation (18) and boundary conditions given in (3) are usedto obtain the approximate solution at end points of the meshas

119906 (1199090 119905119896+1) = 1198861119863119896+1minus1 + 1198862119863119896+10 + 1198861119863119896+11 = 1198921 (119905119896+1) 119906 (119909119899 119905119896+1) = 1198861119863119896+1119899minus1 + 1198862119863119896+1119899 + 1198861119863119896+1119899+1 = 1198922 (119905119896+1) (21)

22 Numerical Solution of the Generalized Burgers-Fisher andGeneralized Burgers-Huxley Equations By utilizing temporaldiscretization and Crank-Nicolson approach (1) can bewritten as

119906119896+1119895 minus 119906119896119895Δ119905 + (120572 (119906120575119906119909)119896+1119895 minus 120598 (119906119909119909)119896+1119895 )2

+ (120572 (119906120575119906119909)119896119895 minus 120598 (119906119909119909)119896119895)2= 120573(119891 (119906119896119895) + 119891 (119906119896+1119895 )2 )

(22)

where 119896 and 119896 + 1 describe successive time positions and119891(119906119896119895 ) = 119906119896119895 (1 minus (119906120575)119896119895 ) for GBF and 119891(119906119896119895 ) = 119906119896119895 (1 minus(119906119896119895 )120575)((119906119896119895 )120575minus120574) for GBH equations After simplification (22)for GBF can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ 05Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)

(23)

and (22) for GBH equation can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)((119906119896+1119895 )120575 minus 120574)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)((119906119896119895)120575 minus 120574)

(24)

The system thus obtained on simplifying (23) for GBFand (24) for GBH problem using (20) consists of 119899 + 1nonlinear equations in 119899 + 3 unknowns 119863119896+1 = (119863119896+1minus1 119863119896+10 119863119896+11 119863119896+1119899+1) at the time level 119905119896+1 Further two equa-tions are included in the resulting system to obtain a uniquesolution of the problem using the boundary conditions givenin (21) The initial vector 1198630 can be obtained from initialcondition given in (2) [59ndash64] Thus the resulting systembecomes amatrix systemof dimension (119899+3)times(119899+3)which isa tridiagonal system that can be solved byThomas algorithm[65ndash67]

3 Stability

This section discusses the Von Neumann criteria to investi-gate the stability of GBF and GBH equations In the productterm consider 119906120575 = 120591 where 120591 is taken [68] as locallyconstant the GBF equation is described as follows

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = (1 minus 120591) 120573119906 (25)

and applying same procedure for nonlinear term as in [64]the GBH equation can be converted to the following

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = minus120574120573119906 (26)

Utilizing same procedure as stated in (22) and setting 1 minus 120591 =120582 the above two GBF and GBH equations take the followingforms

1198871119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 1198874119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

11988611119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 11988622119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

(27)

where 11988611 = 2 + 120573120574Δ119905 11988622 = 2 minus 120573120574Δ119905 1198871 = 2 minus 120573120582Δ119905 1198872 =120572120591Δ119905 1198873 = 120598Δ119905 1198874 = 2 + 120573120582Δ119905Substitute (23) into (27) and simplifying yield

1199021119863119896+1119895minus1 + 1199022119863119896+1119895 + 1199023119863119896+1119895+1= 1199024119863119896119895minus1 + 1199025119863119896119895 + 1199026119863119896119895+1

1199011119863119896+1119895minus1 + 1199012119863119896+1119895 + 1199013119863119896+1119895+1= 1199014119863119896119895minus1 + 1199015119863119896119895 + 1199016119863119896119895+1

(28)

where

1199011 = 119886111198861 + 11988721198863 minus 119887311988651199012 = 119886111198862 minus 119887311988661199013 = 119886111198861 + 11988721198864 minus 119887311988651199014 = 119886221198861 minus 11988721198863 + 119887311988651199015 = 119886221198862 + 11988731198866

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 2: Hybrid B-Spline Collocation Method for Solving the ...

2 Mathematical Problems in Engineering

The exact solution of GBF is

119906 (119909 119905) = (1205742 + 1205742 tanh (1198891 (119909 minus 1198892119905)))1120575 (5)

subject to the initial condition

119906 (119909 0) = (1205742 + 1205742 tanh (1198891119909))1120575 (6)

and the boundary conditions

119906 (119886 119905) = (1205742 + 1205742 tanh (1198891 (119886 minus 1198892119905)))1120575119906 (119887 119905) = (1205742 + 1205742 tanh (1198891 (119887 minus 1198892119905)))1120575

(7)

where

1198891 = minus120572 + radic1205722 + 4120573 (1 + 120575)4 (1 + 120575) 1205741205751198892 = 120574120572(1 + 120575) minus

(1 + 120575 minus 120574) (minus120572 + radic1205722 + 4120573 (1 + 120575))2 (1 + 120575)

(8)

Wang et al [11] investigated the exact solution of GBHequation with the help of nonlinear transformations

12 Model II Satsuma et al [12] investigated the GBH equa-tion in 1987 This equation reduces to the Huxley equation[11] when 120572 = 0 120575 = 1 120598 = 1 which describes nerve pulsepropagation in nerve fibres and wall motion in liquid crystals[13 14] It can be expressed as follows

119906119905 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 120574) (9)

By considering a well-known experiment in liquid crystals asimilarity between the motion of a wall in liquid crystals andnerve propagation was discussed in [14] These models havebeen studied widely in the last decades due to their impor-tance in neurobiology Hodgkin and Huxley [15] suggestedtheir famous Hodgkin-Huxley model for nerve propagationin 1952 It takes the form of Burgers equation by considering120573 = 0 120575 = 1 120598 = 1 In nonlinear dissipative systems[16] it describes the far field of wave propagation and can beexpressed as follows

119906119905 + 120572119906119906119909 minus 119906119909119909 = 0 (10)

It becomes a FitzHugh-Nagumo (FN) equation when 120572 = 0120573 = 1 120575 = 1 120598 = 1 are chosen Basically it is reactiondiffusion equation utilized in circuit theory and biology [17]and its mathematical form is

119906119905 minus 119906119909119909 = 119906 (1 minus 119906) (119906 minus 120574) (11)

When 120572 = 0 120573 = 0 120575 = 1 this equation also reducesto prototype model named as Burgers-Huxley equationIt describes the interaction between diffusion transportsconvection and reaction mechanisms [18] and is given by

119906119905 + 120572119906119906119909 minus 120598119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 120574) (12)

The exact solution of GBH equation can be written as follows

119906 (119909 119905) = (05 + 05 tanh (1205901 (119909 minus 1205902119905)))1120575 (13)

subject to the initial condition

119906 (119909 0) = (05 + 05 tanh (1205901119909))1120575 (14)

and the boundary conditions

119906 (119886 119905) = (05 + 05 tanh (1205901 (119886 minus 1205902119905)))1120575 119906 (119887 119905) = (05 + 05 tanh (1205901 (119887 minus 1205902119905)))1120575 (15)

where

1205901 = minus1205721205752 (1 + 120575) 1205902 = 120572(1 + 120575) + 120573 (1 + 120575)120572 (16)

This solution was investigated by Xinyi and Yuekai [19] whichis the generalization of the preceding results

Several numerical techniques have been developed to findthe numerical solution of GBF and GBH equations Javidipresented the numerical solution of GBH equation usingspectral collocation method [20] and pseudospectral andpreconditioning [21] and Chebyshev polynomials to developa new domain decomposition algorithm [22] Golbabai andJavidi [23] applied a spectral domain decomposition tech-nique for the numerical solution of GBF equation Darvishi etal [24] investigated the numerical solution of GBH equationby adopting a spectral collocation method and Darvishi etalrsquos preconditioning Sari et al [25] presented the numericalsolution of GBF equation by applying a compact finitedifference scheme Hammad and El-Azab [26] computed thenumerical solution of two types of equations namely GBFand GBH using 2119873 order compact finite difference schemeA computationalmeshlessmethodwas developed by Khattak[27] for solving the GBH equation

Sari and Gurarslan [28] obtained the numerical solu-tion of the GBH equation using a polynomial differentialquadrature method Malik et al [29] developed a heuristicscheme for the numerical solution of the GBF equation basedon the hybridization of Exp-function method with natureinspired algorithm The problem was converted into a non-linear ordinary differential equation (ODE) by substitutionThe travelling wave solution was approximated by the Exp-function method with unknown parameters Dehghan etal [30] developed two numerical methods based on theinterpolating scaling functions and mixed collocation finitedifference schemes for the numerical solution of the GBHequation

Zhang et al [31] developed a new kind of exact finitedifference scheme for solving Burgers equation and Burgers-Fisher equation using the solitary wave solution Biazar andMuhammadi [32] solved GBH equation using differentialtransform method (DTM) Bratsos [33 34] solved GBHequation using a modified predictor-corrector method based

Mathematical Problems in Engineering 3

on a second- and fourth-order time finite difference schemeZarebnia and Aliniya [35] used a mesh-free collocationmethod based on sinc functions for solving the Burgers-Huxley equation Batiha et al [36] applied Hersquos variationaliteration method (VIM) without any discretization to solvethe GBH equation Morufu [37] developed an improvedalgorithm for solving GBF equation based on a Maple codeHashim et al [38] applied Adomian decomposition method(ADM) to get rapidly convergent analytical series solution ofthe GBH equation

Zhao et al [39] approximated the GBF equation using thepseudospectral method based on Crank-Nicolsonleapfrogscheme The approximate solutions were obtained for theGBH and GBF equations using the Adomian and discreteAdomian decompositionmethods [40 41] Inan and Bahadir[42] obtained a numerical solution of the GBH equationusing implicit exponential finite difference method Celik[43] proposed aChebyshevwavelet collocationmethod basedon truncated Chebyshev wavelet series for the solution ofGBH equation Moreover the numerical solution of pro-posed GBH equation was obtained using several numericalmethods named Galerkin method [44] implicit and fullyimplicit exponential finite difference methods [45] Haarwavelet method [46] conditionally bounded and symmetry-preserving method [47] linearly implicit compact scheme[48] positive and bounded finite element method [49]explicit solution scheme [50] exponential time differencingscheme [51] and higher order finite difference schemes [52]

The B-spline collocation scheme is a well-known inter-polating or approximating scheme which provides a goodapproximation rate computationally fast numerically con-sistent and has ability to reproduce the shape of the datawith second order of continuity as compared to polynomialsRecently several numerical schemes based on different typesof B-spline functions were applied to find the numericalsolutions of the differential equations Mittal and Tripathi[53] proposed a numerical scheme based on modified cubicB-spline functions to get the approximate solutions ofGBF and GBH equations Mittal and Jain [54] obtained anumerical solution of nonlinear Burgers equation using amodified cubic B-spline collocation method Singh et al [55]developed a numerical scheme for solving the GBH equationusingmodified cubic B-spline differential quadraturemethod(MCB-DQM) and numerical results can be obtained usingSSP-RK43 scheme Reza [56] implemented the cubic B-splinecollocation scheme based on the finite difference scheme forsolving the GBH equation Reza [57] developed a numericalmethod based on exponential B-spline with finite differenceapproximations to solve the GBF equation Recently Bukhariet al [58] applied local radial basis functions differentialcollocation (LRBDQ) method to compute the numericalsolution of GBH equation

13 Motivation of the Study The finite difference schemeis not the only tool for computing approximations to thesolution of boundary value problems There are variousapproximation techniques which have been examined bymany researchers Spline interpolation method is one ofthe most effective approximation methods on account of its

= 2

= 1

= 0

= 15

= minus102

04

06

08

10

12

xj+2 xj+3 xj+4xj+1

Figure 1 Hybrid B-spline functions with parameter 120583 = 2 15 1 0and minus1simplicity and practicality The main advantage of using thismethod is that it is able to approximate the analytical curveup to certain smoothness Therefore the spline method hasthe flexibility to get the approximation at any point in thedomain with more accurate results compared to the usualfinite difference method This thus provides the motivationfor this study on examining the accuracy of hybrid B-splineon solving nonlinear partial differential equation Howeverone of the limitations of classical B-spline interpolation isthat it does not possess any free parameter for the curvemodification Therefore the shape of the curve is incapableof being altered once the control points are determined Onthe other hand spline interpolation is a global interpolationany changes of the data point will require solving all the linearsystems again The advantage of using hybrid B-spline is thatit possesses a free parameter 120583 to control the global shape ofcurve An appropriate choice of the parameter rises the orderof accuracy of the scheme Hybrid B-spline basis functionreduces to cubic trigonometric B-spline and cubic B-splinefunctionwhen120583 = 0 and 1 respectivelyThis research focuseson the value of 120583 gt 1 Figure 1 depicts the graph of cubictrigonometric B-spline when 120583 = 0 cubic B-spline functionwhen 120583 = 1 and the effect of parameter 120583 for proposedhybrid B-spline function Therefore the superiority of thisspline interpolation method on proposed problem is to beexamined

Although a finite difference scheme is only able to give theapproximations at selected points this method is relativelysimple and very much easy to implement Hence an idea ofcombining finite difference approach with hybrid B-splineinterpolation method for solving proposed problem alsonaturally arose Here hybrid B-spline is used to model thesolution curve at each level of time Thus it is applied tointerpolate the solutions at time 119905 while finite differencescheme is used to discretize the time derivativeThe obtainedresults are more accurate than some available methods inthe literature Stability analysis of the proposed method ispresented and shown to be unconditionally stable withoutany restriction on the choice of step sizes ℎ andΔ119905 An advan-tage of the proposed hybrid B-spline collocation method(HBSCM) outlined in this study is that it produces a splinefunction on each new time line which can be used to obtainthe solution at any intermediate point in the spatial direction

