Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 760542 12 pageshttpdxdoiorg1011552013760542

Research ArticleA Jacobi Collocation Method for Solving NonlinearBurgers-Type Equations

E H Doha1 D Baleanu23 A H Bhrawy45 and M A Abdelkawy5

1 Department of Mathematics Faculty of Science Cairo University Giza 12613 Egypt2 Department of Chemical and Materials Engineering Faculty of Engineering King Abdulaziz University Jeddah 21589 Saudi Arabia3 Department of Mathematics and Computer Sciences Faculty of Arts and Sciences Cankaya University Eskiehir Yolu 29 Km06810 Ankara Turkey

4Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia5 Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef 62511 Egypt

Correspondence should be addressed to M A Abdelkawy melkawyyahoocom

Received 24 July 2013 Accepted 15 August 2013

Academic Editor Soheil Salahshour

Copyright copy 2013 E H Doha 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

We solve three versions of nonlinear time-dependent Burgers-type equations The Jacobi-Gauss-Lobatto points are used ascollocation nodes for spatial derivativesThis approach has the advantage of obtaining the solution in terms of the Jacobi parameters120572 and 120573 In addition the problem is reduced to the solution of the system of ordinary differential equations (SODEs) in time Thissystemmay be solved by any standard numerical techniques Numerical solutions obtained by thismethodwhen comparedwith theexact solutions reveal that the obtained solutions produce high-accurate results Numerical results show that the proposed methodis of high accuracy and is efficient to solve the Burgers-type equation Also the results demonstrate that the proposed method is apowerful algorithm to solve the nonlinear partial differential equations

1 Introduction

Spectral methods (see eg [1ndash3] and the references therein)are techniques used in applied mathematics and scientificcomputing to numerically solve linear and nonlinear dif-ferential equations There are three well-known versions ofspectral methods namely Galerkin tau and collocationmethods Spectral collocation method is characterized bythe fact of providing highly accurate solutions to nonlineardifferential equations [3ndash6] also it has become increasinglypopular for solving fractional differential equations [7ndash9]Bhrawy et al [5] proposed a new Bernoulli matrix methodfor solving high-order Fredholm integro-differential equa-tions with piecewise intervals Saadatmandi and Dehghan[10] developed the Sinc-collocation approach for solvingmultipoint boundary value problems in this approach thecomputation of solution of such problems is reduced to solvesome algebraic equations Bhrawy and Alofi [4] proposedthe spectral-shifted Jacobi-Gauss collocation method to find

an accurate solution of the Lane-Emden-type equationMoreover Doha et al [11] developed the shifted Jacobi-Gausscollocation method to solve nonlinear high-order multipointboundary value problems To the best of our knowledge thereare no results on Jacobi-Gauss-Lobatto collocation methodfor solving Burgers-type equations arising in mathematicalphysicsThis partially motivated our interest in suchmethod

For time-dependent partial differential equations spec-tral methods have been studied in some articles for severaldecades In [12] Ierley et al investigated spectral methods tonumerically solve time-dependent class of parabolic partialdifferential equations subject to periodic boundary condi-tions Tal-Ezer [13 14] introduced spectral methods usingpolynomial approximation of the evolution operator in theChebyshev Least-Squares sense for time-dependent parabolicand hyperbolic equations respectively Moreover Coutsiaset al [15] developed spectral integration method to solvesome time-dependent partial differential equations Zhang[16] applied the Fourier spectral scheme in spatial together

2 Abstract and Applied Analysis

with the Legendre spectral method to solve time-dependentpartial differential equations and gave error estimates of themethod Tang and Ma [17] introduced the Legendre spectralmethod together with the Fourier approximation in spatialfor time-dependent first-order hyperbolic equations withperiodic boundary conditions Recently the author of [18]proposed an accurate numerical algorithm to solve the gen-eralized Fitzhugh-Nagumo equation with time-dependentcoefficients

In [20] Bateman introduced the one-dimensional quasi-linear parabolic partial differential equation while Burgers[21] developed it as mathematical modeling of turbulenceand it is referred as one-dimensional Burgersrsquo equationMany authors gave different solutions for Burgersrsquo equationby using various methods Kadalbajoo and Awasthi [22]and Gulsu [23] used a finite-difference approach method tofind solutions of one-dimensional Burgersrsquo equation Crank-Nicolson scheme for Burgersrsquo equation is developed by Kim[24] Nguyen and Reynen [25 26] Gardner et al [27 28]and Kutluay et al [29] used methods based on the Petrov-Galerkin Least-Squares finite-elements and B-spline finiteelement methods to solve Burgersrsquo equation A methodbased on collocation of modified cubic B-splines over finiteelements has been investigated by Mittal and Jain in [30]

In this work we propose a J-GL-Cmethod to numericallysolve the following three nonlinear time-dependent Burgersrsquo-type equations

(1) time-dependent 1D Burgersrsquo equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (1)

(2) time-dependent 1D generalized Burger-Fisher equa-tion

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(2)

(3) time-dependent 1D generalized Burgers-Huxleyequation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin [119860 119861] times [0 119879]

(3)

In order to obtain the solution in terms of the Jacobiparameters 120572 and 120573 the use of the Jacobi polynomials forsolving differential equations has gained increasing popu-larity in recent years (see [31ndash35]) The main concern ofthis paper is to extend the application of J-GL-C methodto solve the three nonlinear time-dependent Burgers-typeequations It would be very useful to carry out a systematicstudy on J-GL-C method with general indexes (120572 120573 gt

minus1)The nonlinear time-dependent Burgersrsquo-type equation iscollocated only for the space variable at (119873 minus 1) points andfor suitable collocation points we use the (119873 minus 1) nodes ofthe Jacobi-Gauss-Lobatto interpolation which depends uponthe two general parameters (120572 120573 gt minus1) these equationstogether with the two-point boundary conditions constitute

the system of (119873 + 1) ordinary differential equations (ODEs)in time This system can be solved by one of the possiblemethods of numerical analysis such as the Euler methodMidpoint method and the Runge-Kutta method Finally theaccuracy of the proposed method is demonstrated by testproblems

The remainder of the paper is organized as followsIn the next section we introduce some properties of theJacobi polynomials In Section 3 the way of constructingthe Gauss-Lobatto collocation technique for nonlinear time-dependent Burgers-type equations is described using theJacobi polynomials and in Section 4 the proposed methodis applied to three problems of nonlinear time-dependentBurgers-type equations Finally some concluding remarksare given in Section 5

2 Some Properties of Jacobi Polynomials

The standard Jacobi polynomials of degree 119896 (119875(120572120573)119896

(119909) 119896 =0 1 ) with the parameters 120572 gt minus1 120573 gt minus1 are satisfyingthe following relations

119875(120572120573)

119896(minus119909) = (minus1)

119896119875(120572120573)

119896(119909)

119875(120572120573)

119896(minus1) =

(minus1)119896Γ (119896 + 120573 + 1)

119896Γ (120573 + 1)

119875(120572120573)

119896(1) =

Γ (119896 + 120572 + 1)

119896Γ (120572 + 1)

(4)

Let 119908(120572120573)(119909) = (1 minus 119909)120572(1 + 119909)120573 then we define the weightedspace 1198712

119908(120572120573) as usual equipped with the following inner

product and norm

(119906 V)119908(120572120573) = int

1

minus1

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573) = (119906 119906)

12

119908(120572120573)

(5)

The set the of Jacobi polynomials forms a complete 1198712119908(120572120573)-

orthogonal system and100381710038171003817100381710038171003817

119875(120572120573)

119896

100381710038171003817100381710038171003817119908(120572120573)

= ℎ119896=

2120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) Γ (119896 + 1) Γ (119896 + 120572 + 120573 + 1)

(6)

Let 119878119873(minus1 1) be the set of polynomials of degree at most

119873 and due to the property of the standard Jacobi-Gaussquadrature it follows that for any 120601 isin 119878

2119873+1(minus1 1)

int

1

minus1

119908(120572120573)

(119909) 120601 (119909) 119889119909 =

119873

sum

119895=0

120603(120572120573)

119873119895120601 (119909(120572120573)

119873119895) (7)

where 119909(120572120573)119873119895

(0 le 119895 le 119873) and 120603(120572120573)

119873119895(0 le 119895 le 119873)

are the nodes and the corresponding Christoffel numbers ofthe Jacobi-Gauss-quadrature formula on the interval (minus1 1)

Abstract and Applied Analysis 3

respectively Now we introduce the following discrete innerproduct and norm

(119906 V)119908(120572120573) =

119873

sum

119895=0

119906 (119909(120572120573)

119873119895) V (119909(120572120573)119873119895

) 120603(120572120573)

119873119895

119906119908(120572120573) = (119906 119906)

12

119908(120572120573)

(8)

For 120572 = 120573 one recovers the ultraspherical polynomials(symmetric Jacobi polynomials) and for 120572 = 120573 = ∓12 120572 =120573 = 0 the Chebyshev of the first and second kindsand the Legendre polynomials respectively and for thenonsymmetric Jacobi polynomials the two important specialcases120572 = minus120573 = plusmn12 (theChebyshev polynomials of the thirdand fourth kinds) are also recovered

3 Jacobi Spectral Collocation Method

Since the collocation method approximates the differentialequations in physical space it is very easy to implement andbe adaptable to various problems including variable coeffi-cient and nonlinear differential equations (see for instance[4 6]) In this section we develop the J-GL-C method tonumerically solve the Burgers-type equations