4 Mathematical Problems in Engineering

whereas the finite difference approach yields the solution onlyat the selected points

This article is organized as follows In Section 2 hybridB-spline collocation method a combination of cubic B-spline function and cubic trigonometric B-spline functionwith one free parameter 120583 is constructed and applied toobtain the numerical solutions of the proposed equationsIn Section 3 the method is proved unconditionally stableby Von Neumann approach In Section 4 several numericalcases of GBF and GBH equations are considered to show thefeasibility of the proposed method Finally the conclusion ofthis study is provided

2 Materials and Methods

This section introduces the hybrid B-spline basis functionand derivation of proposed HBSCM for solving the GBF andGBH equations

21 Hybrid B-Spline Basis Function For the discretization ofthe grid region [119886 119887] times [0 119879] an equally divided meshΩ withgrid points (119909119895 119905119896) is considered Here 119909119895 = 119886 + 119895ℎ 119905119896 = 119896Δ119905where 119895 = 0 1 119899 and 119896 = 0 1 119873 while ℎ and Δ119905are spatial size and time step respectively Hybrid B-splinecollocation basis function can be written as follows

1198674119895 (119909)

=

1205836ℎ3 (119909 minus 119909119895)3 + (1 minus 120583)120577 1199033 (119909119895) 119909 isin [119909119895 119909119895+1]1205836ℎ3 (ℎ3 + 3ℎ2 (119909 minus 119909119895+1) + 3ℎ (119909 minus 119909119895+1)2 minus 3 (119909 minus 119909119895+1)3) + (1 minus 120583)120577 (119903 (119909119895) (119903 (119909119895) 119904 (119909119895+2) + 119904 (119909119895+3) 119903 (119909119895+1)) + 119904 (119909119895+4) 1199032 (119909119895+1)) 119909 isin [119909119895+1 119909119895+2]1205836ℎ3 (ℎ3 + 3ℎ2 (119909119895+3 minus 119909) + 3ℎ (119909119895+3 minus 119909)2 minus 3 (119909119895+3 minus 119909)3) + (1 minus 120583)120577 (119904 (119909119895+4) (119903 (119909119895+1) 119904 (119909119895+3) + 119904 (119909119895+4) 119903 (119909119895+2)) + 119903 (119909119895) 1199042 (119909119895+3)) 119909 isin [119909119895+2 119909119895+3]1205836ℎ3 (119909119895+4 minus 119909)3 + (1 minus 120583)120577 1199043 (119909119895+4) 119909 isin [119909119895+3 119909119895+4]

0 otherwise

(17)

where 119903(119909119895) = sin ((119909 minus 119909119895)2) 119904(119909119895) = sin ((119909119895 minus 119909)2) 120577 =sin (ℎ2) sin (ℎ) sin (3ℎ2) and 120583 isin 119877

The approximate solution 119906(119909119895 119905119896) to the exact solution119906exc(119909 119905) can be expressed as follows [59ndash64]

119906119896119895 (119909 119905) = 119895+1sum119898=119895minus1

119863119896119898 (119905)1198674119898 (119909) (18)

where 119863119896119898(119905) are time-dependent unknowns to be deter-mined

The values of1198674119895 (119909) and its derivatives at node 119909 = 119909119895 aregiven by

1198674119898 (119909119895) =

1198861 = 1205836 + (1 minus 120583) sin2 (ℎ2) cosec (ℎ) cosec(3ℎ2 ) 119898 = 119895 plusmn 11198862 = 21205833 + (1 minus 120583) 21 + 2 cos (ℎ) 119898 = 1198950 otherwise

1198891198891199091198674119898 (119909119895) =

1198863 = minus( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 minus 11198864 = ( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 + 10 otherwise

119889211988911990921198674119898 (119909119895) =

1198865 = 120583ℎ2 + (1 minus 120583) 3 (1 + 3 cos (ℎ)) cosec2 (ℎ2)16 (2 cos (ℎ2) + cos (3ℎ2)) 119898 = 119895 plusmn 11198866 = minus(2120583ℎ2 + (1 minus 120583) 3cos2 (ℎ2) cosec2 (ℎ2)(2 + 4 cos (ℎ)) ) 119898 = 1198950 otherwise

(19)

Mathematical Problems in Engineering 5

From (17)ndash(19) the values of 119906119896119895 (119909 119905) and their derivatives atthe knots are calculated in terms of time parameters119863119896119895 (119905) asfollows

119906119896119895 = 1198861119863119896119895minus1 + 1198862119863119896119895 + 1198861119863119896119895+1(119906119909)119896119895 = 1198863119863119896119895minus1 + 0119863119896119895 + 1198864119863119896119895+1(119906119909119909)119896119895 = 1198865119863119896119895minus1 + 1198866119863119896119895 + 1198865119863119896119895+1

(20)

Equation (18) and boundary conditions given in (3) are usedto obtain the approximate solution at end points of the meshas

119906 (1199090 119905119896+1) = 1198861119863119896+1minus1 + 1198862119863119896+10 + 1198861119863119896+11 = 1198921 (119905119896+1) 119906 (119909119899 119905119896+1) = 1198861119863119896+1119899minus1 + 1198862119863119896+1119899 + 1198861119863119896+1119899+1 = 1198922 (119905119896+1) (21)

22 Numerical Solution of the Generalized Burgers-Fisher andGeneralized Burgers-Huxley Equations By utilizing temporaldiscretization and Crank-Nicolson approach (1) can bewritten as

119906119896+1119895 minus 119906119896119895Δ119905 + (120572 (119906120575119906119909)119896+1119895 minus 120598 (119906119909119909)119896+1119895 )2

+ (120572 (119906120575119906119909)119896119895 minus 120598 (119906119909119909)119896119895)2= 120573(119891 (119906119896119895) + 119891 (119906119896+1119895 )2 )

(22)

where 119896 and 119896 + 1 describe successive time positions and119891(119906119896119895 ) = 119906119896119895 (1 minus (119906120575)119896119895 ) for GBF and 119891(119906119896119895 ) = 119906119896119895 (1 minus(119906119896119895 )120575)((119906119896119895 )120575minus120574) for GBH equations After simplification (22)for GBF can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ 05Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)

(23)

and (22) for GBH equation can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)((119906119896+1119895 )120575 minus 120574)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)((119906119896119895)120575 minus 120574)

(24)

The system thus obtained on simplifying (23) for GBFand (24) for GBH problem using (20) consists of 119899 + 1nonlinear equations in 119899 + 3 unknowns 119863119896+1 = (119863119896+1minus1 119863119896+10 119863119896+11 119863119896+1119899+1) at the time level 119905119896+1 Further two equa-tions are included in the resulting system to obtain a uniquesolution of the problem using the boundary conditions givenin (21) The initial vector 1198630 can be obtained from initialcondition given in (2) [59ndash64] Thus the resulting systembecomes amatrix systemof dimension (119899+3)times(119899+3)which isa tridiagonal system that can be solved byThomas algorithm[65ndash67]

3 Stability

This section discusses the Von Neumann criteria to investi-gate the stability of GBF and GBH equations In the productterm consider 119906120575 = 120591 where 120591 is taken [68] as locallyconstant the GBF equation is described as follows

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = (1 minus 120591) 120573119906 (25)

and applying same procedure for nonlinear term as in [64]the GBH equation can be converted to the following

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = minus120574120573119906 (26)

Utilizing same procedure as stated in (22) and setting 1 minus 120591 =120582 the above two GBF and GBH equations take the followingforms

1198871119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 1198874119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

11988611119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 11988622119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

(27)

where 11988611 = 2 + 120573120574Δ119905 11988622 = 2 minus 120573120574Δ119905 1198871 = 2 minus 120573120582Δ119905 1198872 =120572120591Δ119905 1198873 = 120598Δ119905 1198874 = 2 + 120573120582Δ119905Substitute (23) into (27) and simplifying yield

1199021119863119896+1119895minus1 + 1199022119863119896+1119895 + 1199023119863119896+1119895+1= 1199024119863119896119895minus1 + 1199025119863119896119895 + 1199026119863119896119895+1

1199011119863119896+1119895minus1 + 1199012119863119896+1119895 + 1199013119863119896+1119895+1= 1199014119863119896119895minus1 + 1199015119863119896119895 + 1199016119863119896119895+1

(28)

where

1199011 = 119886111198861 + 11988721198863 minus 119887311988651199012 = 119886111198862 minus 119887311988661199013 = 119886111198861 + 11988721198864 minus 119887311988651199014 = 119886221198861 minus 11988721198863 + 119887311988651199015 = 119886221198862 + 11988731198866

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 3: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 3

on a second- and fourth-order time finite difference schemeZarebnia and Aliniya [35] used a mesh-free collocationmethod based on sinc functions for solving the Burgers-Huxley equation Batiha et al [36] applied Hersquos variationaliteration method (VIM) without any discretization to solvethe GBH equation Morufu [37] developed an improvedalgorithm for solving GBF equation based on a Maple codeHashim et al [38] applied Adomian decomposition method(ADM) to get rapidly convergent analytical series solution ofthe GBH equation

Zhao et al [39] approximated the GBF equation using thepseudospectral method based on Crank-Nicolsonleapfrogscheme The approximate solutions were obtained for theGBH and GBF equations using the Adomian and discreteAdomian decompositionmethods [40 41] Inan and Bahadir[42] obtained a numerical solution of the GBH equationusing implicit exponential finite difference method Celik[43] proposed aChebyshevwavelet collocationmethod basedon truncated Chebyshev wavelet series for the solution ofGBH equation Moreover the numerical solution of pro-posed GBH equation was obtained using several numericalmethods named Galerkin method [44] implicit and fullyimplicit exponential finite difference methods [45] Haarwavelet method [46] conditionally bounded and symmetry-preserving method [47] linearly implicit compact scheme[48] positive and bounded finite element method [49]explicit solution scheme [50] exponential time differencingscheme [51] and higher order finite difference schemes [52]

The B-spline collocation scheme is a well-known inter-polating or approximating scheme which provides a goodapproximation rate computationally fast numerically con-sistent and has ability to reproduce the shape of the datawith second order of continuity as compared to polynomialsRecently several numerical schemes based on different typesof B-spline functions were applied to find the numericalsolutions of the differential equations Mittal and Tripathi[53] proposed a numerical scheme based on modified cubicB-spline functions to get the approximate solutions ofGBF and GBH equations Mittal and Jain [54] obtained anumerical solution of nonlinear Burgers equation using amodified cubic B-spline collocation method Singh et al [55]developed a numerical scheme for solving the GBH equationusingmodified cubic B-spline differential quadraturemethod(MCB-DQM) and numerical results can be obtained usingSSP-RK43 scheme Reza [56] implemented the cubic B-splinecollocation scheme based on the finite difference scheme forsolving the GBH equation Reza [57] developed a numericalmethod based on exponential B-spline with finite differenceapproximations to solve the GBF equation Recently Bukhariet al [58] applied local radial basis functions differentialcollocation (LRBDQ) method to compute the numericalsolution of GBH equation

13 Motivation of the Study The finite difference schemeis not the only tool for computing approximations to thesolution of boundary value problems There are variousapproximation techniques which have been examined bymany researchers Spline interpolation method is one ofthe most effective approximation methods on account of its

= 2

= 1

= 0

= 15

= minus102

04

06

08

10

12

xj+2 xj+3 xj+4xj+1

Figure 1 Hybrid B-spline functions with parameter 120583 = 2 15 1 0and minus1simplicity and practicality The main advantage of using thismethod is that it is able to approximate the analytical curveup to certain smoothness Therefore the spline method hasthe flexibility to get the approximation at any point in thedomain with more accurate results compared to the usualfinite difference method This thus provides the motivationfor this study on examining the accuracy of hybrid B-splineon solving nonlinear partial differential equation Howeverone of the limitations of classical B-spline interpolation isthat it does not possess any free parameter for the curvemodification Therefore the shape of the curve is incapableof being altered once the control points are determined Onthe other hand spline interpolation is a global interpolationany changes of the data point will require solving all the linearsystems again The advantage of using hybrid B-spline is thatit possesses a free parameter 120583 to control the global shape ofcurve An appropriate choice of the parameter rises the orderof accuracy of the scheme Hybrid B-spline basis functionreduces to cubic trigonometric B-spline and cubic B-splinefunctionwhen120583 = 0 and 1 respectivelyThis research focuseson the value of 120583 gt 1 Figure 1 depicts the graph of cubictrigonometric B-spline when 120583 = 0 cubic B-spline functionwhen 120583 = 1 and the effect of parameter 120583 for proposedhybrid B-spline function Therefore the superiority of thisspline interpolation method on proposed problem is to beexamined

Although a finite difference scheme is only able to give theapproximations at selected points this method is relativelysimple and very much easy to implement Hence an idea ofcombining finite difference approach with hybrid B-splineinterpolation method for solving proposed problem alsonaturally arose Here hybrid B-spline is used to model thesolution curve at each level of time Thus it is applied tointerpolate the solutions at time 119905 while finite differencescheme is used to discretize the time derivativeThe obtainedresults are more accurate than some available methods inthe literature Stability analysis of the proposed method ispresented and shown to be unconditionally stable withoutany restriction on the choice of step sizes ℎ andΔ119905 An advan-tage of the proposed hybrid B-spline collocation method(HBSCM) outlined in this study is that it produces a splinefunction on each new time line which can be used to obtainthe solution at any intermediate point in the spatial direction

4 Mathematical Problems in Engineering

whereas the finite difference approach yields the solution onlyat the selected points