31 (1 + 1)-Dimensional Burgersrsquo Equation In 1939 Burg-ers has simplified the Navier-Stokes equation by droppingthe pressure term to obtain his one-dimensional Burgersrsquoequation This equation has many applications in appliedmathematics such as modeling of gas dynamics [36 37]modeling of fluid dynamics turbulence boundary layerbehavior shock wave formation and traffic flow [38] In thissubsection we derive a J-GL-C method to solve numericallythe (1 + 1)-dimensional Burgersrsquo model problem

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin 119863 times [0 119879] (9)

where

119863 = 119909 minus1 le 119909 le 1 (10)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) 119905 isin [0 119879] (11)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (12)

Now we assume that

119906 (119909 119905) =

119873

sum

119895=0

119886119895(119905) 119875(120572120573)

119895(119909) (13)

and if we make use of (6)ndash(8) then we find

119886119895(119905) =

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905) (14)

and accordingly (14) takes the form

119906 (119909 119905)

=

119873

sum

119895=0

(

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905))119875

(120572120573)

119895(119909)

(15)

or equivalently takes the form

119906 (119909 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) 119875(120572120573)

119895(119909) 120603(120572120573)

119873119894)119906 (119909

(120572120573)

119873119894 119905)

(16)

The spatial partial derivatives with respect to 119909 in (9) can becomputed at the J-GL-C points to give

119906119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119860119899119894119906 (119909(120572120573)

119873119894 119905) 119899 = 0 1 119873

119906119909119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119861119899119894119906(119909(120572120573)

119873119894 119905)

(17)

where

119860119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894

119861119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894

(18)

Making use of (17) and (18) enables one to rewrite (9) in theform

119899(119905) + ]119906

119899(119905)

119873

sum

119894=0

119860119899119894119906119894(119905) minus 120583

119873

sum

119894=0

119861119899119894119906119894(119905) = 0

119899 = 1 119873 minus 1

(19)

where

119906119899(119905) = 119906 (119909

(120572120573)

119873119899 119905) (20)

4 Abstract and Applied Analysis

Using Equation (19) and using the two-point boundaryconditions (11) generate a system of (119873 minus 1) ODEs in time

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

(21)

where

119889119899(119905) = 119860

1198991199001198921(119905) + 119860

1198991198731198922(119905)

119889119899(119905) = 119861

1198991199001198921(119905) + 119861

1198991198731198922(119905)

(22)

Then the problem (9)ndash(12) transforms to the SODEs

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+ ]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

119906119899(0) = 119891 (119909

(120572120573)

119873119899)

(23)

which may be written in the following matrix form

u (119905) = F (119905 119906 (119905))

u (0) = f (24)

where

u (119905) = [1(119905) 2(119905)

119873minus1(119905)]119879

f = [119891 (1199091198731) 119891 (119909

1198732) 119891 (119909

119873119873minus1)]119879

F (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865

2(119905 119906 (119905)) 119865

119873minus1(119905 119906 (119905))]

119879

119865119899(119905 119906 (119905)) = minus]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) + 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

minus ]119889119899(119905) + 120583

119889119899(119905) 119899 = 1 119873 minus 1

(25)

The SODEs (24) in time may be solved using any standardtechnique like the implicit Runge-Kutta method

32 (1 + 1)-Dimensional Burger-Fisher Equation TheBurger-Fisher equation is a combined form of Fisher and Burgersrsquoequations The Fisher equation was firstly introduced byFisher in [39] to describe the propagation of a mutantgene This equation has a wide range of applications in alarge number of the fields of chemical kinetics [40] logisticpopulation growth [41] flame propagation [42] populationin one-dimensional habitual [43] neutron population in anuclear reaction [44] neurophysiology [45] autocatalyticchemical reactions [19] branching the Brownian motionprocesses [40] and nuclear reactor theory [46] Moreoverthe Burger-Fisher equation has awide range of applications invarious fields of financial mathematics applied mathematics

and physics applications gas dynamic and traffic flow TheBurger-Fisher equation can be written in the following form

119906119905= 119906119909119909minus ]119906119906

119909+ 120574119906 (1 minus 119906) (119909 119905) isin 119863 times [0 119879] (26)

where

119863 = 119909 minus1 lt 119909 lt 1 (27)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (28)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (29)

The same procedure of Section 31 can be used to reduce(26)ndash(29) to the system of nonlinear differential equationsin the unknown expansion coefficients of the sought-forsemianalytical solution This system is solved by using theimplicit Runge-Kutta method

33 (1 + 1)-Dimensional Generalized Burgers-Huxley Equa-tion The Huxley equation is a nonlinear partial differentialequation of second order of the form

119906119905minus 119906119909119909minus 119906 (119896 minus 119906) (119906 minus 1) = 0 119896 = 0 (30)

It is an evolution equation that describes the nerve propaga-tion [47] in biology from which molecular CB properties canbe calculated It also gives a phenomenological description ofthe behavior of the myosin heads II In addition to this non-linear evolution equation combined forms of this equationand Burgersrsquo equation will be investigated It is interesting topoint out that this equation includes the convection term 119906

119909

and the dissipation term119906119909119909in addition to other terms In this

subsection we derive J-GL-C method to solve numericallythe (1+1)-dimensional generalizedBurgers-Huxley equation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin 119863 times [0 119879]

(31)

where

119863 = 119909 minus1 lt 119909 lt 1 (32)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (33)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (34)

The same procedure of Sections 31 and 32 is used to solvenumerically (30)ndash(34)

Abstract and Applied Analysis 5

4 Numerical Results

To illustrate the effectiveness of the proposed method in thepresent paper three test examples are carried out in thissection The comparison of the results obtained by variouschoices of the Jacobi parameters 120572 and 120573 reveals that thepresentmethod is very effective and convenient for all choicesof 120572 and 120573 We consider the following three examples

Example 1 Consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120578119906 (1 minus 119906

120575) (119906120575minus 120574)

(119909 119905) isin [119860 119861] times [0 119879]

(35)

subject to the boundary conditions

119906 (119860 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119860 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1 + 120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

119906 (119861 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119861 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1+120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

(36)

and the initial condition

119906 (119909 0)

=[

[

[

120574

2

+

120574

2

tanh[[

[

119909120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

]

]

]

]

]

]

1120575

119909 isin [119860 119861]

(37)

The exact solution of (35) is

119906 (119909 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times(119909minus

]120574 (1+120575minus120574)(radic]2+4120578(1+120575)minus])

2(120575+1)2

119905)]

]

]

]

]

]

1120575

(38)

The difference between themeasured value of the approx-imate solution and its actual value (absolute error) given by

119864 (119909 119905) = |119906 (119909 119905) minus (119909 119905)| (39)

where 119906(119909 119905) and (119909 119905) is the exact solution and theapproximate solution at the point (119909 119905) respectively

In the cases of 120574 = 10minus3 ] = 120578 = 120575 = 1 and 119873 = 4

Table 1 lists the comparison of absolute errors of problem(35) subject to (36) and (37) using the J-GL-C method fordifferent choices of 120572 and 120573 with references [19] in theinterval [0 1] Moreover in Tables 2 and 3 the absolute errorsof this problem with 120572 = 120573 = 12 and various choices of119909 119905 for 120575 = 1 (3) in both intervals [0 1] and [minus1 1] aregiven respectively In Table 4 maximum absolute errors withvarious choices of (120572 120573) for both values of 120575 = 1 3 are givenwhere ] = 120574 = 120578 = 0001 in both intervals [0 1] and [minus1 1]Moreover the absolute errors of problem (35) are shown inFigures 1 2 and 3 for 120575 = 1 2 and 3with values of parameterslisted in their captions respectively while in Figure 4 weplotted the approximate solution of this problemwhere120572 = 0120573 = 1 ] = 120578 = 120574 = 10

minus3 and 119873 = 12 for 120575 = 1 Thesefigures demonstrate the good accuracy of this algorithm forall choices of 120572 120573 and119873 and moreover in any interval

Example 2 Consider the nonlinear time-dependent onedimensional Burgers-type equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (40)

6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Mathematical PhysicsAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Abstract and Applied Analysis

with the Legendre spectral method to solve time-dependentpartial differential equations and gave error estimates of themethod Tang and Ma [17] introduced the Legendre spectralmethod together with the Fourier approximation in spatialfor time-dependent first-order hyperbolic equations withperiodic boundary conditions Recently the author of [18]proposed an accurate numerical algorithm to solve the gen-eralized Fitzhugh-Nagumo equation with time-dependentcoefficients

In [20] Bateman introduced the one-dimensional quasi-linear parabolic partial differential equation while Burgers[21] developed it as mathematical modeling of turbulenceand it is referred as one-dimensional Burgersrsquo equationMany authors gave different solutions for Burgersrsquo equationby using various methods Kadalbajoo and Awasthi [22]and Gulsu [23] used a finite-difference approach method tofind solutions of one-dimensional Burgersrsquo equation Crank-Nicolson scheme for Burgersrsquo equation is developed by Kim[24] Nguyen and Reynen [25 26] Gardner et al [27 28]and Kutluay et al [29] used methods based on the Petrov-Galerkin Least-Squares finite-elements and B-spline finiteelement methods to solve Burgersrsquo equation A methodbased on collocation of modified cubic B-splines over finiteelements has been investigated by Mittal and Jain in [30]

In this work we propose a J-GL-Cmethod to numericallysolve the following three nonlinear time-dependent Burgersrsquo-type equations

(1) time-dependent 1D Burgersrsquo equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (1)

(2) time-dependent 1D generalized Burger-Fisher equa-tion

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(2)

(3) time-dependent 1D generalized Burgers-Huxleyequation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin [119860 119861] times [0 119879]

(3)

In order to obtain the solution in terms of the Jacobiparameters 120572 and 120573 the use of the Jacobi polynomials forsolving differential equations has gained increasing popu-larity in recent years (see [31ndash35]) The main concern ofthis paper is to extend the application of J-GL-C methodto solve the three nonlinear time-dependent Burgers-typeequations It would be very useful to carry out a systematicstudy on J-GL-C method with general indexes (120572 120573 gt

minus1)The nonlinear time-dependent Burgersrsquo-type equation iscollocated only for the space variable at (119873 minus 1) points andfor suitable collocation points we use the (119873 minus 1) nodes ofthe Jacobi-Gauss-Lobatto interpolation which depends uponthe two general parameters (120572 120573 gt minus1) these equationstogether with the two-point boundary conditions constitute

the system of (119873 + 1) ordinary differential equations (ODEs)in time This system can be solved by one of the possiblemethods of numerical analysis such as the Euler methodMidpoint method and the Runge-Kutta method Finally theaccuracy of the proposed method is demonstrated by testproblems

The remainder of the paper is organized as followsIn the next section we introduce some properties of theJacobi polynomials In Section 3 the way of constructingthe Gauss-Lobatto collocation technique for nonlinear time-dependent Burgers-type equations is described using theJacobi polynomials and in Section 4 the proposed methodis applied to three problems of nonlinear time-dependentBurgers-type equations Finally some concluding remarksare given in Section 5

2 Some Properties of Jacobi Polynomials

The standard Jacobi polynomials of degree 119896 (119875(120572120573)119896

(119909) 119896 =0 1 ) with the parameters 120572 gt minus1 120573 gt minus1 are satisfyingthe following relations

119875(120572120573)

119896(minus119909) = (minus1)

119896119875(120572120573)

119896(119909)

119875(120572120573)

119896(minus1) =

(minus1)119896Γ (119896 + 120573 + 1)

119896Γ (120573 + 1)

119875(120572120573)

119896(1) =

Γ (119896 + 120572 + 1)

119896Γ (120572 + 1)

(4)

Let 119908(120572120573)(119909) = (1 minus 119909)120572(1 + 119909)120573 then we define the weightedspace 1198712

119908(120572120573) as usual equipped with the following inner

product and norm

(119906 V)119908(120572120573) = int

1

minus1

119906 (119909) V (119909) 119908(120572120573) (119909) 119889119909

119906119908(120572120573) = (119906 119906)

12

119908(120572120573)

(5)

The set the of Jacobi polynomials forms a complete 1198712119908(120572120573)-

orthogonal system and100381710038171003817100381710038171003817

119875(120572120573)

119896

100381710038171003817100381710038171003817119908(120572120573)

= ℎ119896=

2120572+120573+1

Γ (119896 + 120572 + 1) Γ (119896 + 120573 + 1)

(2119896 + 120572 + 120573 + 1) Γ (119896 + 1) Γ (119896 + 120572 + 120573 + 1)

(6)

Let 119878119873(minus1 1) be the set of polynomials of degree at most

119873 and due to the property of the standard Jacobi-Gaussquadrature it follows that for any 120601 isin 119878

2119873+1(minus1 1)

int

1

minus1

119908(120572120573)

(119909) 120601 (119909) 119889119909 =

119873

sum

119895=0

120603(120572120573)

119873119895120601 (119909(120572120573)

119873119895) (7)

where 119909(120572120573)119873119895

(0 le 119895 le 119873) and 120603(120572120573)

119873119895(0 le 119895 le 119873)

are the nodes and the corresponding Christoffel numbers ofthe Jacobi-Gauss-quadrature formula on the interval (minus1 1)

Abstract and Applied Analysis 3

respectively Now we introduce the following discrete innerproduct and norm

(119906 V)119908(120572120573) =

119873

sum

119895=0

119906 (119909(120572120573)

119873119895) V (119909(120572120573)119873119895

) 120603(120572120573)

119873119895

119906119908(120572120573) = (119906 119906)

12

119908(120572120573)

(8)

For 120572 = 120573 one recovers the ultraspherical polynomials(symmetric Jacobi polynomials) and for 120572 = 120573 = ∓12 120572 =120573 = 0 the Chebyshev of the first and second kindsand the Legendre polynomials respectively and for thenonsymmetric Jacobi polynomials the two important specialcases120572 = minus120573 = plusmn12 (theChebyshev polynomials of the thirdand fourth kinds) are also recovered

3 Jacobi Spectral Collocation Method

Since the collocation method approximates the differentialequations in physical space it is very easy to implement andbe adaptable to various problems including variable coeffi-cient and nonlinear differential equations (see for instance[4 6]) In this section we develop the J-GL-C method tonumerically solve the Burgers-type equations

31 (1 + 1)-Dimensional Burgersrsquo Equation In 1939 Burg-ers has simplified the Navier-Stokes equation by droppingthe pressure term to obtain his one-dimensional Burgersrsquoequation This equation has many applications in appliedmathematics such as modeling of gas dynamics [36 37]modeling of fluid dynamics turbulence boundary layerbehavior shock wave formation and traffic flow [38] In thissubsection we derive a J-GL-C method to solve numericallythe (1 + 1)-dimensional Burgersrsquo model problem

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin 119863 times [0 119879] (9)

where

119863 = 119909 minus1 le 119909 le 1 (10)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) 119905 isin [0 119879] (11)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (12)

Now we assume that

119906 (119909 119905) =

119873

sum

119895=0

119886119895(119905) 119875(120572120573)

119895(119909) (13)

and if we make use of (6)ndash(8) then we find

119886119895(119905) =

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905) (14)

and accordingly (14) takes the form

119906 (119909 119905)

=

119873

sum

119895=0

(

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905))119875

(120572120573)

119895(119909)

(15)

or equivalently takes the form

119906 (119909 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) 119875(120572120573)

119895(119909) 120603(120572120573)

119873119894)119906 (119909

(120572120573)

119873119894 119905)

(16)

The spatial partial derivatives with respect to 119909 in (9) can becomputed at the J-GL-C points to give

119906119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119860119899119894119906 (119909(120572120573)

119873119894 119905) 119899 = 0 1 119873

119906119909119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119861119899119894119906(119909(120572120573)

119873119894 119905)

(17)

where

119860119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894

119861119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894

(18)

Making use of (17) and (18) enables one to rewrite (9) in theform

119899(119905) + ]119906

119899(119905)

119873

sum

119894=0

119860119899119894119906119894(119905) minus 120583

119873

sum

119894=0

119861119899119894119906119894(119905) = 0

119899 = 1 119873 minus 1

(19)

where

119906119899(119905) = 119906 (119909

(120572120573)

119873119899 119905) (20)

4 Abstract and Applied Analysis

Using Equation (19) and using the two-point boundaryconditions (11) generate a system of (119873 minus 1) ODEs in time

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

(21)

where

119889119899(119905) = 119860

1198991199001198921(119905) + 119860

1198991198731198922(119905)

119889119899(119905) = 119861

1198991199001198921(119905) + 119861

1198991198731198922(119905)

(22)

Then the problem (9)ndash(12) transforms to the SODEs

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+ ]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

119906119899(0) = 119891 (119909

(120572120573)

119873119899)

(23)

which may be written in the following matrix form

u (119905) = F (119905 119906 (119905))

u (0) = f (24)

where

u (119905) = [1(119905) 2(119905)

119873minus1(119905)]119879

f = [119891 (1199091198731) 119891 (119909

1198732) 119891 (119909

119873119873minus1)]119879

F (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865

2(119905 119906 (119905)) 119865

119873minus1(119905 119906 (119905))]

119879

119865119899(119905 119906 (119905)) = minus]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) + 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

minus ]119889119899(119905) + 120583

119889119899(119905) 119899 = 1 119873 minus 1

(25)

The SODEs (24) in time may be solved using any standardtechnique like the implicit Runge-Kutta method

32 (1 + 1)-Dimensional Burger-Fisher Equation TheBurger-Fisher equation is a combined form of Fisher and Burgersrsquoequations The Fisher equation was firstly introduced byFisher in [39] to describe the propagation of a mutantgene This equation has a wide range of applications in alarge number of the fields of chemical kinetics [40] logisticpopulation growth [41] flame propagation [42] populationin one-dimensional habitual [43] neutron population in anuclear reaction [44] neurophysiology [45] autocatalyticchemical reactions [19] branching the Brownian motionprocesses [40] and nuclear reactor theory [46] Moreoverthe Burger-Fisher equation has awide range of applications invarious fields of financial mathematics applied mathematics

and physics applications gas dynamic and traffic flow TheBurger-Fisher equation can be written in the following form

119906119905= 119906119909119909minus ]119906119906

119909+ 120574119906 (1 minus 119906) (119909 119905) isin 119863 times [0 119879] (26)

where

119863 = 119909 minus1 lt 119909 lt 1 (27)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (28)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (29)

The same procedure of Section 31 can be used to reduce(26)ndash(29) to the system of nonlinear differential equationsin the unknown expansion coefficients of the sought-forsemianalytical solution This system is solved by using theimplicit Runge-Kutta method

33 (1 + 1)-Dimensional Generalized Burgers-Huxley Equa-tion The Huxley equation is a nonlinear partial differentialequation of second order of the form

119906119905minus 119906119909119909minus 119906 (119896 minus 119906) (119906 minus 1) = 0 119896 = 0 (30)

It is an evolution equation that describes the nerve propaga-tion [47] in biology from which molecular CB properties canbe calculated It also gives a phenomenological description ofthe behavior of the myosin heads II In addition to this non-linear evolution equation combined forms of this equationand Burgersrsquo equation will be investigated It is interesting topoint out that this equation includes the convection term 119906