This article is organized as follows In Section 2 hybridB-spline collocation method a combination of cubic B-spline function and cubic trigonometric B-spline functionwith one free parameter 120583 is constructed and applied toobtain the numerical solutions of the proposed equationsIn Section 3 the method is proved unconditionally stableby Von Neumann approach In Section 4 several numericalcases of GBF and GBH equations are considered to show thefeasibility of the proposed method Finally the conclusion ofthis study is provided

2 Materials and Methods

This section introduces the hybrid B-spline basis functionand derivation of proposed HBSCM for solving the GBF andGBH equations

21 Hybrid B-Spline Basis Function For the discretization ofthe grid region [119886 119887] times [0 119879] an equally divided meshΩ withgrid points (119909119895 119905119896) is considered Here 119909119895 = 119886 + 119895ℎ 119905119896 = 119896Δ119905where 119895 = 0 1 119899 and 119896 = 0 1 119873 while ℎ and Δ119905are spatial size and time step respectively Hybrid B-splinecollocation basis function can be written as follows

1198674119895 (119909)

=

1205836ℎ3 (119909 minus 119909119895)3 + (1 minus 120583)120577 1199033 (119909119895) 119909 isin [119909119895 119909119895+1]1205836ℎ3 (ℎ3 + 3ℎ2 (119909 minus 119909119895+1) + 3ℎ (119909 minus 119909119895+1)2 minus 3 (119909 minus 119909119895+1)3) + (1 minus 120583)120577 (119903 (119909119895) (119903 (119909119895) 119904 (119909119895+2) + 119904 (119909119895+3) 119903 (119909119895+1)) + 119904 (119909119895+4) 1199032 (119909119895+1)) 119909 isin [119909119895+1 119909119895+2]1205836ℎ3 (ℎ3 + 3ℎ2 (119909119895+3 minus 119909) + 3ℎ (119909119895+3 minus 119909)2 minus 3 (119909119895+3 minus 119909)3) + (1 minus 120583)120577 (119904 (119909119895+4) (119903 (119909119895+1) 119904 (119909119895+3) + 119904 (119909119895+4) 119903 (119909119895+2)) + 119903 (119909119895) 1199042 (119909119895+3)) 119909 isin [119909119895+2 119909119895+3]1205836ℎ3 (119909119895+4 minus 119909)3 + (1 minus 120583)120577 1199043 (119909119895+4) 119909 isin [119909119895+3 119909119895+4]

0 otherwise

(17)

where 119903(119909119895) = sin ((119909 minus 119909119895)2) 119904(119909119895) = sin ((119909119895 minus 119909)2) 120577 =sin (ℎ2) sin (ℎ) sin (3ℎ2) and 120583 isin 119877

The approximate solution 119906(119909119895 119905119896) to the exact solution119906exc(119909 119905) can be expressed as follows [59ndash64]

119906119896119895 (119909 119905) = 119895+1sum119898=119895minus1

119863119896119898 (119905)1198674119898 (119909) (18)

where 119863119896119898(119905) are time-dependent unknowns to be deter-mined

The values of1198674119895 (119909) and its derivatives at node 119909 = 119909119895 aregiven by

1198674119898 (119909119895) =

1198861 = 1205836 + (1 minus 120583) sin2 (ℎ2) cosec (ℎ) cosec(3ℎ2 ) 119898 = 119895 plusmn 11198862 = 21205833 + (1 minus 120583) 21 + 2 cos (ℎ) 119898 = 1198950 otherwise

1198891198891199091198674119898 (119909119895) =

1198863 = minus( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 minus 11198864 = ( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 + 10 otherwise

119889211988911990921198674119898 (119909119895) =

1198865 = 120583ℎ2 + (1 minus 120583) 3 (1 + 3 cos (ℎ)) cosec2 (ℎ2)16 (2 cos (ℎ2) + cos (3ℎ2)) 119898 = 119895 plusmn 11198866 = minus(2120583ℎ2 + (1 minus 120583) 3cos2 (ℎ2) cosec2 (ℎ2)(2 + 4 cos (ℎ)) ) 119898 = 1198950 otherwise

(19)

Mathematical Problems in Engineering 5

From (17)ndash(19) the values of 119906119896119895 (119909 119905) and their derivatives atthe knots are calculated in terms of time parameters119863119896119895 (119905) asfollows

119906119896119895 = 1198861119863119896119895minus1 + 1198862119863119896119895 + 1198861119863119896119895+1(119906119909)119896119895 = 1198863119863119896119895minus1 + 0119863119896119895 + 1198864119863119896119895+1(119906119909119909)119896119895 = 1198865119863119896119895minus1 + 1198866119863119896119895 + 1198865119863119896119895+1

(20)

Equation (18) and boundary conditions given in (3) are usedto obtain the approximate solution at end points of the meshas

119906 (1199090 119905119896+1) = 1198861119863119896+1minus1 + 1198862119863119896+10 + 1198861119863119896+11 = 1198921 (119905119896+1) 119906 (119909119899 119905119896+1) = 1198861119863119896+1119899minus1 + 1198862119863119896+1119899 + 1198861119863119896+1119899+1 = 1198922 (119905119896+1) (21)

22 Numerical Solution of the Generalized Burgers-Fisher andGeneralized Burgers-Huxley Equations By utilizing temporaldiscretization and Crank-Nicolson approach (1) can bewritten as

119906119896+1119895 minus 119906119896119895Δ119905 + (120572 (119906120575119906119909)119896+1119895 minus 120598 (119906119909119909)119896+1119895 )2

+ (120572 (119906120575119906119909)119896119895 minus 120598 (119906119909119909)119896119895)2= 120573(119891 (119906119896119895) + 119891 (119906119896+1119895 )2 )

(22)

where 119896 and 119896 + 1 describe successive time positions and119891(119906119896119895 ) = 119906119896119895 (1 minus (119906120575)119896119895 ) for GBF and 119891(119906119896119895 ) = 119906119896119895 (1 minus(119906119896119895 )120575)((119906119896119895 )120575minus120574) for GBH equations After simplification (22)for GBF can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ 05Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)

(23)

and (22) for GBH equation can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)((119906119896+1119895 )120575 minus 120574)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)((119906119896119895)120575 minus 120574)

(24)

The system thus obtained on simplifying (23) for GBFand (24) for GBH problem using (20) consists of 119899 + 1nonlinear equations in 119899 + 3 unknowns 119863119896+1 = (119863119896+1minus1 119863119896+10 119863119896+11 119863119896+1119899+1) at the time level 119905119896+1 Further two equa-tions are included in the resulting system to obtain a uniquesolution of the problem using the boundary conditions givenin (21) The initial vector 1198630 can be obtained from initialcondition given in (2) [59ndash64] Thus the resulting systembecomes amatrix systemof dimension (119899+3)times(119899+3)which isa tridiagonal system that can be solved byThomas algorithm[65ndash67]

3 Stability

This section discusses the Von Neumann criteria to investi-gate the stability of GBF and GBH equations In the productterm consider 119906120575 = 120591 where 120591 is taken [68] as locallyconstant the GBF equation is described as follows

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = (1 minus 120591) 120573119906 (25)

and applying same procedure for nonlinear term as in [64]the GBH equation can be converted to the following

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = minus120574120573119906 (26)

Utilizing same procedure as stated in (22) and setting 1 minus 120591 =120582 the above two GBF and GBH equations take the followingforms

1198871119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 1198874119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

11988611119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 11988622119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

(27)

where 11988611 = 2 + 120573120574Δ119905 11988622 = 2 minus 120573120574Δ119905 1198871 = 2 minus 120573120582Δ119905 1198872 =120572120591Δ119905 1198873 = 120598Δ119905 1198874 = 2 + 120573120582Δ119905Substitute (23) into (27) and simplifying yield

1199021119863119896+1119895minus1 + 1199022119863119896+1119895 + 1199023119863119896+1119895+1= 1199024119863119896119895minus1 + 1199025119863119896119895 + 1199026119863119896119895+1

1199011119863119896+1119895minus1 + 1199012119863119896+1119895 + 1199013119863119896+1119895+1= 1199014119863119896119895minus1 + 1199015119863119896119895 + 1199016119863119896119895+1

(28)

where

1199011 = 119886111198861 + 11988721198863 minus 119887311988651199012 = 119886111198862 minus 119887311988661199013 = 119886111198861 + 11988721198864 minus 119887311988651199014 = 119886221198861 minus 11988721198863 + 119887311988651199015 = 119886221198862 + 11988731198866

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 4: Hybrid B-Spline Collocation Method for Solving the ...

4 Mathematical Problems in Engineering

whereas the finite difference approach yields the solution onlyat the selected points

This article is organized as follows In Section 2 hybridB-spline collocation method a combination of cubic B-spline function and cubic trigonometric B-spline functionwith one free parameter 120583 is constructed and applied toobtain the numerical solutions of the proposed equationsIn Section 3 the method is proved unconditionally stableby Von Neumann approach In Section 4 several numericalcases of GBF and GBH equations are considered to show thefeasibility of the proposed method Finally the conclusion ofthis study is provided

2 Materials and Methods

This section introduces the hybrid B-spline basis functionand derivation of proposed HBSCM for solving the GBF andGBH equations

21 Hybrid B-Spline Basis Function For the discretization ofthe grid region [119886 119887] times [0 119879] an equally divided meshΩ withgrid points (119909119895 119905119896) is considered Here 119909119895 = 119886 + 119895ℎ 119905119896 = 119896Δ119905where 119895 = 0 1 119899 and 119896 = 0 1 119873 while ℎ and Δ119905are spatial size and time step respectively Hybrid B-splinecollocation basis function can be written as follows

1198674119895 (119909)

=

1205836ℎ3 (119909 minus 119909119895)3 + (1 minus 120583)120577 1199033 (119909119895) 119909 isin [119909119895 119909119895+1]1205836ℎ3 (ℎ3 + 3ℎ2 (119909 minus 119909119895+1) + 3ℎ (119909 minus 119909119895+1)2 minus 3 (119909 minus 119909119895+1)3) + (1 minus 120583)120577 (119903 (119909119895) (119903 (119909119895) 119904 (119909119895+2) + 119904 (119909119895+3) 119903 (119909119895+1)) + 119904 (119909119895+4) 1199032 (119909119895+1)) 119909 isin [119909119895+1 119909119895+2]1205836ℎ3 (ℎ3 + 3ℎ2 (119909119895+3 minus 119909) + 3ℎ (119909119895+3 minus 119909)2 minus 3 (119909119895+3 minus 119909)3) + (1 minus 120583)120577 (119904 (119909119895+4) (119903 (119909119895+1) 119904 (119909119895+3) + 119904 (119909119895+4) 119903 (119909119895+2)) + 119903 (119909119895) 1199042 (119909119895+3)) 119909 isin [119909119895+2 119909119895+3]1205836ℎ3 (119909119895+4 minus 119909)3 + (1 minus 120583)120577 1199043 (119909119895+4) 119909 isin [119909119895+3 119909119895+4]

0 otherwise

(17)

where 119903(119909119895) = sin ((119909 minus 119909119895)2) 119904(119909119895) = sin ((119909119895 minus 119909)2) 120577 =sin (ℎ2) sin (ℎ) sin (3ℎ2) and 120583 isin 119877

The approximate solution 119906(119909119895 119905119896) to the exact solution119906exc(119909 119905) can be expressed as follows [59ndash64]

119906119896119895 (119909 119905) = 119895+1sum119898=119895minus1

119863119896119898 (119905)1198674119898 (119909) (18)

where 119863119896119898(119905) are time-dependent unknowns to be deter-mined

The values of1198674119895 (119909) and its derivatives at node 119909 = 119909119895 aregiven by

1198674119898 (119909119895) =

1198861 = 1205836 + (1 minus 120583) sin2 (ℎ2) cosec (ℎ) cosec(3ℎ2 ) 119898 = 119895 plusmn 11198862 = 21205833 + (1 minus 120583) 21 + 2 cos (ℎ) 119898 = 1198950 otherwise

1198891198891199091198674119898 (119909119895) =

1198863 = minus( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 minus 11198864 = ( 1205832ℎ + (1 minus 120583) 34cosec(3ℎ2 )) 119898 = 119895 + 10 otherwise

119889211988911990921198674119898 (119909119895) =

1198865 = 120583ℎ2 + (1 minus 120583) 3 (1 + 3 cos (ℎ)) cosec2 (ℎ2)16 (2 cos (ℎ2) + cos (3ℎ2)) 119898 = 119895 plusmn 11198866 = minus(2120583ℎ2 + (1 minus 120583) 3cos2 (ℎ2) cosec2 (ℎ2)(2 + 4 cos (ℎ)) ) 119898 = 1198950 otherwise

(19)

Mathematical Problems in Engineering 5

From (17)ndash(19) the values of 119906119896119895 (119909 119905) and their derivatives atthe knots are calculated in terms of time parameters119863119896119895 (119905) asfollows

119906119896119895 = 1198861119863119896119895minus1 + 1198862119863119896119895 + 1198861119863119896119895+1(119906119909)119896119895 = 1198863119863119896119895minus1 + 0119863119896119895 + 1198864119863119896119895+1(119906119909119909)119896119895 = 1198865119863119896119895minus1 + 1198866119863119896119895 + 1198865119863119896119895+1

(20)

Equation (18) and boundary conditions given in (3) are usedto obtain the approximate solution at end points of the meshas

119906 (1199090 119905119896+1) = 1198861119863119896+1minus1 + 1198862119863119896+10 + 1198861119863119896+11 = 1198921 (119905119896+1) 119906 (119909119899 119905119896+1) = 1198861119863119896+1119899minus1 + 1198862119863119896+1119899 + 1198861119863119896+1119899+1 = 1198922 (119905119896+1) (21)