119909

and the dissipation term119906119909119909in addition to other terms In this

subsection we derive J-GL-C method to solve numericallythe (1+1)-dimensional generalizedBurgers-Huxley equation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin 119863 times [0 119879]

(31)

where

119863 = 119909 minus1 lt 119909 lt 1 (32)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (33)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (34)

The same procedure of Sections 31 and 32 is used to solvenumerically (30)ndash(34)

Abstract and Applied Analysis 5

4 Numerical Results

To illustrate the effectiveness of the proposed method in thepresent paper three test examples are carried out in thissection The comparison of the results obtained by variouschoices of the Jacobi parameters 120572 and 120573 reveals that thepresentmethod is very effective and convenient for all choicesof 120572 and 120573 We consider the following three examples

Example 1 Consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120578119906 (1 minus 119906

120575) (119906120575minus 120574)

(119909 119905) isin [119860 119861] times [0 119879]

(35)

subject to the boundary conditions

119906 (119860 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119860 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1 + 120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

119906 (119861 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119861 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1+120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

(36)

and the initial condition

119906 (119909 0)

=[

[

[

120574

2

+

120574

2

tanh[[

[

119909120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

]

]

]

]

]

]

1120575

119909 isin [119860 119861]

(37)

The exact solution of (35) is

119906 (119909 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times(119909minus

]120574 (1+120575minus120574)(radic]2+4120578(1+120575)minus])

2(120575+1)2

119905)]

]

]

]

]

]

1120575

(38)

The difference between themeasured value of the approx-imate solution and its actual value (absolute error) given by

119864 (119909 119905) = |119906 (119909 119905) minus (119909 119905)| (39)

where 119906(119909 119905) and (119909 119905) is the exact solution and theapproximate solution at the point (119909 119905) respectively

In the cases of 120574 = 10minus3 ] = 120578 = 120575 = 1 and 119873 = 4

Table 1 lists the comparison of absolute errors of problem(35) subject to (36) and (37) using the J-GL-C method fordifferent choices of 120572 and 120573 with references [19] in theinterval [0 1] Moreover in Tables 2 and 3 the absolute errorsof this problem with 120572 = 120573 = 12 and various choices of119909 119905 for 120575 = 1 (3) in both intervals [0 1] and [minus1 1] aregiven respectively In Table 4 maximum absolute errors withvarious choices of (120572 120573) for both values of 120575 = 1 3 are givenwhere ] = 120574 = 120578 = 0001 in both intervals [0 1] and [minus1 1]Moreover the absolute errors of problem (35) are shown inFigures 1 2 and 3 for 120575 = 1 2 and 3with values of parameterslisted in their captions respectively while in Figure 4 weplotted the approximate solution of this problemwhere120572 = 0120573 = 1 ] = 120578 = 120574 = 10

minus3 and 119873 = 12 for 120575 = 1 Thesefigures demonstrate the good accuracy of this algorithm forall choices of 120572 120573 and119873 and moreover in any interval

Example 2 Consider the nonlinear time-dependent onedimensional Burgers-type equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (40)

6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 3

respectively Now we introduce the following discrete innerproduct and norm

(119906 V)119908(120572120573) =

119873

sum

119895=0

119906 (119909(120572120573)

119873119895) V (119909(120572120573)119873119895

) 120603(120572120573)

119873119895

119906119908(120572120573) = (119906 119906)

12

119908(120572120573)

(8)

For 120572 = 120573 one recovers the ultraspherical polynomials(symmetric Jacobi polynomials) and for 120572 = 120573 = ∓12 120572 =120573 = 0 the Chebyshev of the first and second kindsand the Legendre polynomials respectively and for thenonsymmetric Jacobi polynomials the two important specialcases120572 = minus120573 = plusmn12 (theChebyshev polynomials of the thirdand fourth kinds) are also recovered

3 Jacobi Spectral Collocation Method

Since the collocation method approximates the differentialequations in physical space it is very easy to implement andbe adaptable to various problems including variable coeffi-cient and nonlinear differential equations (see for instance[4 6]) In this section we develop the J-GL-C method tonumerically solve the Burgers-type equations

31 (1 + 1)-Dimensional Burgersrsquo Equation In 1939 Burg-ers has simplified the Navier-Stokes equation by droppingthe pressure term to obtain his one-dimensional Burgersrsquoequation This equation has many applications in appliedmathematics such as modeling of gas dynamics [36 37]modeling of fluid dynamics turbulence boundary layerbehavior shock wave formation and traffic flow [38] In thissubsection we derive a J-GL-C method to solve numericallythe (1 + 1)-dimensional Burgersrsquo model problem

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin 119863 times [0 119879] (9)

where

119863 = 119909 minus1 le 119909 le 1 (10)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) 119905 isin [0 119879] (11)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (12)

Now we assume that

119906 (119909 119905) =

119873

sum

119895=0

119886119895(119905) 119875(120572120573)

119895(119909) (13)

and if we make use of (6)ndash(8) then we find

119886119895(119905) =

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905) (14)

and accordingly (14) takes the form

119906 (119909 119905)

=

119873

sum

119895=0

(

1

ℎ119895

119873

sum

119894=0

119875(120572120573)

119895(119909(120572120573)

119873119894) 120603(120572120573)

119873119894119906 (119909(120572120573)

119873119894 119905))119875

(120572120573)

119895(119909)

(15)

or equivalently takes the form

119906 (119909 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) 119875(120572120573)

119895(119909) 120603(120572120573)

119873119894)119906 (119909

(120572120573)

119873119894 119905)

(16)

The spatial partial derivatives with respect to 119909 in (9) can becomputed at the J-GL-C points to give

119906119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119860119899119894119906 (119909(120572120573)

119873119894 119905) 119899 = 0 1 119873

119906119909119909(119909(120572120573)

119873119899 119905)

=

119873

sum

119894=0

(

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894)119906(119909

(120572120573)

119873119894 119905)

=

119873

sum

119894=0

119861119899119894119906(119909(120572120573)

119873119894 119905)

(17)

where

119860119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

1015840

120603(120572120573)

119873119894

119861119899119894=

119873

sum

119895=0

1

ℎ119895

119875(120572120573)

119895(119909(120572120573)

119873119894) (119875(120572120573)

119895(119909(120572120573)

119873119899))

10158401015840

120603(120572120573)

119873119894

(18)

Making use of (17) and (18) enables one to rewrite (9) in theform

119899(119905) + ]119906

119899(119905)

119873

sum

119894=0

119860119899119894119906119894(119905) minus 120583

119873

sum

119894=0

119861119899119894119906119894(119905) = 0

119899 = 1 119873 minus 1

(19)

where

119906119899(119905) = 119906 (119909

(120572120573)

119873119899 119905) (20)

4 Abstract and Applied Analysis

Using Equation (19) and using the two-point boundaryconditions (11) generate a system of (119873 minus 1) ODEs in time

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

(21)

where

119889119899(119905) = 119860

1198991199001198921(119905) + 119860

1198991198731198922(119905)

119889119899(119905) = 119861

1198991199001198921(119905) + 119861

1198991198731198922(119905)

(22)

Then the problem (9)ndash(12) transforms to the SODEs

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+ ]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

119906119899(0) = 119891 (119909

(120572120573)

119873119899)

(23)

which may be written in the following matrix form

u (119905) = F (119905 119906 (119905))

u (0) = f (24)

where

u (119905) = [1(119905) 2(119905)

119873minus1(119905)]119879

f = [119891 (1199091198731) 119891 (119909

1198732) 119891 (119909

119873119873minus1)]119879

F (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865

2(119905 119906 (119905)) 119865

119873minus1(119905 119906 (119905))]

119879

119865119899(119905 119906 (119905)) = minus]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) + 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

minus ]119889119899(119905) + 120583

119889119899(119905) 119899 = 1 119873 minus 1

(25)

The SODEs (24) in time may be solved using any standardtechnique like the implicit Runge-Kutta method

32 (1 + 1)-Dimensional Burger-Fisher Equation TheBurger-Fisher equation is a combined form of Fisher and Burgersrsquoequations The Fisher equation was firstly introduced byFisher in [39] to describe the propagation of a mutantgene This equation has a wide range of applications in alarge number of the fields of chemical kinetics [40] logisticpopulation growth [41] flame propagation [42] populationin one-dimensional habitual [43] neutron population in anuclear reaction [44] neurophysiology [45] autocatalyticchemical reactions [19] branching the Brownian motionprocesses [40] and nuclear reactor theory [46] Moreoverthe Burger-Fisher equation has awide range of applications invarious fields of financial mathematics applied mathematics

and physics applications gas dynamic and traffic flow TheBurger-Fisher equation can be written in the following form

119906119905= 119906119909119909minus ]119906119906

119909+ 120574119906 (1 minus 119906) (119909 119905) isin 119863 times [0 119879] (26)

where

119863 = 119909 minus1 lt 119909 lt 1 (27)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (28)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (29)

The same procedure of Section 31 can be used to reduce(26)ndash(29) to the system of nonlinear differential equationsin the unknown expansion coefficients of the sought-forsemianalytical solution This system is solved by using theimplicit Runge-Kutta method

33 (1 + 1)-Dimensional Generalized Burgers-Huxley Equa-tion The Huxley equation is a nonlinear partial differentialequation of second order of the form

119906119905minus 119906119909119909minus 119906 (119896 minus 119906) (119906 minus 1) = 0 119896 = 0 (30)

It is an evolution equation that describes the nerve propaga-tion [47] in biology from which molecular CB properties canbe calculated It also gives a phenomenological description ofthe behavior of the myosin heads II In addition to this non-linear evolution equation combined forms of this equationand Burgersrsquo equation will be investigated It is interesting topoint out that this equation includes the convection term 119906