22 Numerical Solution of the Generalized Burgers-Fisher andGeneralized Burgers-Huxley Equations By utilizing temporaldiscretization and Crank-Nicolson approach (1) can bewritten as

119906119896+1119895 minus 119906119896119895Δ119905 + (120572 (119906120575119906119909)119896+1119895 minus 120598 (119906119909119909)119896+1119895 )2

+ (120572 (119906120575119906119909)119896119895 minus 120598 (119906119909119909)119896119895)2= 120573(119891 (119906119896119895) + 119891 (119906119896+1119895 )2 )

(22)

where 119896 and 119896 + 1 describe successive time positions and119891(119906119896119895 ) = 119906119896119895 (1 minus (119906120575)119896119895 ) for GBF and 119891(119906119896119895 ) = 119906119896119895 (1 minus(119906119896119895 )120575)((119906119896119895 )120575minus120574) for GBH equations After simplification (22)for GBF can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ 05Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)

(23)

and (22) for GBH equation can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)((119906119896+1119895 )120575 minus 120574)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)((119906119896119895)120575 minus 120574)

(24)

The system thus obtained on simplifying (23) for GBFand (24) for GBH problem using (20) consists of 119899 + 1nonlinear equations in 119899 + 3 unknowns 119863119896+1 = (119863119896+1minus1 119863119896+10 119863119896+11 119863119896+1119899+1) at the time level 119905119896+1 Further two equa-tions are included in the resulting system to obtain a uniquesolution of the problem using the boundary conditions givenin (21) The initial vector 1198630 can be obtained from initialcondition given in (2) [59ndash64] Thus the resulting systembecomes amatrix systemof dimension (119899+3)times(119899+3)which isa tridiagonal system that can be solved byThomas algorithm[65ndash67]

3 Stability

This section discusses the Von Neumann criteria to investi-gate the stability of GBF and GBH equations In the productterm consider 119906120575 = 120591 where 120591 is taken [68] as locallyconstant the GBF equation is described as follows

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = (1 minus 120591) 120573119906 (25)

and applying same procedure for nonlinear term as in [64]the GBH equation can be converted to the following

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = minus120574120573119906 (26)

Utilizing same procedure as stated in (22) and setting 1 minus 120591 =120582 the above two GBF and GBH equations take the followingforms

1198871119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 1198874119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

11988611119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 11988622119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

(27)

where 11988611 = 2 + 120573120574Δ119905 11988622 = 2 minus 120573120574Δ119905 1198871 = 2 minus 120573120582Δ119905 1198872 =120572120591Δ119905 1198873 = 120598Δ119905 1198874 = 2 + 120573120582Δ119905Substitute (23) into (27) and simplifying yield

1199021119863119896+1119895minus1 + 1199022119863119896+1119895 + 1199023119863119896+1119895+1= 1199024119863119896119895minus1 + 1199025119863119896119895 + 1199026119863119896119895+1

1199011119863119896+1119895minus1 + 1199012119863119896+1119895 + 1199013119863119896+1119895+1= 1199014119863119896119895minus1 + 1199015119863119896119895 + 1199016119863119896119895+1

(28)

where

1199011 = 119886111198861 + 11988721198863 minus 119887311988651199012 = 119886111198862 minus 119887311988661199013 = 119886111198861 + 11988721198864 minus 119887311988651199014 = 119886221198861 minus 11988721198863 + 119887311988651199015 = 119886221198862 + 11988731198866

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 5: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 5

From (17)ndash(19) the values of 119906119896119895 (119909 119905) and their derivatives atthe knots are calculated in terms of time parameters119863119896119895 (119905) asfollows

119906119896119895 = 1198861119863119896119895minus1 + 1198862119863119896119895 + 1198861119863119896119895+1(119906119909)119896119895 = 1198863119863119896119895minus1 + 0119863119896119895 + 1198864119863119896119895+1(119906119909119909)119896119895 = 1198865119863119896119895minus1 + 1198866119863119896119895 + 1198865119863119896119895+1

(20)

Equation (18) and boundary conditions given in (3) are usedto obtain the approximate solution at end points of the meshas

119906 (1199090 119905119896+1) = 1198861119863119896+1minus1 + 1198862119863119896+10 + 1198861119863119896+11 = 1198921 (119905119896+1) 119906 (119909119899 119905119896+1) = 1198861119863119896+1119899minus1 + 1198862119863119896+1119899 + 1198861119863119896+1119899+1 = 1198922 (119905119896+1) (21)

22 Numerical Solution of the Generalized Burgers-Fisher andGeneralized Burgers-Huxley Equations By utilizing temporaldiscretization and Crank-Nicolson approach (1) can bewritten as

119906119896+1119895 minus 119906119896119895Δ119905 + (120572 (119906120575119906119909)119896+1119895 minus 120598 (119906119909119909)119896+1119895 )2

+ (120572 (119906120575119906119909)119896119895 minus 120598 (119906119909119909)119896119895)2= 120573(119891 (119906119896119895) + 119891 (119906119896+1119895 )2 )

(22)

where 119896 and 119896 + 1 describe successive time positions and119891(119906119896119895 ) = 119906119896119895 (1 minus (119906120575)119896119895 ) for GBF and 119891(119906119896119895 ) = 119906119896119895 (1 minus(119906119896119895 )120575)((119906119896119895 )120575minus120574) for GBH equations After simplification (22)for GBF can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ 05Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)

(23)

and (22) for GBH equation can be written as follows

119906119896+1119895 + 05120572Δ119905 (119906120575119906119909)119896+1119895 minus 05120598Δ119905 (119906119909119909)119896+1119895minus 05Δ119905120573119906119896+1119895 (1 minus (119906120575)119896+1

119895)((119906119896+1119895 )120575 minus 120574)

= 119906119896119895 minus 05120572Δ119905 (119906120575119906119909)119896119895 + 05120598Δ119905 (119906119909119909)119896119895+ Δ119905120573119906119896119895 (1 minus (119906120575)119896

119895)((119906119896119895)120575 minus 120574)

(24)

The system thus obtained on simplifying (23) for GBFand (24) for GBH problem using (20) consists of 119899 + 1nonlinear equations in 119899 + 3 unknowns 119863119896+1 = (119863119896+1minus1 119863119896+10 119863119896+11 119863119896+1119899+1) at the time level 119905119896+1 Further two equa-tions are included in the resulting system to obtain a uniquesolution of the problem using the boundary conditions givenin (21) The initial vector 1198630 can be obtained from initialcondition given in (2) [59ndash64] Thus the resulting systembecomes amatrix systemof dimension (119899+3)times(119899+3)which isa tridiagonal system that can be solved byThomas algorithm[65ndash67]

3 Stability

This section discusses the Von Neumann criteria to investi-gate the stability of GBF and GBH equations In the productterm consider 119906120575 = 120591 where 120591 is taken [68] as locallyconstant the GBF equation is described as follows

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = (1 minus 120591) 120573119906 (25)

and applying same procedure for nonlinear term as in [64]the GBH equation can be converted to the following

119906119905 + 120572120591119906119909 minus 120598119906119909119909 = minus120574120573119906 (26)

Utilizing same procedure as stated in (22) and setting 1 minus 120591 =120582 the above two GBF and GBH equations take the followingforms

1198871119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 1198874119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

11988611119906119896+1119895 + 1198872 (119906119909)119896+1119895 minus 1198873 (119906119909119909)119896+1119895= 11988622119906119896119895 minus 1198872 (119906119909)119896119895 + 1198873 (119906119909119909)119896119895

(27)

where 11988611 = 2 + 120573120574Δ119905 11988622 = 2 minus 120573120574Δ119905 1198871 = 2 minus 120573120582Δ119905 1198872 =120572120591Δ119905 1198873 = 120598Δ119905 1198874 = 2 + 120573120582Δ119905Substitute (23) into (27) and simplifying yield

1199021119863119896+1119895minus1 + 1199022119863119896+1119895 + 1199023119863119896+1119895+1= 1199024119863119896119895minus1 + 1199025119863119896119895 + 1199026119863119896119895+1

1199011119863119896+1119895minus1 + 1199012119863119896+1119895 + 1199013119863119896+1119895+1= 1199014119863119896119895minus1 + 1199015119863119896119895 + 1199016119863119896119895+1

(28)

where

1199011 = 119886111198861 + 11988721198863 minus 119887311988651199012 = 119886111198862 minus 119887311988661199013 = 119886111198861 + 11988721198864 minus 119887311988651199014 = 119886221198861 minus 11988721198863 + 119887311988651199015 = 119886221198862 + 11988731198866

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 6: Hybrid B-Spline Collocation Method for Solving the ...

6 Mathematical Problems in Engineering

1199016 = 119886221198861 minus 11988721198864 + 119887311988651199021 = 11988711198861 + 11988721198863 minus 119887311988651199022 = 11988711198862 minus 119887311988661199023 = 11988711198861 + 11988721198864 minus 119887311988651199024 = 11988741198861 minus 11988721198863 + 119887311988651199025 = 11988741198862 + 119887311988661199026 = 11988741198861 minus 11988721198864 + 11988731198865Now substituting 119863119896119895 = 120588119896119890119894120596119895ℎ into (28) After simplificationdividing both sides by 120588119896119890119894120596119895ℎ we obtain the following expres-sions given in (29) and (30) for GBF and GBH equationsrespectively

120588 = 1198881 + 11989411988911198882 + 1198941198892 119894 = radicminus1 (29)

120588 = 1198971 + 11989411989811198972 + 1198941198982 119894 = radicminus1 (30)

where

1198881 = cos (120601)(1199026 + 1199024) + 11990251198891 = sin (120601)(1199026 minus 1199024)1198882 = cos (120601)(1199023 + 1199021) + 11990221198892 = sin (120601)(1199023 minus 1199021)1198971 = cos (120601)(1199016 + 1199014) + 11990151198981 = sin (120601)(1199016 minus 1199014)1198972 = cos (120601)(1199013 + 1199011) + 11990121198982 = sin (120601)(1199013 minus 1199011)Since the wave number 120596 = 2120587120582 where 120582 is the wave lengthso 120601 = 120596ℎ = 2120587119873 for 0 lt 120601 lt 120587 The amplification factor 120588is a complex number therefore the stability condition |120588| le 1yields the following relation by adopting same procedure asin [60]

For GBF equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11988811198882 + 1198891119889211988822 + 11988922 )2 + (11988911198882 minus 1198881119889211988822 + 11988922 )2 le 1 (31)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816 = radic(11989711198972 + 1198981119898211989722 + 11989822 )2 + (11989811198972 minus 1198971119898211989722 + 11989822 )2 le 1 (32)

Substituting the values into (31) and (32) we obtain thefollowing expressions for GBF equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990224 + 11990225 + 11990226 + 21199025 (1199024 + 1199026) cos (120601) + 211990241199026 cos (2120601)11990221 + 11990222 + 11990223 + 21199022 (1199021 + 1199023) cos (120601) + 211990211199023 cos (2120601)

(33)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= radic 11990124 + 11990125 + 11990126 + 21199015 (1199014 + 1199016) cos (120601) + 211990141199016 cos (2120601)11990121 + 11990122 + 11990123 + 21199012 (1199011 + 1199013) cos (120601) + 211990111199013 cos (2120601)

(34)

Taking extreme value 120601 = 120587 (33) and (34) become

10038161003816100381610038161205881003816100381610038161003816 = radic (1199024 minus 1199025 + 1199026)2(1199021 minus 1199022 + 1199023)2 10038161003816100381610038161205881003816100381610038161003816 = radic (1199014 minus 1199015 + 1199016)2(1199011 minus 1199012 + 1199013)2

(35)

By substituting the values from (28) into (35) amplificationfactor 120588 for GBF equation is

10038161003816100381610038161205881003816100381610038161003816= (1205852 + 1205855 + 1205859) minus (1205851 + 1205853 + 1205854 + 1205856 + 1205857 + 1205858 + 12058510)(1205851 + 1205852 + 1205853 + 1205854 + 1205855 + 1205856 + 1205857 + 12058510) minus (1205858 + 1205859)

(36)

and for GBH equation

10038161003816100381610038161205881003816100381610038161003816= (1205755 + 1205752) minus (1205751 + 1205753 + 1205754 + 1205756 + 1205757 + 3ℎ2 (1 minus 120583) (8 + (3 minus 4120573120574) Δ119905 + (minus8 + (9 + 4120573120574) Δ119905) cos (ℎ)) csc2 (ℎ2) sec (ℎ2))1205751 + 1205752 + 1205753 + 1205754 + 1205755 + 1205756 + 1205757 + 6ℎ2 (1 minus 120583) ((minus8 minus (9 + 4120573120574) Δ119905) + 6Δ119905csc2 (ℎ2)) sec (ℎ2) (37)

where 1205851 = 96Δ1199051205831205981205852 = 16ℎ2(6 minus 5120583)1205853 = 8ℎ2120573Δ119905(6 minus 5120583)1205821205854 = 192Δ119905120598120583 cos (ℎ)

1205855 = 32ℎ2120583 cos (ℎ)1205856 = 16ℎ2120573Δ119905120583120582 cos (ℎ)1205857 = 36ℎ2120598Δ119905(1 minus 120583)cot2 (ℎ2)1205858 = 48ℎ2(1 minus 120583) sec (ℎ2)1205859 = 24ℎ2(1 minus 120583)120573Δ119905120582 sec (ℎ2)

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 7: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 7

Table 1 Absolute errors of GBH equation at different values of time and parameter 120583 taking Δ119905 = 00001 for case 1119909 119905 120583 = 11 120583 = 15 120583 = 1901