119909

and the dissipation term119906119909119909in addition to other terms In this

subsection we derive J-GL-C method to solve numericallythe (1+1)-dimensional generalizedBurgers-Huxley equation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin 119863 times [0 119879]

(31)

where

119863 = 119909 minus1 lt 119909 lt 1 (32)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (33)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (34)

The same procedure of Sections 31 and 32 is used to solvenumerically (30)ndash(34)

Abstract and Applied Analysis 5

4 Numerical Results

To illustrate the effectiveness of the proposed method in thepresent paper three test examples are carried out in thissection The comparison of the results obtained by variouschoices of the Jacobi parameters 120572 and 120573 reveals that thepresentmethod is very effective and convenient for all choicesof 120572 and 120573 We consider the following three examples

Example 1 Consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120578119906 (1 minus 119906

120575) (119906120575minus 120574)

(119909 119905) isin [119860 119861] times [0 119879]

(35)

subject to the boundary conditions

119906 (119860 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119860 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1 + 120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

119906 (119861 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119861 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1+120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

(36)

and the initial condition

119906 (119909 0)

=[

[

[

120574

2

+

120574

2

tanh[[

[

119909120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

]

]

]

]

]

]

1120575

119909 isin [119860 119861]

(37)

The exact solution of (35) is

119906 (119909 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times(119909minus

]120574 (1+120575minus120574)(radic]2+4120578(1+120575)minus])

2(120575+1)2

119905)]

]

]

]

]

]

1120575

(38)

The difference between themeasured value of the approx-imate solution and its actual value (absolute error) given by

119864 (119909 119905) = |119906 (119909 119905) minus (119909 119905)| (39)

where 119906(119909 119905) and (119909 119905) is the exact solution and theapproximate solution at the point (119909 119905) respectively

In the cases of 120574 = 10minus3 ] = 120578 = 120575 = 1 and 119873 = 4

Table 1 lists the comparison of absolute errors of problem(35) subject to (36) and (37) using the J-GL-C method fordifferent choices of 120572 and 120573 with references [19] in theinterval [0 1] Moreover in Tables 2 and 3 the absolute errorsof this problem with 120572 = 120573 = 12 and various choices of119909 119905 for 120575 = 1 (3) in both intervals [0 1] and [minus1 1] aregiven respectively In Table 4 maximum absolute errors withvarious choices of (120572 120573) for both values of 120575 = 1 3 are givenwhere ] = 120574 = 120578 = 0001 in both intervals [0 1] and [minus1 1]Moreover the absolute errors of problem (35) are shown inFigures 1 2 and 3 for 120575 = 1 2 and 3with values of parameterslisted in their captions respectively while in Figure 4 weplotted the approximate solution of this problemwhere120572 = 0120573 = 1 ] = 120578 = 120574 = 10

minus3 and 119873 = 12 for 120575 = 1 Thesefigures demonstrate the good accuracy of this algorithm forall choices of 120572 120573 and119873 and moreover in any interval

Example 2 Consider the nonlinear time-dependent onedimensional Burgers-type equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (40)

6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

Using Equation (19) and using the two-point boundaryconditions (11) generate a system of (119873 minus 1) ODEs in time

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

(21)

where

119889119899(119905) = 119860

1198991199001198921(119905) + 119860

1198991198731198922(119905)

119889119899(119905) = 119861

1198991199001198921(119905) + 119861

1198991198731198922(119905)

(22)

Then the problem (9)ndash(12) transforms to the SODEs

119899(119905) + ]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) minus 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

+ ]119889119899(119905) minus 120583

119889119899(119905) = 0 119899 = 1 119873 minus 1

119906119899(0) = 119891 (119909

(120572120573)

119873119899)

(23)

which may be written in the following matrix form

u (119905) = F (119905 119906 (119905))

u (0) = f (24)

where

u (119905) = [1(119905) 2(119905)

119873minus1(119905)]119879

f = [119891 (1199091198731) 119891 (119909

1198732) 119891 (119909

119873119873minus1)]119879

F (119905 119906 (119905)) = [1198651(119905 119906 (119905)) 119865

2(119905 119906 (119905)) 119865

119873minus1(119905 119906 (119905))]

119879

119865119899(119905 119906 (119905)) = minus]119906

119899(119905)

119873minus1

sum

119894=1

119860119899119894119906119894(119905) + 120583

119873minus1

sum

119894=1

119861119899119894119906119894(119905)

minus ]119889119899(119905) + 120583

119889119899(119905) 119899 = 1 119873 minus 1

(25)

The SODEs (24) in time may be solved using any standardtechnique like the implicit Runge-Kutta method

32 (1 + 1)-Dimensional Burger-Fisher Equation TheBurger-Fisher equation is a combined form of Fisher and Burgersrsquoequations The Fisher equation was firstly introduced byFisher in [39] to describe the propagation of a mutantgene This equation has a wide range of applications in alarge number of the fields of chemical kinetics [40] logisticpopulation growth [41] flame propagation [42] populationin one-dimensional habitual [43] neutron population in anuclear reaction [44] neurophysiology [45] autocatalyticchemical reactions [19] branching the Brownian motionprocesses [40] and nuclear reactor theory [46] Moreoverthe Burger-Fisher equation has awide range of applications invarious fields of financial mathematics applied mathematics

and physics applications gas dynamic and traffic flow TheBurger-Fisher equation can be written in the following form

119906119905= 119906119909119909minus ]119906119906

119909+ 120574119906 (1 minus 119906) (119909 119905) isin 119863 times [0 119879] (26)

where

119863 = 119909 minus1 lt 119909 lt 1 (27)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (28)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (29)

The same procedure of Section 31 can be used to reduce(26)ndash(29) to the system of nonlinear differential equationsin the unknown expansion coefficients of the sought-forsemianalytical solution This system is solved by using theimplicit Runge-Kutta method

33 (1 + 1)-Dimensional Generalized Burgers-Huxley Equa-tion The Huxley equation is a nonlinear partial differentialequation of second order of the form

119906119905minus 119906119909119909minus 119906 (119896 minus 119906) (119906 minus 1) = 0 119896 = 0 (30)

It is an evolution equation that describes the nerve propaga-tion [47] in biology from which molecular CB properties canbe calculated It also gives a phenomenological description ofthe behavior of the myosin heads II In addition to this non-linear evolution equation combined forms of this equationand Burgersrsquo equation will be investigated It is interesting topoint out that this equation includes the convection term 119906

119909

and the dissipation term119906119909119909in addition to other terms In this

subsection we derive J-GL-C method to solve numericallythe (1+1)-dimensional generalizedBurgers-Huxley equation

119906119905+ ]119906120575119906

119909minus 119906119909119909minus 120578119906 (1 minus 119906

120575) (119906120575minus 120574) = 0

(119909 119905) isin 119863 times [0 119879]

(31)

where

119863 = 119909 minus1 lt 119909 lt 1 (32)

subject to the boundary conditions

119906 (minus1 119905) = 1198921(119905) 119906 (1 119905) = 119892

2(119905) (33)

and the initial condition

119906 (119909 0) = 119891 (119909) 119909 isin 119863 (34)

The same procedure of Sections 31 and 32 is used to solvenumerically (30)ndash(34)

Abstract and Applied Analysis 5

4 Numerical Results

To illustrate the effectiveness of the proposed method in thepresent paper three test examples are carried out in thissection The comparison of the results obtained by variouschoices of the Jacobi parameters 120572 and 120573 reveals that thepresentmethod is very effective and convenient for all choicesof 120572 and 120573 We consider the following three examples

Example 1 Consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120578119906 (1 minus 119906

120575) (119906120575minus 120574)

(119909 119905) isin [119860 119861] times [0 119879]

(35)

subject to the boundary conditions

119906 (119860 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119860 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1 + 120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

119906 (119861 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119861 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1+120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

(36)

and the initial condition

119906 (119909 0)

=[

[

[

120574

2

+

120574

2

tanh[[

[

119909120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

]

]

]

]

]

]

1120575

119909 isin [119860 119861]

(37)

The exact solution of (35) is

119906 (119909 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times(119909minus

]120574 (1+120575minus120574)(radic]2+4120578(1+120575)minus])

2(120575+1)2

119905)]

]

]

]

]

]

1120575

(38)

The difference between themeasured value of the approx-imate solution and its actual value (absolute error) given by

119864 (119909 119905) = |119906 (119909 119905) minus (119909 119905)| (39)

where 119906(119909 119905) and (119909 119905) is the exact solution and theapproximate solution at the point (119909 119905) respectively

In the cases of 120574 = 10minus3 ] = 120578 = 120575 = 1 and 119873 = 4

Table 1 lists the comparison of absolute errors of problem(35) subject to (36) and (37) using the J-GL-C method fordifferent choices of 120572 and 120573 with references [19] in theinterval [0 1] Moreover in Tables 2 and 3 the absolute errorsof this problem with 120572 = 120573 = 12 and various choices of119909 119905 for 120575 = 1 (3) in both intervals [0 1] and [minus1 1] aregiven respectively In Table 4 maximum absolute errors withvarious choices of (120572 120573) for both values of 120575 = 1 3 are givenwhere ] = 120574 = 120578 = 0001 in both intervals [0 1] and [minus1 1]Moreover the absolute errors of problem (35) are shown inFigures 1 2 and 3 for 120575 = 1 2 and 3with values of parameterslisted in their captions respectively while in Figure 4 weplotted the approximate solution of this problemwhere120572 = 0120573 = 1 ] = 120578 = 120574 = 10

minus3 and 119873 = 12 for 120575 = 1 Thesefigures demonstrate the good accuracy of this algorithm forall choices of 120572 120573 and119873 and moreover in any interval

Example 2 Consider the nonlinear time-dependent onedimensional Burgers-type equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (40)

6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 5

4 Numerical Results

To illustrate the effectiveness of the proposed method in thepresent paper three test examples are carried out in thissection The comparison of the results obtained by variouschoices of the Jacobi parameters 120572 and 120573 reveals that thepresentmethod is very effective and convenient for all choicesof 120572 and 120573 We consider the following three examples

Example 1 Consider the nonlinear time-dependent one-dimensional generalized Burgers-Huxley equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120578119906 (1 minus 119906

120575) (119906120575minus 120574)

(119909 119905) isin [119860 119861] times [0 119879]

(35)

subject to the boundary conditions

119906 (119860 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119860 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1 + 120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

119906 (119861 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times (119861 minus (]120574 (1+120575minus120574) (radic]2+4120578 (1+120575)minus]))

times (2(120575 + 1)2)

minus1

119905)]

]

]

]

]

]

1120575

(36)

and the initial condition

119906 (119909 0)

=[

[

[

120574

2

+

120574

2

tanh[[

[

119909120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

]

]

]

]

]

]

1120575

119909 isin [119860 119861]

(37)

The exact solution of (35) is

119906 (119909 119905)

=[

[

[

120574

2

+

120574

2

times tanh[[

[

120574

minus]120575 + 120575radic]2 + 4120578 (1 + 120575)

4 (120575 + 1)

times(119909minus

]120574 (1+120575minus120574)(radic]2+4120578(1+120575)minus])

2(120575+1)2

119905)]

]

]

]

]

]

1120575

(38)

The difference between themeasured value of the approx-imate solution and its actual value (absolute error) given by

119864 (119909 119905) = |119906 (119909 119905) minus (119909 119905)| (39)

where 119906(119909 119905) and (119909 119905) is the exact solution and theapproximate solution at the point (119909 119905) respectively

In the cases of 120574 = 10minus3 ] = 120578 = 120575 = 1 and 119873 = 4

Table 1 lists the comparison of absolute errors of problem(35) subject to (36) and (37) using the J-GL-C method fordifferent choices of 120572 and 120573 with references [19] in theinterval [0 1] Moreover in Tables 2 and 3 the absolute errorsof this problem with 120572 = 120573 = 12 and various choices of119909 119905 for 120575 = 1 (3) in both intervals [0 1] and [minus1 1] aregiven respectively In Table 4 maximum absolute errors withvarious choices of (120572 120573) for both values of 120575 = 1 3 are givenwhere ] = 120574 = 120578 = 0001 in both intervals [0 1] and [minus1 1]Moreover the absolute errors of problem (35) are shown inFigures 1 2 and 3 for 120575 = 1 2 and 3with values of parameterslisted in their captions respectively while in Figure 4 weplotted the approximate solution of this problemwhere120572 = 0120573 = 1 ] = 120578 = 120574 = 10

minus3 and 119873 = 12 for 120575 = 1 Thesefigures demonstrate the good accuracy of this algorithm forall choices of 120572 120573 and119873 and moreover in any interval

Example 2 Consider the nonlinear time-dependent onedimensional Burgers-type equation

119906119905+ ]119906119906

119909minus 120583119906119909119909= 0 (119909 119905) isin [119860 119861] times [0 119879] (40)

6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

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6 Abstract and Applied Analysis

Table 1 Comparison of absolute errors of Example 1 with results from different articles where119873 = 4 120574 = 10minus3 and ] = 120578 = 120575 = 1

(119909 119905) ADM [19]Our method for various values of (120572 120573) with119873 = 4

(0 0) (minus12 minus12) (12 12) (minus12 12) (0 1)

(01 005) 194 times 10minus7

299 times 10minus8

128 times 10minus9

232 times 10minus8

128 times 10minus9

134 times 10minus8

(01 01) 387 times 10minus7

598 times 10minus8

256 times 10minus9

463 times 10minus8

256 times 10minus9

268 times 10minus8

(01 1) 3875 times 10minus7

597 times 10minus7

246 times 10minus8

463 times 10minus7

246 times 10minus8

268 times 10minus7

(05 005) 194 times 10minus7

102 times 10minus8

526 times 10minus8

106 times 10minus8

526 times 10minus8

364 times 10minus8

(05 01) 387 times 10minus7

203 times 10minus8

105 times 10minus7

213 times 10minus8

105 times 10minus7

728 times 10minus8

(05 1) 3857 times 10minus7

204 times 10minus7

1054 times 10minus7

213 times 10minus7

1054 times 10minus7

728 times 10minus7

(09 005) 194 times 10minus7

793 times 10minus9

522 times 10minus8

449 times 10minus9

522 times 10minus8

307 times 10minus8

(09 01) 387 times 10minus7

159 times 10minus8

104 times 10minus7

899 times 10minus9

104 times 10minus7

615 times 10minus8

(09 1) 3876 times 10minus7

159 times 10minus7

1046 times 10minus7

901 times 10minus8

1046 times 10minus7

614 times 10minus7

Table 2 Absolute errors with 120572 = 120573 = 12 120575 = 1 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 704 times 10minus12

minus1 1 247 times 10minus11

01 561 times 10minus11

419 times 10minus12

02 729 times 10minus11

701 times 10minus12

03 826 times 10minus12

459 times 10minus11

04 437 times 10minus11

574 times 10minus11

05 247 times 10minus11

144 times 10minus11

06 701 times 10minus12

216 times 10minus11

07 574 times 10minus11

374 times 10minus11

08 215 times 10minus11

103 times 10minus10

09 103 times 10minus10

125 times 10minus10

1 464 times 10minus12

464 times 10minus12

00 02 0 1 0001 0001 0001 12 133 times 10minus11

minus1 1 492 times 10minus11

01 111 times 10minus10

833 times 10minus12

02 146 times 10minus10

142 times 10minus11

03 165 times 10minus11

920 times 10minus11

04 874 times 10minus11

115 times 10minus10

05 492 times 10minus11

287 times 10minus11

06 142 times 10minus11

430 times 10minus11

07 114 times 10minus10

749 times 10minus11

08 430 times 10minus11

205 times 10minus10

09 205 times 10minus10

251 times 10minus10

1 109 times 10minus11

109 times 10minus11

subject to the boundary conditions

119906 (119860 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119860 minus 119888119905)]

119906 (119861 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119861 minus 119888119905)]

(41)

and the initial condition

119906 (119909 0) =

119888

]minus

119888

]tanh [ 119888

2120583

119909] 119909 isin [119860 119861] (42)

If we apply the generalized tanh method [48] then we findthat the analytical solution of (40) is

119906 (119909 119905) =

119888

]minus

119888

]tanh [ 119888

2120583

(119909 minus 119888119905)] (43)

InTable 5 themaximumabsolute errors of (40) subject to(41) and (42) are introduced using the J-GL-C method withvarious choices of (120572 120573) in both intervals [0 1] and [minus1 1]Absolute errors between exact and numerical solutions of thisproblem are introduced in Table 6 using the J-GL-C methodfor 120572 = 120573 = 12 with 119873 = 20 and ] = 10 120583 = 01 and119888 = 01 in both intervals [0 1] and [minus1 1] In Figures 5 6and 7 we displayed the absolute errors of problem (40) for

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 7

Table 3 Absolute errors with 120572 = 120573 = 12 120575 = 3 and various choices of 119909 119905 for Example 1

119909 119905 119860 119861 ] 120574 120578 119873 119864 119860 119861 119864

00 01 0 1 0001 0001 0001 12 562 times 10minus8

minus1 1 832 times 10minus9

01 650 times 10minus6

299 times 10minus6

02 467 times 10minus6

165 times 10minus6

03 308 times 10minus6

244 times 10minus6

04 165 times 10minus6

309 times 10minus6

05 833 times 10minus9

197 times 10minus6

06 165 times 10minus6

467 times 10minus6

07 309 times 10minus6

379 times 10minus6

08 467 times 10minus6

653 times 10minus6

09 653 times 10minus6

3532 times 10minus6

1 566 times 10minus8

566 times 10minus8

00 02 0 1 0001 0001 0001 12 227 times 10minus7

minus1 1 254 times 10minus8

01 1298 times 10minus6

597 times 10minus6

02 932 times 10minus6

332 times 10minus6

03 616 times 10minus6

489 times 10minus6

04 330 times 10minus6

621 times 10minus6

05 254 times 10minus8

393 times 10minus6

06 332 times 10minus6

936 times 10minus6

07 621 times 10minus6

758 times 10minus6

08 936 times 10minus6

1310 times 10minus6

09 1310 times 10minus6

7074 times 10minus6

1 227 times 10minus7

227 times 10minus7

Table 4 Maximum absolute errors with various choices of (120572 120573) for both values of 120575 = 1 3 Example 3

120572 120573 119860 119861 ] 120574 120578 120575 119873 119872119864

119860 119861 119872119864

0 0 0 1 0001 0001 0001 1 12 139 times 10minus9

minus1 1 255 times 10minus9

12 12 638 times 10minus10

172 times 10minus9

minus12 minus12 383 times 10minus9

465 times 10minus9

minus12 12 504 times 10minus9

458 times 10minus9

0 1 216 times 10minus9

242 times 10minus9

0 0 0 1 0001 0001 0001 3 12 184 times 10minus4

minus1 1 1475 times 10minus4

12 12 223 times 10minus5

413 times 10minus4

minus12 minus12 3344 times 10minus4

8117 times 10minus4

minus12 12 5696 times 10minus4

9097 times 10minus4

0 1 524 times 10minus4

1646 times 10minus4

different numbers of collation points and different choicesof 120572 and 120573 in interval [0 1] with values of parameters beinglisted in their captions Moreover in Figure 8 we see thatthe approximate solution and the exact solution are almostcoincided for different values of 119905 (0 05 and 09) of problem(40) where ] = 10 120583 = 01 119888 = 01 120572 = 120573 = minus05 and119873 = 20

in interval [minus1 1]