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E ndash 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1685E minus 08 1681E minus 08 1677E minus 0810 1685E minus 08 1681E minus 08 1677E minus 08

05

005 1734E minus 08 1730E minus 08 1726E minus 0801 2881E minus 08 2874E minus 08 2867E minus 0810 4682E minus 08 4670E minus 08 4658E minus 0850 4682E minus 08 4670E minus 08 4658E minus 0810 4682E minus 08 4670E minus 08 4658E minus 08

09

005 7722E minus 09 7703E minus 09 7684E minus 0901 1129E minus 08 1126E minus 08 1123E minus 0810 1685E minus 08 1681E minus 08 1677E minus 0850 1686E minus 08 1681E minus 08 1677E minus 0810 1686E minus 08 1681E minus 08 1677E minus 08

12058510 = 18ℎ2(1minus120583)120598Δ119905(1+3 cos (ℎ))(1minuscos (ℎ)) sec (ℎ2)1205751 = 96Δ1199051205831205752 = 16ℎ2(6 minus 5120583)1205753 = 8ℎ2120573120574Δ119905(6 minus 5120583)1205754 = 192Δ119905120583 cos (ℎ)1205755 = 32ℎ2120583 cos (ℎ)1205756 = 16ℎ2120573120574Δ119905120583 cos (ℎ)1205757 = 36ℎ2Δ119905(1 minus 120583)cot2 (ℎ2)Since the numerator is less than the denominator in (36)and (37) so amplification factor |120588| le 1 for both GBF andGBH equations which clearly demonstrate that the proposedscheme is unconditionally stable

4 Numerical Results and Discussion

This section presents the numerical results of GBF and GBHequations with initial (2) and boundary conditions (3) byHBSCM To test the accuracy of present method severalnumerical tests for different values of 120572 120573 120574 120575 are providedwhere 119871infin 1198712 and order of convergence 119901 are calculated by

119871infin = max 10038161003816100381610038161003816119906119895 minus 119906exc11989510038161003816100381610038161003816 1198712 = radicsum119899119895=1 10038161003816100381610038161003816119906119895 minus 119906exc119895100381610038161003816100381610038162

radicsum119899119895=1 10038161003816100381610038161003816119906exc119895100381610038161003816100381610038162

119901 = log (119871infin (119899) 119871infin (2119899))log (2)

(38)

Numerical results obtained by HBSCM are compared withgiven exact solutions and the approximate methods existing

in the literature The programming of the proposed problemis carried out in Matlab R2015b Numerical results arecomputed at different time levels with smaller storage whichare tabulated in different Tables

Consider the following numerical test cases for GBH andGBF equations to show the accuracy of proposed method

41 Numerical Test Cases for Model I

(1) When 120572 = 1 120573 = 1 120574 = 0001 120598 = 1 GBH equationcan be described as follows

119906119905 + 119906120575119906119909 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (39)

The obtained errors at 120575 = 1 corresponding todifferent values of parameter 120583 are listed in Table 1Table 2 shows a comparison of the absolute errors at120583 = 39 120575 = 1 2 3 obtained by proposed methodHBSCM with the methods existing in the literaturenamed compact finite difference scheme (CFDS)[26] a fourth-order improved numerical scheme(FONS) [33] variational iteration method (VIM)[36] Adomian decomposition method (ADM) [3840] implicit exponential finite difference method(IEFM) [42] and modified cubic B-spline (MCBS)[55] The obtained results are compared with Haarwavelet method (HWM) [46] at 119905 = 08 in Table 3while a comparison between HBSCM and a newdomain decomposition method (NDDA) [22] canbe observed in Table 4 The results of the pro-posed method in terms of errors comparative toLocal Radial Basis Function Differential Collocationmethod (LRBFDQ) [58] is provided in Table 5 It canbe observed that increase in Δ119905 did not disturb theaccuracy of HBSCM and our method still approx-imates the exact solution quite adequately due tohybrid parameter The graphical representations oferror terms at different time levels have been carriedout in Figures 2 and 3 A prominent difference in the

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 8: Hybrid B-Spline Collocation Method for Solving the ...

8 Mathematical Problems in Engineering

Table 2 Comparison of absolute errors calculated by HBSCM with the existing methods at 120575 = 1 120583 = 39 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 7587E minus 09 1704E ndash 08 7587E minus 09 1109E minus 08 2830E minus 08 1109E minus 08 1656E minus 08 4599E minus 08 1656E minus 08CFDS [26] 7700E minus 09 1728E ndash 08 7700E minus 09 1126E minus 08 2873E minus 08 1126E minus 08 1686E minus 08 4684E minus 08 1686E minus 08FONS [33] 1264E minus 09 1977E ndash 08 4602E minus 08 6395E minus 09 3996E minus 08 7663E minus 08 3292E minus 07 3792E minus 07 4292E minus 07VIM [36] 1874E minus08 1874E ndash 08 1874E minus 08 3748E minus 08 1374E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [38] 1874E minus 08 1874E ndash 08 1874E minus 08 3748E minus 08 3748E minus 08 3748E minus 08 3748E minus 07 3748E minus 07 3748E minus 07ADM [40] 1937E minus 07 1937E ndash 07 1937E minus 07 3874E minus 07 3874E minus 07 3874E minus 07 3875E minus 06 3875E minus 06 3875E minus 06IEFM [42] 1544E minus 08 3469E ndash 08 1544E minus 08 2258E minus 08 5764E minus 08 2259E minus 08 3372E minus 08 9369E minus 08 3373E minus 08MCBS [55] mdash mdash mdash 1111E minus 08 2870E minus 08 1111E minus 08 1668E minus 08 4665E minus 08 1668E minus 08120575 = 2HBSCM 3545E minus 07 7960E minus 07 3544E minus 07 5182E minus 07 1322E minus 06 5182E minus 07 7734E minus 07 2148E minus 06 7734E minus 07VIM [36] mdash mdash mdash 5515E minus 05 5510E minus 05 mdash mdash mdash mdashADM [38] mdash mdash mdash 5515E minus 05 5511E minus 05 mdash mdash mdash mdashIEFM [42] 1402E minus 06 3150E minus 06 1402E minus 06 2051E minus 06 5233E minus 06 2051E minus 06 3056E minus 06 8490E minus 06 3056E minus 06120575 = 3HBSCM 1294E minus 06 2906E minus 06 1293E minus 06 1892E minus 06 4828E minus 06 1892E minus 06 2823E minus 06 7841E minus 06 2823E minus 06IEFM [42] 8789E minus 06 1973E minus 05 8788E minus 06 1284E minus 05 3278E minus 05 1284E minus 05 1902E minus 05 5285E minus 05 1902E minus 05

Table 3 Comparison of absolute errors of GBH equation at 120583 = 39 Δ119905 = 00001 taking different values of 120575 for case 1119905 = 08 120575 = 1 120575 = 8119909 01 05 09 01 05 09HBSCM 5870E minus 09 1342E minus 08 5870E minus 09 6805E minus 06 1555E minus 05 6800E minus 06HWM [46] 1586E minus 08 4662E minus 08 1586E minus 08 1404E minus 05 4123E minus 05 1404E minus 05

Table 4 Comparison of maximum errors of GBH equation at 120583 = 39 Δ119905 = 000005 taking different values of 120575 for case 1119905 = 10 120575 = 1 120575 = 4 120575 = 8HBSCM 1342E minus 08 5325E minus 06 1555E minus 05NDDA [22] 4685E minus 08 1532E minus 05 4141E minus 05

Table 5 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 1119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 848E minus 11 191E minus 10 858E minus 11 123E minus 10 318E minus 10 125E minus 10 187E minus 10 522E minus 10 188E minus 10LRBDQ [58] 797E minus 10 187E minus 08 383E minus 08 159E minus 09 375E minus 08 766E minus 08 160E minus 08 375E minus 07 765E minus 07120575 = 2HBSCM 114E minus 08 257E minus 08 114E minus 08 167E minus 08 426E minus 08 166E minus 08 244E minus 08 677E minus 08 243E minus 08LRBDQ [58] 115E minus 09 870E minus 07 175E minus 06 226E minus 09 175E minus 06 350E minus 06 137E minus 08 175E minus 05 349E minus 05120575 = 3HBSCM 769E minus 08 172E minus 07 765E minus 08 111E minus 07 285E minus 07 111E minus 07 163E minus 07 454E minus 07 163E minus 07LRBDQ [58] 940E minus 07 319E minus 06 730E minus 06 189E minus 06 683E minus 06 147E minus 05 189E minus 05 638E minus 05 146E minus 04

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

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Page 9: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 9

Table 6 Comparison of error norms at 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001 120598 = 1 120575 = 2 for case 2119905 HBSCM LRBF [58]119871infin 1198712 119871infin 119871200005 59337E minus 13 88011E minus 14 22152E minus 11 13610E minus 110001 11867E minus 12 23285E minus 13 44301E minus 11 27153E minus 1105 14722E minus 10 87302E minus 10 22152E minus 08 13565E minus 081 14831E minus 10 13634E minus 09 44304E minus 08 27130E minus 085 14832E minus 10 32646E minus 09 22152E minus 07 13536E minus 07

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

t = 005

t = 01

t = 05

t = 1

times10minus8

Figure 2 Absolute errors at different time levels when Δ119905 = 00001120583 = 39 120575 = 1 for case 1absolute errors can be visualized in the figures andtables due to the hybrid parameter It is pertinent toclaim that the proposed method provides accurateand improves results as compared to others

(2) When 120572 = 0001 120573 = 0001 120574 = 0001 Δ119905 = 00001120598 = 1 120575 = 2 GBH equation can be [58] described asfollows119906119905 + 00011199062119906119909 minus 119906119909119909

= 0001119906 (1 minus 1199062) (1199062 minus 0001) (40)

In Table 6 the error norms are calculated at differenttime levels for Δ119905 = 00001 and compared withLRBFDQ [58] Figures 4 and 5 depict the error normsat different time levels It can be concluded that theproposed method is more accurate than LRBFDQ[58]

(3) When 120572 = 01 120573 = 0001 120574 = 00001 120598 = 1 GBHequation is expressed as follows

119906119905 + 01119906120575119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906120575) (119906120575 minus 00001) (41)

A comparison of the absolute errors calculated byHBSCM at 120575 = 1 2 8 is presented in Table 7 with the

Error at different time levels

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Abso

lute

erro

rs

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 3 Absolute errors at different time levels when Δ119905 = 001120583 = 26 120575 = 1 for case 1existing methods named spectral collocation method(SCM) [20] and IEFM [42]The order of convergencecalculated numerically by HBSCM is compared withIEFM [42] in Table 8 It can be concluded that presentmethod has a rapid convergence Figure 6 depictsthe absolute errors which clearly demonstrate thatincrease in time does not disturb the accuracy of theobtained results

(4) When 120572 = 0 120573 = 1 120574 = 0001 120583 = 165 Δ119905 = 0001120598 = 1 GBH can be represented as follows

119906119905 minus 119906119909119909 = 119906 (1 minus 119906120575) (119906120575 minus 0001) (42)

In Table 9 we tabulate a comparison betweenHBSCM and the existing methods CFDS [26] FONS[33] ADM [38] Galerkin method (GM) [44] higherorder finite difference method (HFDM) [52] MCBS[55] cubic B-spline algorithm (CBSA) [56] andoptimal homotopy asymptotic method (OHAM)[69] at 120575 = 1 3 Computations of the absolute errorsand two different types of error norms have beenmentioned in Tables 10 and 11 respectively Figure 7exhibits the computed results at different time levels

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 10: Hybrid B-Spline Collocation Method for Solving the ...