Example 3 Consider the nonlinear time-dependent one-dimensional generalized Burger-Fisher-type equation

119906119905= 119906119909119909minus ]119906120575119906

119909+ 120574119906 (1 minus 119906

120575) (119909 119905) isin [119860 119861] times [0 119879]

(44)

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Abstract and Applied Analysis

00

05

10

x

00

05

10

t

02468

E

times10minus7

Figure 1 The absolute error of problem (35) where 120572 = 120573 = 0] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 1

0204

0608

x

00

05

10

t

01234

E

times10minus7

Figure 2 The absolute error of problem (35) where 120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 2

0204

0608

10

x

00

05

10

t

0

1

2

E

times10minus6

Figure 3 The absolute error of problem (35) where minus120572 = 120573 = 12] = 120578 = 120574 = 10minus3 and119873 = 4 for 120575 = 3

05

10

x

0

5

10

t

000049995

000050000

000050005

minus5

minus10

u(xt)

Figure 4 The approximate solution of problem (35) where 120572 = 0120573 = 1 ] = 120578 = 120574 = 10minus3 and119873 = 12 for 120575 = 1

00

05

10

x

00

05

10

t

05

1

15

E

times10minus9

Figure 5The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 minus120572 = 120573 = 12 and119873 = 12

00

05

10

x

00

05

10

t

01234

E

times10minus10

Figure 6The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = 12 and119873 = 20

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 9

Table 5Maximum absolute errors with various choices of (120572 120573) forExample 2

120572 120573 119860 119861 120583 ] 119873 119872119864

119860 119861 119872119864

0 0 0 1 01 10 4 149 times 10minus6

minus1 1 562 times 10minus7

12 12 245 times 10minus6

825 times 10minus7

minus12 minus12 651 times 10minus7

651 times 10minus7

minus12 12 384 times 10minus6

467 times 10minus7

12 minus12 120 times 10minus6

120 times 10minus6

0 0 0 1 01 10 16 143 times 10minus9

minus1 1 106 times 10minus9

12 12 162 times 10minus9

118 times 10minus9

minus12 minus12 670 times 10minus10

445 times 10minus10

minus12 12 668 times 10minus10

449 times 10minus10

12 minus12 667 times 10minus10

423 times 10minus10

00

05

10

x

00

05

10

t

0051

15

E

times10minus9

Figure 7The absolute error of problem (40) where ] = 10 120583 = 01119888 = 01 120572 = 120573 = minus12 and119873 = 16

subject to the boundary conditions

119906 (119860 119905)

= [

1

2

minus

1

2

timestanh [ ]120575

2 (120575 + 1)

(119860minus(

]

120575+1

+

120574 (120575+1)

]) 119905)]]

1120575

119906 (119861 119905)

= [

1

2

minus

1

2

times tanh [ ]120575

2 (120575 + 1)

(119861 minus (

]

120575 + 1

+

120574 (120575 + 1)

]) 119905)]]

1120575

(45)

Table 6 Absolute errors with 120572 = 120573 = 12 and various choices of119909 119905 for Example 2

119909 119905 119860 119861 120583 ] 119873 119864 119860 119861 119864

00 01 0 1 01 10 20 134 times 10minus12

minus1 1 921 times 10minus11

01 871 times 10minus11

870 times 10minus11

02 107 times 10minus10

816 times 10minus11

03 107 times 10minus10

761 times 10minus11

04 101 times 10minus10

701 times 10minus11

05 921 times 10minus11

638 times 10minus11

06 816 times 10minus11

570 times 10minus11

07 701 times 10minus11

488 times 10minus11

08 570 times 10minus11

385 times 10minus11

09 385 times 10minus11

230 times 10minus11

1 134 times 10minus12

134 times 10minus12

00 02 0 1 01 10 20 177 times 10minus12

minus1 1 335 times 10minus10

01 271 times 10minus10

317 times 10minus10

02 361 times 10minus10

298 times 10minus10

03 378 times 10minus10

277 times 10minus10

04 364 times 10minus10

255 times 10minus10

05 334 times 10minus10

231 times 10minus10

06 298 times 10minus10

203 times 10minus10

07 255 times 10minus10

171 times 10minus10

08 202 times 10minus10

130 times 10minus10

09 130 times 10minus10

770 times 10minus11

1 177 times 10minus12

177 times 10minus12

and the initial condition

119906 (119909 0) = [

1

2

minus

1

2

tanh [ ]120575

2 (120575 + 1)

(119909)]]

1120575

119909 isin [119860 119861]

(46)

The exact solution of (44) is

119906 (119909 119905)

= [

1

2

minus

1

2

tanh [ ]120575

2 (120575+1)

(119909minus(

]

120575 + 1

+

120574 (120575+1)

]) 119905)]]

1120575

(47)

In Table 7 we listed a comparison of absolute errorsof problem (44) subject to (45) and (46) using the J-GL-C method with [19] Absolute errors between exact andnumerical solutions of (44) subject to (45) and (46) areintroduced in Table 8 using the J-GL-Cmethod for 120572 = 120573 = 0with119873 = 16 respectively and ] = 120574 = 10minus2 In Figures 9 and10 we displayed the absolute errors of problem (44) where] = 120574 = 10minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) ininterval [minus1 1] respectively Moreover in Figures 11 and 12we see that in interval [minus1 1] the approximate solution andthe exact solution are almost coincided for different valuesof 119905 (0 05 and 09) of problem (44) where ] = 120574 = 10

minus2 at119873 = 20 and (120572 = 120573 = 0 and 120572 = 120573 = minus12) respectivelyThis asserts that the obtained numerical results are accurateand can be compared favorably with the analytical solution

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Abstract and Applied Analysis

Table 7 Comparison of absolute errors of Example 3 with results from [19] where119873 = 4 120572 = 120573 = 0 and various choices of 119909 119905

119909 119905 120574 ] 120575 [19] 119864 119909 119905 120574 ] 120575 [19] 119864

01 0005 0001 0001 1 969 times 10minus6

185 times 10minus6 01 00005 1 1 2 140 times 10

minus3383 times 10

minus5

0001 194 times 10minus6

372 times 10minus7 00001 280 times 10

minus4388 times 10

minus5

001 194 times 10minus5

372 times 10minus6 0001 280 times 10

minus3376 times 10

minus5

05 0005 969 times 10minus6

704 times 10minus6 05 00005 135 times 10

minus3232 times 10

minus5

0001 194 times 10minus6

141 times 10minus6 00001 269 times 10

minus4238 times 10

minus5

001 194 times 10minus5

141 times 10minus5 0001 269 times 10

minus3225 times 10

minus5

09 0005 969 times 10minus6

321 times 10minus6 09 00005 128 times 10

minus3158 times 10

minus5

0001 194 times 10minus6

642 times 10minus7 00001 255 times 10

minus4155 times 10

minus5

001 194 times 10minus5

642 times 10minus6 0001 255 times 10

minus3161 times 10

minus5

minus04 minus02 00 02 040008

0009

00010

00011

00012

00013

x

uandu

u(x 00)

u(x 05)

u(x 09)u(x 00)

u(x 05)

u(x 09)

Figure 8 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (40) where ] = 10 120583 = 01 119888 = 01120572 = 120573 = minus05 and119873 = 20

minus10minus05

00

05

10

x

00

05

10

t

0

2

4

6

E

times10minus10

Figure 9 The absolute error of problem (44) where 120572 = 120573 = 0 and] = 120574 = 10minus2 at119873 = 20

Table 8 Absolute errors with 120572 = 120573 = 0 120575 = 1 and various choicesof 119909 119905 for Example 3

119909 119905 120574 ] 120575 119873 119864 119909 119905 120574 ] 120575 119873 119864

00 01 1 1 1 16 326 times 10minus9 00 02 1 1 1 16 357 times 10

minus11

01 382 times 10minus9 01 775 times 10

minus11

02 428 times 10minus9 02 196 times 10

minus10

03 466 times 10minus9 03 352 times 10

minus10

04 493 times 10minus9 04 562 times 10

minus10

05 505 times 10minus9 05 731 times 10

minus10

06 493 times 10minus9 06 787 times 10

minus10

07 449 times 10minus9 07 823 times 10

minus10

08 361 times 10minus9 08 817 times 10

minus10

09 226 times 10minus9 09 805 times 10

minus10

10 113 times 10minus11 10 109 times 10

minus11

minus10minus05

0005

10

x

00

05

10

t

02468

E

times10minus10

Figure 10 The absolute error of problem (44) where 120572 = 120573 = 12and ] = 120574 = 10minus2 at119873 = 20

5 Conclusion

An efficient and accurate numerical scheme based on the J-GL-C spectral method is proposed to solve nonlinear time-dependent Burgers-type equations The problem is reducedto the solution of a SODEs in the expansion coefficient ofthe solution Numerical examples were given to demonstrate

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Abstract and Applied Analysis 11

minus10 minus05 00 05 10

05005

05010

05015

05020

05025

05030

x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 11 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 0 and ] = 120574 =10minus2 at119873 = 20

00 05 10

05005

05010

05015

05020

05025

05030

minus10 minus05x

u(x 06)

u(x 09)

u(x 13)u(x 06)

u(x 09)

u(x 13)

uandu

Figure 12 The approximate and exact solutions for different valuesof 119905 (0 05 and 09) of problem (44) where 120572 = 120573 = 12 and ] = 120574 =10minus2 at119873 = 20

the validity and applicability of the methodThe results showthat the J-GL-C method is simple and accurate In fact byselecting few collocation points excellent numerical resultsare obtained