10 Mathematical Problems in Engineering

Table 7 Comparison of absolute errors at different values of 120575 and parameter 120583 = 19 for case 3119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1315E minus 13 3569E minus 13 1315E minus 13 1490E minus 13 4135E minus 13 1490E minus 13 1499E minus 13 4165E minus 13 1499E minus 13SCM [20] mdash mdash mdash 8381E minus 12 4879E minus 11 2859E minus 11 mdash mdash mdashIEFM [42] 1561E minus 13 4238E minus 13 1561E minus 13 1770E minus 13 4911E minus 13 1770E minus 13 1780E minus 13 4946E minus 13 1780E minus 13120575 = 2HBSCM 1928E minus 11 5232E minus 11 1928E minus 11 21847E minus 11 6062E minus 11 2184E minus 11 2198E minus 11 6105E minus 11 2198E minus 11SCM [20] mdash mdash mdash 5554E minus 10 3815E minus 09 2236E minus 09 mdash mdash mdashIEFM [42] 3838E minus 11 1041E minus 10 3838E minus 11 4349E minus 11 1206E minus 10 4349E minus 11 4375E minus 11 1215E minus 10 4375E minus 11120575 = 8HBSCM 8877E minus 10 2408E minus 09 8877E minus 10 1005E minus 09 2790E minus 09 1005E minus 09 1011E minus 09 2810E minus 09 1011E minus 09SCM [20] mdash mdash mdash 3395E minus 08 1976E minus 07 1158E minus 07 mdash mdash mdashIEFM [42] 1743E minus 08 4732E minus 08 1743E minus 08 1976E minus 08 5483E minus 08 1976E minus 08 1988E minus 08 5522E minus 08 1988E minus 08

Table 8 Order of convergence (119901) at 119905 = 1 for case 3

119899 2 4 8 16 32 64119871infin 3054E minus 06 7008E minus 07 1761E minus 07 4874E minus 08 1710E minus 08 9212E minus 09HBSCM (119901) mdash 2123 1992 1853 1510 0892IEFM [42] mdash 0175608438 0078691220 0041403087 0021266315 0011014546

Maximum error

01 02 03 04 05 06 07 08 09 10x

0

1

2

3

4

5

6

Max

imum

erro

r

times10minus13 Maximum error at t = 00005

Figure 4 119871infin errors at 120575 = 2 119905 = 00005 for case 2

which enable us to claim that the proposed schemeis superior to the existing method in the terms ofaccuracy

(5) When 120572 = 5 120575 = 2 120573 = 10 120583 = 16 Δ119905 = 00001120598 = 1 the following GBH can be obtained

119906119905 + 51199062119906119909 minus 119906119909119909 = 10119906 (1 minus 1199062) (1199062 minus 120574) (43)

L2 error at t = 00005

L2 error

0

1

2

3

4

5

6

7

8

9

L2

erro

r

01 02 03 04 05 06 07 08 09 10x

times10minus14

Figure 5 1198712 errors at 120575 = 2 119905 = 00005 for case 2

The absolute errors are recorded in Table 12 at dif-ferent values of 120574 = 10minus3 10minus4 10minus5 calculated byHBSCM and compared with IEFM [42] Figure 8explains these errors graphically at 120574 = 10minus3 Aprominent difference in the accuracy can be seen inthe figure and table due to the hybrid parameter

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 11: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 11

Table 9 Comparison of absolute errors computed by HBSCM with other methods at 120572 = 0 120573 = 1 120574 = 0001 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08CFDS [26] 1026E ndash 08 2304E minus 08 1026E minus 08 1502E minus 08 3832E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08FONS [33] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07ADM [40] 1874E ndash 07 1874E minus 07 1875E minus 07 3749E minus 07 3749E minus 07 3750E minus 07 3750E minus 07 3750E minus 07 3750E minus 07GM [44] 1069E ndash 08 9259E minus 09 7892E minus 09 2318E minus 08 2174E minus 08 2038E minus 08 2487E minus 07 2472E minus 07 2459E minus 07HFDM [52] 1030E ndash 08 2313E minus 08 1030E minus 08 1506E minus 08 3844E minus 08 1506E minus 08 2248E minus 08 6246E minus 08 2248E minus 08MCBS [55] 1004E ndash 08 2304E minus 08 1004E minus 08 1479E minus 08 3825E minus 08 1479E minus 08 2220E minus 08 6216E minus 08 2220E minus 08CBSA [56] 1029E ndash 08 2313E minus 08 1029E minus 08 1502E minus 08 3843E minus 08 1502E minus 08 2248E minus 08 6246E minus 08 2248E minus 08OHAM [69] 2498E ndash 08 2498E minus 08 2498E minus 08 4997E minus 08 4997E minus 08 4997E minus 08 4997E minus 07 4997E minus 07 4997E minus 07120575 = 3HBSCM 1635E minus 06 3673E minus 06 1635E minus 06 2391E minus 06 6102E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06CFDS [26] 1630E minus 06 3658E minus 06 1629E minus 06 2385E minus 06 6083E minus 06 2385E minus 06 3567E minus 06 9907E minus 06 3566E minus 06FONS [33] 3967E minus 06 3966E minus 06 3965E minus 06 7934E minus 06 7933E minus 06 7931E minus 06 7934E minus 06 7933E minus 06 7931E minus 06ADM [40] 1984E minus 06 1983E minus 06 1983E minus 06 3968E minus 06 3967E minus 06 3966E minus 06 3966E minus 06 3965E minus 06 3964E minus 06HFDM [52] 1635E minus 06 3672E minus 06 1635E minus 06 2391E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06MCBS [55] 1594E minus 06 3658E minus 06 1594E minus 06 2347E minus 06 6072E minus 06 2347E minus 06 3522E minus 06 9861E minus 06 3521E minus 06CBSA [56] 1634E minus 06 3672E minus 06 1634E minus 06 2390E minus 06 6101E minus 06 2390E minus 06 3567E minus 06 9907E minus 06 3566E minus 06

Table 10 Comparison of absolute errors calculated by HBSCM with the existing method at Δ119905 = 001 120583 = 26 for case 4119905 = 005 119905 = 01 119905 = 10119909 01 05 09 01 05 09 01 05 09120575 = 1

HBSCM 250E minus 09 563E minus 09 250E minus 09 365E minus 09 933E minus 09 365E minus 09 541E minus 09 150E minus 08 541E minus 09LRBDQ [58] 137E minus 09 250E minus 08 137E minus 09 270E minus 09 500E minus 08 270E minus 09 270E minus 08 500E minus 07 270E minus 08120575 = 2HBSCM 11E minus 07 25E minus 07 11E minus 07 16E minus 07 41E minus 07 16E minus 07 24E minus 07 67E minus 07 24E minus 07LRBDQ [58] 61E minus 08 11E minus 06 61E minus 08 12E minus 07 22E minus 06 12E minus 07 11E minus 06 22E minus 05 12E minus 06120575 = 3HBSCM 398E minus 07 893E minus 07 397E minus 07 579E minus 07 148E minus 06 579E minus 07 856E minus 07 237E minus 06 855E minus 07LRBDQ [58] 217E minus 07 397E minus 06 219E minus 07 433E minus 07 793E minus 06 437E minus 07 432E minus 06 793E minus 05 436E minus 06

Table 11 Comparison of error norms for 120572 = 0 120573 = 1 120574 = 0001 Δ119905 = 001 120598 = 1 120575 = 2 of case 4119905 HBSCM LRBF [58]119871infin 1198712 119871infin 1198712005 25183E minus 07 12105E minus 07 51347E minus 08 30711E minus 0801 41746E minus 07 28221E minus 07 10269E minus 07 61418E minus 081 67151E minus 07 19621E minus 06 10262E minus 06 61359E minus 075 13669E minus 06 46487E minus 06 51155E minus 06 30649E minus 0650 20079E minus 05 13669E minus 05 49458E minus 05 30125E minus 05

(6) When120572 = 1120575 = 1 120574 = 00001120583 = 135Δ119905 = 00001120598 = 1 GBH equation is

119906119905 + 119906119906119909 minus 119906119909119909 = 120573119906 (1 minus 119906) (119906 minus 00001) (44)

The computed results by HBSCM at 120573 = 1 10 100 arerecorded in Table 13 and compared with the existingmethod IEFM [42] Figure 9 illustrates the behaviourof these errors graphically at 120573 = 1

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 12: Hybrid B-Spline Collocation Method for Solving the ...

12 Mathematical Problems in Engineering

Table 12 Comparison of absolute errors obtained by HBSCM with existing methods at different values of 120574 for case 5119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120574 = 10minus3

HBSCM 6230E ndash 06 1690E minus 06 6231E minus 06 7056E minus 06 1958E minus 05 7057E minus 06 7094E minus 06 1970E minus 05 7095E minus 06IEFM [42] 2075E minus 05 5633E minus 05 2076E minus 05 2339E minus 05 6495E minus 05 2340E minus 05 2340E minus 05 6503E minus 05 2341E minus 05120574 = 10minus4HBSCM 1971E ndash 07 5350E minus 07 1971E minus 07 2233E minus 07 6198E minus 07 2233E minus 07 2247E minus 07 6242E minus 07 2247E minus 07IEFM [42] 6580E ndash 07 1785E minus 06 6580E minus 07 7453E minus 07 2068E minus 06 7453E minus 07 7494E minus 07 2081E minus 06 7495E minus 07120574 = 10minus5HBSCM 6235E ndash 09 1692E minus 08 6235E minus 09 7065E minus 09 1960E minus 08 7065E minus 09 7108E minus 09 1974E minus 08 7108E minus 09IEFM [42] 2081E ndash 08 5648E minus 08 2081E minus 08 2358E minus 08 6545E minus 08 2358E minus 08 2372E minus 08 6591E minus 08 2373E minus 08

Table 13 Comparison of absolute errors by HBSCM with other methods at 120572 = 1 120574 = 10minus4 120575 = 1 for case 6119905 = 02 119905 = 05 119905 = 08119909 01 05 09 01 05 09 01 05 09120573 = 1

HBSCM 1479E minus 10 4015E minus 10 1479E minus 10 1676E minus 10 4653E minus 10 1676E minus 10 1687E minus 10 4686E minus 10 1687E minus 10IEFM [42] 2959E minus 10 8029E minus 10 2959E minus 10 3353E minus 10 9304E minus 10 3353E minus 10 3373E minus 10 9370E minus 10 3373E minus 10120573 = 10HBSCM 1775E ndash 09 4818E minus 09 1775E minus 09 2012E minus 09 5583E minus 09 2012E minus 09 2024E minus 09 5622E minus 09 2024E minus 09IEFM [42] 4142E ndash 09 1124E minus 08 4142E minus 09 4694E minus 09 1302E minus 08 4694E minus 09 4723E minus 09 1311E minus 08 4723E minus 09120573 = 100HBSCM 1905E ndash 08 5171E minus 08 1905E minus 08 2159E minus 08 5991E minus 08 2159E minus 08 2172E minus 08 6034E minus 08 2172E minus 08IEFM [42] 4662E ndash 08 1265E minus 07 4662E minus 08 5283E minus 08 1466E minus 07 5283E minus 08 5314E minus 08 1476E minus 07 5314E minus 08

Error at different time levels

0

05

1

15

2

25

3

35

4

45

Abso

lute

erro

rs

0 02 03 04 05 06 07 08 09 101x

times10minus13

t = 005

t = 01

t = 05

t = 1

Figure 6 Absolute errors at different time levels when 120575 = 1 for case3

42 Numerical Test Cases for Model II

(7) When 120572 = 1 120573 = 1 Δ119905 = 000001 120598 = 1 120575 = 1GBF equation in the domain [minus1 1] can be written asfollows

119906119905 + 119906119906119909 minus 119906119909119909 = 119906 (1 minus 119906) (45)

Error at different time levels

0

1

2

3

4

5

6

7

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus8

t = 005

t = 01

t = 05

t = 1

Figure 7 Absolute errors at 120572 = 0 120573 = 1 120574 = 0001 120575 = 1 for case4

In Table 14 we record the absolute errors obtained byHBSCM for three different values of hybrid parameter120583 and compare them with the cubic B-spline method(CBSM) [53] Table 15 establishes a comparison oferror norms 1198712 and 119871infin calculated by HBSCM at

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

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Page 13: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 13

Table 14 Comparison of absolute errors with [53] 120572 = 1 120573 = 1 120575 = 1 for case 7119909 119905 120583 = 11 120583 = 15 120583 = 19 CBSM [53]

minus0505 3142E minus 06 1463E minus 06 4226E minus 07 6108E minus 0610 3408E minus 06 1177E minus 06 4825E minus 07 5496E minus 0620 4505E minus 06 1766E minus 06 5298E minus 07 3758E minus 0640 4149E minus 06 3644E minus 06 7470E minus 07 6071E minus 07

00

05 4329E minus 06 2196E minus 06 5588E minus 07 3072E minus 0610 4495E minus 06 1621E minus 06 6447E minus 07 4118E minus 0620 5650E minus 06 2109E minus 06 6744E minus 07 4559E minus 0640 6247E minus 06 4725E minus 06 9711E minus 07 3117E minus 07

05

05 3514E minus 06 1893E minus 06 1339E minus 07 1159E minus 0610 3437E minus 06 1279E minus 06 4973E minus 07 3680E minus 0620 4175E minus 06 1499E minus 06 5035E minus 07 5724E minus 0640 6525E minus 06 3595E minus 06 7400E minus 07 1672E minus 07

09

05 9634E minus 07 5331E minus 07 0456E minus 07 1315E minus 0610 8952E minus 07 3360E minus 07 1303E minus 07 4758E minus 0620 1081E minus 06 3814E minus 07 1308E minus 07 7098E minus 0640 9281E minus 06 9499E minus 07 1950E minus 07 2364E minus 07

Table 15 Comparison of error norms 119871infin and 1198712 calculated by HBSCM at 120583 = 19 with [53] for case 7

119879 HBSCM CBSM [53]119871infin 1198712 119871infin 119871205 1343E minus 07 1750E minus 07 9172E minus 06 6953E minus 0610 1347E minus 07 1688E minus 07 6852E minus 06 6131E minus 0620 1345E minus 07 1059E minus 07 7304E minus 06 7228E minus 0640 1593E minus 07 9761E minus 08 6443E minus 07 6702E minus 07

Error at different time levels

0

02

04

06

08

1

12

14

16

18

2

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus5

t = 005

t = 01

t = 05

t = 1

Figure 8 Absolute errors at 120572 = 5 120573 = 10 120574 = 0001 120575 = 2 for case5

120583 = 15 with CBSM [53] The behaviour of theabsolute errors is shown in Figure 10 The exact andapproximate solutions are presented graphically inFigure 11 at different time levels Figures 12 and 13represent the pictorial view of 1198712 and 119871infin errors

Error at different time levels

0

05

1

15

2

25

3

35

4

45

5

Abso

lute

erro

rs

01 02 03 04 05 06 07 08 09 10x

times10minus10

t = 005

t = 01

t = 05

t = 1

Figure 9 Absolute errors at 120572 = 1 120573 = 1 120574 = 00001 120575 = 1 for case6

respectively Figures 14 and 15 indicate that the pro-posed HBSCM approximates the exact solution quitenicely It can be concluded from these figures andtables that the proposedmethod ismore accurate thanexisting methods

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

Applied MathematicsJournal of

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Page 14: Hybrid B-Spline Collocation Method for Solving the ...