References

[1] C Canuto M Y Hussaini A Quarteroni and T A ZangSpectral Methods Fundamentals in Single Domains SpringerNew York NY USA 2006

[2] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers ampMathematics with Applications vol 64 no 4 pp 558ndash571 2012

[3] S Nguyen and C Delcarte ldquoA spectral collocation method tosolve Helmholtz problems with boundary conditions involvingmixed tangential and normal derivativesrdquo Journal of Computa-tional Physics vol 200 no 1 pp 34ndash49 2004

[4] A H Bhrawy and A S Alofi ldquoA Jacobi-Gauss collocationmethod for solving nonlinear Lane-Emden type equationsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 1 pp 62ndash70 2012

[5] A H Bhrawy E Tohidi and F Soleymani ldquoA new Bernoullimatrix method for solving high-order linear and nonlinearFredholm integro-differential equations with piecewise inter-valsrdquo Applied Mathematics and Computation vol 219 no 2 pp482ndash497 2012

[6] A H Bhrawy and M Alshomrani ldquoA shifted legendre spectralmethod for fractional-ordermulti-point boundary value prob-lemsrdquoAdvances inDifference Equations vol 2012 article 8 2012

[7] E H Doha A H Bhrawy D Baleanu and S S Ezz-Eldien ldquoOnshifted Jacobi spectral approximations for solving fractionaldifferential equationsrdquo Applied Mathematics and Computationvol 219 no 15 pp 8042ndash8056 2013

[8] L Zhu and Q Fan ldquoSolving fractional nonlinear Fredholmintegro-differential equations by the second kind Chebyshevwaveletrdquo Communications in Nonlinear Science and NumericalSimulation vol 17 no 6 pp 2333ndash2341 2012

[9] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoA newJacobi operational matrix an application for solving fractionaldifferential equationsrdquoAppliedMathematical Modelling vol 36no 10 pp 4931ndash4943 2012

[10] A Saadatmandi and M Dehghan ldquoThe use of sinc-collocationmethod for solving multi-point boundary value problemsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 2 pp 593ndash601 2012

[11] E H Doha A H Bhrawy and R M Hafez ldquoOn shifted Jacobispectral method for high-order multi-point boundary valueproblemsrdquoCommunications inNonlinear Science andNumericalSimulation vol 17 no 10 pp 3802ndash3810 2012

[12] G Ierley B Spencer and R Worthing ldquoSpectral methods intime for a class of parabolic partial differential equationsrdquoJournal of Computational Physics vol 102 no 1 pp 88ndash97 1992

[13] H Tal-Ezer ldquoSpectral methods in time for hyperbolic equa-tionsrdquo SIAM Journal on Numerical Analysis vol 23 no 1 pp11ndash26 1986

[14] H Tal-Ezer ldquoSpectral methods in time for parabolic problemsrdquoSIAM Journal onNumerical Analysis vol 26 no 1 pp 1ndash11 1989

[15] E A Coutsias T Hagstrom J S Hesthaven and D Tor-res ldquoIntegration preconditioners for differential operators inspectral tau-methodsrdquo in Proceedings of the 3rd InternationalConference on Spectral andHighOrderMethods A V Ilin and LR Scott Eds pp 21ndash38 University of Houston Houston TexUSA 1996 Houston Journal of Mathematics

[16] F-y Zhang ldquoSpectral and pseudospectral approximations intime for parabolic equationsrdquo Journal of Computational Mathe-matics vol 16 no 2 pp 107ndash120 1998

[17] J-G Tang and H-P Ma ldquoA Legendre spectral method intime for first-order hyperbolic equationsrdquo Applied NumericalMathematics vol 57 no 1 pp 1ndash11 2007

[18] A H Bhrawy ldquoA Jacobi-Gauss-Lobatto collocation methodfor solving generalized Fitzhugh-Nagumo equation with time-dependent coefficientsrdquoAppliedMathematics andComputationvol 222 pp 255ndash264 2013

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Abstract and Applied Analysis

[19] H N A Ismail K Raslan and A A A Rabboh ldquoAdo-mian decomposition method for Burgerrsquos-Huxley and Burgerrsquos-Fisher equationsrdquo Applied Mathematics and Computation vol159 no 1 pp 291ndash301 2004

[20] H Bateman ldquoSome recent researchers on the motion of fluidsrdquoMonthly Weather Review vol 43 pp 163ndash170 1915

[21] J M Burgers ldquoA mathematical model illustrating the theoryof turbulencerdquo in Advances in Applied Mechanics pp 171ndash199Academic Press New York NY USA 1948

[22] M K Kadalbajoo and A Awasthi ldquoA numerical method basedon Crank-Nicolson scheme for Burgersrsquo equationrdquo AppliedMathematics and Computation vol 182 no 2 pp 1430ndash14422006

[23] M Gulsu ldquoA finite difference approach for solution of Burgersrsquoequationrdquo Applied Mathematics and Computation vol 175 no2 pp 1245ndash1255 2006

[24] P Kim ldquoInvariantization of the Crank-Nicolson method forBurgersrsquo equationrdquo Physica D vol 237 no 2 pp 243ndash254 2008

[25] H Nguyen and J Reynen ldquoA space-time least-square finiteelement scheme for advection-diffusion equationsrdquo ComputerMethods in Applied Mechanics and Engineering vol 42 no 3pp 331ndash342 1984

[26] HNguyen and J Reynen ldquoA space-timefinite element approachto Burgersrsquo equationrdquo in Numerical Methods for Non-LinearProblems C Taylor EHintonD R JOwen andEOnate Edsvol 2 pp 718ndash728 1984

[27] L R T Gardner G A Gardner and A Dogan ldquoA Petrov-Galerkin finite element scheme for Burgersrsquo equationrdquo TheArabian Journal for Science and Engineering C vol 22 no 2 pp99ndash109 1997

[28] L R T Gardner G A Gardner and A Dogan in A Least-Squares Finite Element Scheme For Burgers Equation Mathe-matics University of Wales Bangor UK 1996

[29] S Kutluay A Esen and I Dag ldquoNumerical solutions ofthe Burgersrsquo equation by the least-squares quadratic B-splinefinite element methodrdquo Journal of Computational and AppliedMathematics vol 167 no 1 pp 21ndash33 2004

[30] 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

[31] E H Doha andA H Bhrawy ldquoEfficient spectral-Galerkin algo-rithms for direct solution of fourth-order differential equationsusing Jacobi polynomialsrdquoApplied Numerical Mathematics vol58 no 8 pp 1224ndash1244 2008

[32] E H Doha and A H Bhrawy ldquoA Jacobi spectral Galerkinmethod for the integrated forms of fourth-order elliptic dif-ferential equationsrdquo Numerical Methods for Partial DifferentialEquations vol 25 no 3 pp 712ndash739 2009

[33] S Kazem ldquoAn integral operational matrix based on Jacobipolynomials for solving fractional-order differential equationsrdquoApplied Mathematical Modelling vol 37 no 3 pp 1126ndash11362013

[34] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 p 104 2013

[35] E H Doha A H Bhrawy and R M Hafez ldquoA Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-orderdifferential equationsrdquo Mathematical and Computer Modellingvol 53 no 9-10 pp 1820ndash1832 2011

[36] A Veksler and Y Zarmi ldquoWave interactions and the analysis ofthe perturbed Burgers equationrdquo Physica D vol 211 no 1-2 pp57ndash73 2005

[37] A Veksler and Y Zarmi ldquoFreedom in the expansion andobstacles to integrability in multiple-soliton solutions of theperturbed KdV equationrdquo Physica D vol 217 no 1 pp 77ndash872006

[38] Z-Y Ma X-F Wu and J-M Zhu ldquoMultisoliton excitations forthe Kadomtsev-Petviashvili equation and the coupled Burgersequationrdquo Chaos Solitons and Fractals vol 31 no 3 pp 648ndash657 2007

[39] R A Fisher ldquoThe wave of advance of advantageous genesrdquoAnnals of Eugenics vol 7 pp 335ndash369 1937

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

[41] N F Britton Reaction-Diffusion Equations and their Applica-tions to Biology Academic Press London UK 1986

[42] D A Frank Diffusion and Heat Exchange in Chemical KineticsPrinceton University Press Princeton NJ USA 1955

[43] A-M Wazwaz ldquoThe extended tanh method for abundantsolitary wave solutions of nonlinear wave equationsrdquo AppliedMathematics and Computation vol 187 no 2 pp 1131ndash11422007

[44] W Malfliet ldquoSolitary wave solutions of nonlinear wave equa-tionsrdquo American Journal of Physics vol 60 no 7 pp 650ndash6541992

[45] Y Tan H Xu and S-J Liao ldquoExplicit series solution oftravelling waves with a front of Fisher equationrdquoChaos Solitonsand Fractals vol 31 no 2 pp 462ndash472 2007

[46] J Canosa ldquoDiffusion in nonlinear multiplicate mediardquo Journalof Mathematical Physics vol 31 pp 1862ndash1869 1969

[47] A-M Wazwaz ldquoTravelling wave solutions of generalizedforms of Burgers Burgers-KdV and Burgers-Huxley equationsrdquoApplied Mathematics and Computation vol 169 no 1 pp 639ndash656 2005

[48] E Fan and Y C Hon ldquoGeneralized tanh method extendedto special types of nonlinear equationsrdquo Zeitschrift fur Natur-forschung A vol 57 no 8 pp 692ndash700 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of