14 Mathematical Problems in Engineering

Table 16 Comparison of absolute errors with existing methods at 120572 = 0001 120573 = 0001 120575 = 1 for case 8119905 = 0001 119905 = 001119909 01 05 09 01 05 09

HBSCM 3056E minus 10 2658E minus 10 3055E minus 10 1962E minus 09 2659E minus 09 1962E minus 09EFM[29] 1970E minus 08 3580E minus 09 1800E minus 08 1970E minus 08 3880E minus 09 1740E minus 08IA [37] 2524E minus 05 2524E minus 05 2524E minus 05 5999E minus 10 5999E minus 10 5999E minus 10ADM [40] 1937E minus 06 1937E minus 06 1937E minus 06 1937E minus 05 1937E minus 05 1937E minus 05OHAM [69] 2250E minus 08 4580E minus 08 4580E minus 08 2250E minus 07 4580E minus 07 4580E minus 07NAT [70] 3466E minus 04 9870E minus 03 9424E minus 05 1178E minus 05 3740E minus 05 6378E minus 08

Table 17 The absolute errors calculated by HBSCM and other methods at 120572 = 1 120573 = 0 120575 = 3 for case 9119905 = 00001 119905 = 0001119909 01 05 09 01 05 09

HBSCM 1260E minus 09 5543E minus 10 4512E minus 11 1147E minus 08 5530E minus 09 5148E minus 10CMM [27] mdash mdash mdash 1900E minus 05 1600E minus 05 1500E minus 05EFM [29] 4550E minus 07 5660E minus 07 7000E minus 07 4600E minus 07 5610E minus 07 6950E minus 07IA [37] 8780E minus 08 4484E minus 07 4175E minus 07 2220E minus 07 2635E minus 07 3645E minus 07ADM [40] 4463E minus 04 1860E minus 03 9318E minus 04 4439E minus 04 1847E minus 03 9042E minus 04NAT [70] 4780E minus 04 1436E minus 04 7659E minus 05 7865E minus 03 5323E minus 04 7115E minus 04

Error at different time levels

0

02

04

06

08

1

12

14

Abso

lute

erro

rs

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

t = 01

t = 03

t = 05

t = 09

Figure 10 Absolute errors at 120572 = 1 120573 = 1 120575 = 1 for case 7

(8) When 120572 = 0001 120573 = 0001 120583 = 13 120598 = 1 120575 = 1GBF equation in the domain [0 1] can be describedas

119906119905 + 0001119906119906119909 minus 119906119909119909 = 0001119906 (1 minus 119906) (46)

The absolute errors calculated at different time lev-els by proposed HBSCM are compared with theexisting methods Exp-function method (EFM) [29]

Solutions at different time levels

ex at t = 1app at t = 1ex at t = 3

app at t = 3ex at t = 5app at t = 5

ex at t = 9app at t =9

04

045

05

055

06

065

07

075

08

085

Exac

t and

appr

oxim

ate s

olut

ions

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

Figure 11 Exact and approximate solutions at 120572 = 1 120573 = 1 120575 = 1for case 7

improved algorithm (IA) [37] ADM [40] OHAM[69] and Nonlinear Analytic Technique (NAT) [70]in Table 16 It is concluded that the proposed methodis more accurate than these methods

(9) When 120572 = 1 120573 = 0 Δ119905 = 000001 120583 = 14 120598 = 1 120575 =3 GBF in the domain [0 1] is represented as follows

119906119905 + 1199063119906119909 minus 119906119909119909 = 0 (47)

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 15: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 15

L2 error at t = 05

L2 error

0

02

04

06

08

1

12

14

16

18

L2

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7

Figure 12 1198712 error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

Maximum error

0

02

04

06

08

1

12

14

Max

imum

erro

r

minus08 minus06 minus04 minus02 0 02 04 06 08 1minus1x

times10minus7 Maximum error at t = 05

Figure 13 119871infin error norm at 120572 = 1 120573 = 1 120575 = 1 for case 7

105 08

x0 06

t04minus05 02minus1 0

0405060708

Exact solution at t = 09

u(x

t)

Figure 14 Exact solution in space at 120572 = 1 120573 = 1 120575 = 1 119905 = 09 forcase 7

105 08

x0 06

t04minus05 02minus1 0

Approximate solution at t t = 09

u(x

t)

0405060708

Figure 15 Approximate solution in space at 120572 = 1 120573 = 1 120575 = 1119905 = 09 for case 7The absolute errors are calculated by proposedHBSCM and compared with the existing methodscomputational meshless method (CMM) [27] EFM[29] IA [37] ADM [40] and NAT [70] at twodifferent time levels which are tabulated in Table 17 Aremarkable difference in accuracy can be noted fromthis table

5 Concluding Remarks

In this paper a hybrid B-spline collocation method is devel-oped for solving one-dimensional GBH and GBF equationswith known initial and boundary conditions A finite dif-ference scheme and hybrid B-spline function are used todiscretize the time and spatial derivatives respectively Thehybrid B-spline method considered in this study is simpleand straight forward by the application point of view Theobtained results are presented in Tables 1ndash17 and graphicallyshown in Figures 1ndash15 It is found that the numerical resultsare in excellent agreement with the analytical solutions Theproposed scheme is not only accurate but also quite differentfrom the schemes [53ndash57] due to presence of parameterThe parameter 120583 gt 1 provides the better approximation tothe exact solution as compared to classical cubic B-splinefunctionThe stability analysis usingVonNeumann approachhas also been presented It is shown that the method isunconditionally stable for any step length

Conflicts of Interest

The authors declare that they have no conflicts of interest

References

[1] W Hu Z Deng S Han and W Zhang ldquoGeneralized multi-symplectic integrators for a class ofHamiltoniannonlinearwavePDEsrdquo Journal of Computational Physics vol 235 pp 394ndash4062013

[2] WHu Z Deng BWang andH Ouyang ldquoChaos in an embed-ded single-walled carbon nanotuberdquo Nonlinear Dynamics vol72 no 1-2 pp 389ndash398 2013

[3] W Hu Z Deng and Y Zhang ldquoMulti-symplectic methodfor peakon-antipeakon collision of quasi-Degasperis-Procesi

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 16: Hybrid B-Spline Collocation Method for Solving the ...

16 Mathematical Problems in Engineering

equationrdquoComputer Physics Communications vol 185 no 7 pp2020ndash2028 2014

[4] E J Parkes and B R Duffy ldquoAn automated tanh-functionmethod for finding solitary wave solutions to non-linear evolu-tion equationsrdquo Computer Physics Communications vol 98 no3 pp 288ndash300 1996

[5] C-G Zhu and W-S Kang ldquoNumerical solution of Burgers-Fisher equation by cubic B-Spline quasi-interpolationrdquo AppliedMathematics and Computation vol 216 no 9 pp 2679ndash26862010

[6] A-MWazwaz ldquoThe tanhmethod for generalized forms of non-linear heat conduction and Burgers-Fisher equationsrdquo AppliedMathematics and Computation vol 169 no 1 pp 321ndash338 2005

[7] A N Kolmogorov I G Petrovskii and N S Piskunov ldquoAstudy of the equation of diffusion with increase in the quantityof matter and its application to a biological problemrdquo in BullMoscow Univ Math Mech vol 1 pp 1ndash26 6 edition 1937

[8] J D Murray Mathematical Biology I An Introduction vol 2Springer NY USA 2002

[9] J Riordan C R Doering and D Ben-Avraham ldquoFluctuationsand stability of sisher wavesrdquo Physical Review Letters vol 75 no3 pp 565ndash568 1995

[10] S Coen M Tlidi P Emplit and M Haelterman ldquoConvectionversus dispersion in optical bistabilityrdquo Physical Review Lettersvol 83 no 12 pp 2328ndash2331 1999

[11] X Y Wang Z S Zhu and Y K Lu ldquoSolitary wave solutions ofthe generalised Burgers-Huxley equationrdquo Journal of Physics AMathematical and General vol 23 no 3 pp 271ndash274 1990

[12] J SatsumaM Ablowitz B Fuchssteiner andM Kruskal ldquoTop-ics in soliton theory and exactly solvable nonlinear equationsrdquoin Physical Review Letters World Scientific Singapore Asia1987

[13] A C Scott Neurophysics Wiley NY USA 1977[14] X-Y Wang ldquoNerve propagation and wall in liquid crystalsrdquo

Physics Letters A vol 112 no 8 pp 402ndash406 1985[15] A L Hodgkin and A F Huxley ldquoA quantitative description

of membrane current and its application to conduction andexcitation in nerverdquoThe Journal of Physiology vol 117 no 4 pp500ndash544 1952

[16] G B Whiteman Linear and Nonlinear Waves Wiley and SonsNY USA 1974

[17] M Dehghan J M Heris and A Saadatmandi ldquoApplicationof semi-analytic methods for the Fitzhugh-Nagumo equationwhich models the transmission of nerve impulsesrdquoMathemati-calMethods in the Applied Sciences vol 33 no 11 pp 1384ndash13982010

[18] J Satsuma ldquoExact solutions of Burgersrsquo equation with reac-tion termsrdquo in Topics in Soliton Theory And Exactly SolvableNonlinear Equations pp 255ndash262 World Scientific PublishingSingapore Asia 1987

[19] W Xinyi and L Yuekai ldquoExact solutions of the extendedburgers-fisher equationrdquoChinese Physics Letters vol 7 no 4 pp145ndash147 1990

[20] M Javidi ldquoA numerical solution of the generalized Burgers-Huxley equation by spectral collocation methodrdquo AppliedMathematics and Computation vol 178 no 2 pp 338ndash3442006

[21] M Javidi ldquoA numerical solution of the generalized Burgerrsquos-Huxley equation by pseudospectral method and Darvishirsquospreconditioningrdquo Applied Mathematics and Computation vol175 no 2 pp 1619ndash1628 2006

[22] M Javidi and A Golbabai ldquoA new domain decompositionalgorithm for generalized Burgerrsquos-Huxley equation based onChebyshev polynomials and preconditioningrdquo Chaos Solitonsamp Fractals vol 39 no 2 pp 849ndash857 2009

[23] A Golbabai and M Javidi ldquoA spectral domain decompositionapproach for the generalized Burgerrsquos-Fisher equationrdquo ChaosSolitons amp Fractals vol 39 no 1 pp 385ndash392 2009

[24] M T Darvishi S Kheybari and F Khani ldquoSpectral collocationmethod and Darvishirsquos preconditionings to solve the general-ized Burgers-Huxley equationrdquo Communications in NonlinearScience and Numerical Simulation vol 13 no 10 pp 2091ndash21032008

[25] M Sari G Gurarslan and I Dag ldquoA compact finite differencemethod for the solution of the generalized Burgers-FisherequationrdquoNumerical Methods for Partial Differential Equationsvol 26 no 1 pp 125ndash134 2010

[26] D A Hammad and M S El-Azab ldquo2N order compact finitedifference scheme with collocation method for solving thegeneralized Burgers-Huxley and Burgers-Fisher equationsrdquoApplied Mathematics and Computation vol 258 pp 296ndash3112015

[27] A J Khattak ldquoA computational meshless method for thegeneralized Burgerrsquos-Huxley equationrdquo Applied MathematicalModelling vol 33 no 9 pp 3718ndash3729 2009

[28] M Sari and G Gurarslan ldquoNumerical solutions of the gen-eralized burgers-huxley equation by a differential quadraturemethodrdquo Mathematical Problems in Engineering vol 2009Article ID 370765 11 pages 2009

[29] S A Malik I M Qureshi M Amir A N Malik and I HaqldquoNumerical Solution to generalized Burgersrsquo-Fisher equationusing exp-function method hybridized with heuristic compu-tationrdquo PLoS ONE vol 10 no 3 Article ID e0121728 2015

[30] M Dehghan B N Saray and M Lakestani ldquoThree methodsbased on the interpolation scaling functions and the mixedcollocation finite difference schemes for the numerical solutionof the nonlinear generalized Burgers-Huxley equationrdquoMathe-matical and Computer Modelling vol 50 no 3-4 pp 1129ndash11422012

[31] L Zhang LWang and X Ding ldquoExact finite difference schemeand nonstandard finite difference scheme for Burgers andBurgers-Fisher equationsrdquo Journal of Applied Mathematics vol2014 Article ID 597926 12 pages 2014

[32] J Biazar and F Muhammadi ldquoApplication of differential trans-form method to the generalized Burgers-Huxley equationrdquoApplications andAppliedMathematics An International Journalvol 5 no 10 pp 1726ndash1740 2010

[33] A G Bratsos ldquoA fourth order improved numerical scheme forthe generalized Burgers-Huxley Equationrdquo American Journal ofComputational Mathematics vol 1 pp 152ndash158 2011

[34] A G Bratsos ldquoAn improved second-order numerical methodfor the generalized burgers-fisher equationrdquo Anziam Journalvol 54 no 3 pp 181ndash199 2013

[35] M Zarebnia and N Aliniya ldquoA collocation method for numer-ical solution of the generalized Burgers-Huxley equationrdquoApplied Mathematics and Computation vol 11 no 8 pp 687ndash701 2014

[36] B Batiha M S M Noorani and I Hashim ldquoApplication ofvariational iteration method to the generalized Burger-HuxleyequationrdquoChaos Solitons amp Fractals vol 36 no 3 pp 660ndash6632008

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 17: Hybrid B-Spline Collocation Method for Solving the ...

Mathematical Problems in Engineering 17

[37] O O Morufu ldquoAn improved algorithm for the solution ofgeneralized burger-Fisher equationrdquo Applied Mathematics andComputation vol 5 pp 1609ndash1614 2014

[38] I Hashim M S Noorani and M R Said Al-Hadidi ldquoSolvingthe generalized Burgers-Huxley equation using the Adomiandecomposition methodrdquo Mathematical and Computer Mod-elling vol 43 no 11-12 pp 1404ndash1411 2006

[39] T Zhao C Li Z Zang and Y Wu ldquoChebyshevndashLegendrepseudo-spectral method for the generalised BurgersndashFisherequationrdquo Applied Mathematical Modelling vol 36 no 3 pp1046ndash1056 2012

[40] H N Ismail K Raslan and A A Rabboh ldquoAdomian decompo-sition method for Burgers-Huxley and equation Burgers-Fisherequationsrdquo Applied Mathematics and Computation vol 159 no1 pp 291ndash301 2004

[41] A M Al-Rozbayani and M O Al-Amr ldquoDiscrete Adomiandecomposition method for solving Burgerrsquos-Huxley equationrdquoInternational Journal of Contemporary Mathematical Sciencesvol 8 no 13-16 pp 623ndash631 2013

[42] B Inan and A R Bahadir ldquoNumerical solutions of the gen-eralized Burger-Huxley equation by implicit exponential finitedifference methodrdquo Journal of Applied Mathematics Statisticsand Informatics vol 11 2015

[43] I Celik ldquoChebyshev Wavelet collocation method for solvinggeneralized Burger-Huxley equationrdquoMathematical Methods inthe Applied Sciences vol 39 no 3 pp 366ndash377 2015

[44] M El-Kady S M El-Sayed and H E Fathy ldquoDevelopment ofGalerkin method for solving the generalized Burgers-Huxleyequationrdquo Mathematical Problems in Engineering vol 2013Article ID 165492 9 pages 2013

[45] B Inan and A R Bahadir ldquoNumerical solution of the one-dimensional Burgersrsquo equation Implicit and fully implicit expo-nential finite differencemethodsrdquo PramanamdashJournal of Physicsvol 81 no 4 pp 547ndash556 2013

[46] I Celik ldquoHaar wavelet method for solving generalized Burgers-Huxley equationrdquo Arab Journal of Mathematical Sciences vol18 no 1 pp 25ndash37 2012

[47] J E Macias-Diaz J Ruiz-Ramirez and J Villa ldquoThe numericalsolution of a generalized burgers-Huxley equation througha conditionally bounder and symmetry-preserving methodrdquoComputers amp Mathematics with Applications An InternationalJournal vol 61 no 11 pp 3330ndash3342 2011

[48] S Zhou and X Cheng ldquoA linearly semi-implicit compactscheme for the Burgers-Huxley equationrdquo International Journalof Computer Mathematics vol 88 no 4 pp 795ndash804 2011

[49] V J Ervin J E Macias-Diaz and J Ruiz-Ramirez ldquoA positiveand bounded finite element approximation of the generalizedBurgers-Huxley equationrdquo Journal of Mathematical Analysisand Applications vol 424 no 2 pp 1143ndash1160 2015

[50] G-WWang X-Q Liu andY-Y Zhang ldquoNew explicit solutionsof the generalized Burgers-Huxley equationrdquo Vietnam Journalof Mathematics vol 41 no 2 pp 161ndash166 2013

[51] KMOwolabi ldquoNumerical solution of the generalized Burgers-Huxley equation by exponential time differencing schemerdquoInternational Journal of Biomedical Engineering and Science vol1 pp 43ndash52 2015

[52] M Sari G Gurarslan and A Zeytinoglu ldquoHigh-order finitedifference schemes for numerical solutions of the generalizedBurgers-Huxley equationrdquo Numerical Methods for Partial Dif-ferential Equations vol 27 no 5 pp 1313ndash1326 2011

[53] R C Mittal and A Tripathi ldquoNumerical solutions of general-ized Burgers-Fisher and generalized Burgers-Huxley equationsusing collocation of cubic B-Splinesrdquo International Journal ofComputer Mathematics vol 92 no 5 pp 1053ndash1077 2015

[54] R C Mittal and R K Jain ldquoNumerical solutions of nonlinearBurgersrsquo equation with modified cubic B-Splines collocationmethodrdquo Applied Mathematics and Computation vol 218 no15 pp 7839ndash7855 2012

[55] B K Singh G Arora and M K Singh ldquoA numerical schemefor the generalized Burgers-Huxley equationrdquo Journal of theEgyptian Mathematical Society vol 24 no 4 pp 629ndash637 2016

[56] M Reza ldquoB-spline collocation algorithm for numerical solutionof the generalized Buegerrsquos-Huxley equationrdquo Numerical Meth-ods for Partial Differential Equations vol 29 no 4 pp 1173ndash11912013

[57] M Reza ldquoSpline solution of the generalized Burgers-Fisherequationrdquo Applicable Analysis vol 91 no 12 pp 2189ndash22152012

[58] M BukhariMArshad S Batool and SM Saqlain ldquoNumericalsolution of generalized Burgerrsquos-Huxley equation using localradial basis functionsrdquo International Journal of Advanced andApplied Sciences vol 4 no 5 pp 1ndash11 2017

[59] MYaseenMAbbas A I Ismail andTNazir ldquoA cubic trigono-metric B-spline collocation approach for the fractional sub-diffusion equationsrdquo Applied Mathematics and Computationvol 293 pp 311ndash319 2017

[60] T Nazir M Abbas A I M Ismail A A Majid and A RashidldquoThe numerical solution of advection-diffusion problems usingnew cubic trigonometric B-splines approachrdquo Applied Mathe-matical Modelling vol 40 no 7-8 pp 4586ndash4611 2016

[61] M Abbas A A Majid A I Md Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 Article IDe83265 2014

[62] M Abbas A A Majid A I Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014

[63] M Abbas A A Majid A I M Ismail and A Rashid ldquoNumer-ical method using cubic trigonometric B-Spline technique fornon-classical diffusion problemrdquoAbstract and Applied Analysisvol 2014 Article ID 849682 10 pages 2014

[64] S Mat Zin A Abd Majid A I M Ismail and M AbbasldquoApplication of hybrid cubic B-spline collocation approachfor solving a generalized nonlinear Klien-Gordon equationrdquoMathematical Problems in Engineering vol 2014 Article ID108560 10 pages 2014

[65] P M Prenter Splines and Variational Methods Wiley ClassicsLibrary John Wiley and Sons NY USA 1989

[66] C De Boor A Practical Guide to Splines (Applied MathematicalSciences) Springer NY USA 1978

[67] D U V Rosenberg Methods for solution of partial differentialequations American Elsevier Publishing NY USA 1969

[68] S S Siddiqi and S Arshed ldquoQuintic B-spline for the numericalsolution of the good Boussinesq equationrdquo Journal of theEgyptian Mathematical Society vol 22 no 2 pp 209ndash213 2014

[69] R Nawaz H Ullah S Islam and M Idrees ldquoApplication ofoptimal homotopy asymptotic method to burger equationsrdquoJournal of Applied Mathematics vol 2013 Article ID 387478 8pages 2013

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 18: Hybrid B-Spline Collocation Method for Solving the ...

18 Mathematical Problems in Engineering

[70] J He ldquoVariational iteration methodmdashA kind of non-linearanalytical technique some examplesrdquo International Journal ofNon-Linear Mechanics vol 34 no 4 pp 699ndash708 1999

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Page 19: Hybrid B-Spline Collocation Method for Solving the ...

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


The convergence of spline collocation for strongly ...arnold/papers/splinecol.pdfThe Spline Collocation for Strongly Elliptic Equations on Curves 321 Once a basis for 5ehd is chosen,

The convergence of spline collocation for strongly ...arnold/papers/splinecol.pdfThe Spline Collocation for Strongly Elliptic Equations on Curves 321 Once a basis for 5ehd is chosen,

Quintic B-spline collocation method for numerical solution ...

Quintic B-spline collocation method for numerical solution ...

HAAR WAVELET COLLOCATION METHOD FOR SOLVING …ijpam.uniud.it/online_issue/201839/31-Singh-Kumar (ijpam).pdf · Haar wavelet method for solving lumped and distributed-parameter systems

HAAR WAVELET COLLOCATION METHOD FOR SOLVING …ijpam.uniud.it/online_issue/201839/31-Singh-Kumar (ijpam).pdf · Haar wavelet method for solving lumped and distributed-parameter systems

Option pricing in jump diffusion models with quadratic spline collocation · 2015. 12. 21. · 1 Option pricing in jump diffusion models with quadratic spline collocation Christina

Option pricing in jump diffusion models with quadratic spline collocation · 2015. 12. 21. · 1 Option pricing in jump diffusion models with quadratic spline collocation Christina

A new numerical approach for the solution of the …Sextic B-spline collocation method1(SBCM1) and Sextic B-spline collocation method 2 (SBCM2) for solving the both BE and MBE. S.

A new numerical approach for the solution of the …Sextic B-spline collocation method1(SBCM1) and Sextic B-spline collocation method 2 (SBCM2) for solving the both BE and MBE. S.

BALL SPLINE STROKE BALL SPLINE...BALL SPLINE ROTARY BALL SPLINE STROKE BALL SPLINE BALL SCREW SPLINE BALL SPLINE STRUCTURE AND ADVANTAGES B-2 TYPES ...

BALL SPLINE STROKE BALL SPLINE...BALL SPLINE ROTARY BALL SPLINE STROKE BALL SPLINE BALL SCREW SPLINE BALL SPLINE STRUCTURE AND ADVANTAGES B-2 TYPES ...

Quintic B-spline Collocation Method For Sixth Order ... · the boundary value problems of type (1)-(2). Lamini et al. [11] used spline collocation method to solve the sixth-order

Quintic B-spline Collocation Method For Sixth Order ... · the boundary value problems of type (1)-(2). Lamini et al. [11] used spline collocation method to solve the sixth-order

Quadratic Non-Polynomial Spline Method for Solving the ...ijisset.org/wp-content/uploads/2015/12/IJISSET-010106.pdf · quadratic polynomial spline function in a,b when w →0, choosing

Quadratic Non-Polynomial Spline Method for Solving the ...ijisset.org/wp-content/uploads/2015/12/IJISSET-010106.pdf · quadratic polynomial spline function in a,b when w →0, choosing

A Chebyshev-Gauss collocation method for the numerical ...the collocation method with ( 1)N Chebyshev-Gauss points as the nodes. Then, we derive a new algorithm for solving an Then,

A Chebyshev-Gauss collocation method for the numerical ...the collocation method with ( 1)N Chebyshev-Gauss points as the nodes. Then, we derive a new algorithm for solving an Then,

BOUNDARY ELEMENT COLLOCATION METHOD FOR SOLVING …

BOUNDARY ELEMENT COLLOCATION METHOD FOR SOLVING …

B-SPLINE COLLOCATION APPROACH FOR SOLVING ...eprints.usm.my/31880/1/SHAZALINA_BINTI_MAT_ZIN_24(NN).pdfPARTIAL DIFFERENTIAL EQUATIONS 5.1 Introduction 69 5.2 Nonlinear Klein-Gordon

B-SPLINE COLLOCATION APPROACH FOR SOLVING ...eprints.usm.my/31880/1/SHAZALINA_BINTI_MAT_ZIN_24(NN).pdfPARTIAL DIFFERENTIAL EQUATIONS 5.1 Introduction 69 5.2 Nonlinear Klein-Gordon

 · CME Collocation Telkomsel, 2016 CM E Collocation Indosat, 2016 Crv1E Collocation Smart, 2016 CME Collocation Smart, 2016 CM E Collocation Smart, 2016 Strengthening 2016 ... Sumut

 · CME Collocation Telkomsel, 2016 CM E Collocation Indosat, 2016 Crv1E Collocation Smart, 2016 CME Collocation Smart, 2016 CM E Collocation Smart, 2016 Strengthening 2016 ... Sumut

Solving the 3D Laplace Equation by Meshless Collocation ...

Solving the 3D Laplace Equation by Meshless Collocation ...

Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Research Article A Jacobi Collocation Method for Solving ...3. Jacobi Spectral Collocation Method Since the collocation method approximates the di erential equationsinphysicalspace,itisveryeasytoimplementand

Research Article A Jacobi Collocation Method for Solving ...3. Jacobi Spectral Collocation Method Since the collocation method approximates the di erential equationsinphysicalspace,itisveryeasytoimplementand

Numerical Solutions of Reaction-Diffusion Equation Systems ...Numerical Solutions of Reaction-Diffusion Equation Systems with Trigonometric Quintic B-spline Collocation Algorithm

Numerical Solutions of Reaction-Diffusion Equation Systems ...Numerical Solutions of Reaction-Diffusion Equation Systems with Trigonometric Quintic B-spline Collocation Algorithm

A collocation method for solving some integral equations ...sapto/5.pdf · A collocation method for solving some integral equations in distributions Sapto W. Indratno Department of

A collocation method for solving some integral equations ...sapto/5.pdf · A collocation method for solving some integral equations in distributions Sapto W. Indratno Department of

B-SPLINE COLLOCATION BASED APPROXIMATIONS FOR STRUCTURAL ...

B-SPLINE COLLOCATION BASED APPROXIMATIONS FOR STRUCTURAL ...

B-Spline Finite Elements and their Efficiency in Solving ... · B-Spline Finite Elements and their Efficiency in Solving Relativistic Mean Field Equations ... The major goal of the

B-Spline Finite Elements and their Efficiency in Solving ... · B-Spline Finite Elements and their Efficiency in Solving Relativistic Mean Field Equations ... The major goal of the

Adaptive and high-order methods for valuing American options · of such efficient high-order spline collocation methods to option pricing, especially to pricing American options,

Adaptive and high-order methods for valuing American options · of such efficient high-order spline collocation methods to option pricing, especially to pricing American options,