Research Article Numerical Method Using Cubic...
12
Research Article Numerical Method Using Cubic Trigonometric B-Spline Technique for Nonclassical Diffusion Problems Muhammad Abbas, 1,2 Ahmad Abd. Majid, 2 Ahmad Izani Md. Ismail, 2 and Abdur Rashid 3 1 Department of Mathematics, University of Sargodha, Sargodha 40100, Pakistan 2 School of Mathematical Sciences, Universiti Sains Malaysia (USM), 11800 Penang, Malaysia 3 Department of Mathematics, Gomal University, Dera Ismail Khan 29050, Pakistan Correspondence should be addressed to Muhammad Abbas; [email protected] Received 17 October 2013; Accepted 13 April 2014; Published 20 May 2014 Academic Editor: Sining Zheng Copyright © 2014 Muhammad Abbas et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of a nonclassical diffusion problem with nonlocal boundary constraints. e standard finite difference approach is applied to discretize the time derivative while cubic trigonometric B-spline is utilized as an interpolating function in the space dimension. e technique is shown to be unconditionally stable using the von Neumann method. Several numerical examples are discussed to exhibit the feasibility and capability of the technique. e 2 and ∞ error norms are also computed at different times for different space size steps to assess the performance of the proposed technique. e technique requires smaller computational time than several other methods and the numerical results are found to be in good agreement with known solutions and with existing schemes in the literature. 1. Introduction is study deals with the numerical solution of a nonclassical diffusion problem with two nonlocal boundary constraints using cubic trigonometric B-splines. is problem arises in several branches of science. In particular, electrochemistry [1], heat conduction process [2], thermoelasticity [3], plasma physics [4], semiconductor modeling [5], biotechnology [6], control theory, and inverse problems [7]. e analysis, development, and implementation of numerical methods for the solution of such diffusion problems have received wide attention in the literature. Consider an insulated rod of length located on the - axis of the interval [0, ]. Let the rod have a source of heat. Let (, ) denote the temperature in the insulated rod with ends held at constant temperatures 1 and 2 , and the initial temperature distribution along the rod is 1 (). e problem is to study the flow of heat in the rod and in this paper the partial differential equation governing the flow of heat in the rod is given by the diffusion equation with specification of energy: (, ) = 2 2 2 (, ) + (, ) 0 ≤ ≤ , 0 ≤ ≤ (1) with the initial constraint (, = 0) = 1 () 0≤≤ (2) and the nonlocal boundary constraints 1 ( = 0, ) + 2 ( = 0, ) =∫ 0 2 () (, ) + ℎ 1 () = 1 , 3 ( = , ) + 4 ( = , ) Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 849682, 11 pages http://dx.doi.org/10.1155/2014/849682
Transcript of Research Article Numerical Method Using Cubic...
Research ArticleNumerical Method Using Cubic Trigonometric B-SplineTechnique for Nonclassical Diffusion Problems
Muhammad Abbas12 Ahmad Abd Majid2
Ahmad Izani Md Ismail2 and Abdur Rashid3
1 Department of Mathematics University of Sargodha Sargodha 40100 Pakistan2 School of Mathematical Sciences Universiti Sains Malaysia (USM) 11800 Penang Malaysia3 Department of Mathematics Gomal University Dera Ismail Khan 29050 Pakistan
Correspondence should be addressed to Muhammad Abbas mabbasuosedupk
Received 17 October 2013 Accepted 13 April 2014 Published 20 May 2014
Academic Editor Sining Zheng
Copyright copy 2014 Muhammad Abbas et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
A new two-time level implicit technique based on cubic trigonometric B-spline is proposed for the approximate solution of anonclassical diffusion problem with nonlocal boundary constraints The standard finite difference approach is applied to discretizethe time derivative while cubic trigonometric B-spline is utilized as an interpolating function in the space dimensionThe techniqueis shown to be unconditionally stable using the von Neumann method Several numerical examples are discussed to exhibit thefeasibility and capability of the technique The 119871
2and 119871
infinerror norms are also computed at different times for different space size
steps to assess the performance of the proposed technique The technique requires smaller computational time than several othermethods and the numerical results are found to be in good agreement with known solutions and with existing schemes in theliterature
1 Introduction
This study deals with the numerical solution of a nonclassicaldiffusion problem with two nonlocal boundary constraintsusing cubic trigonometric B-splines This problem arises inseveral branches of science In particular electrochemistry[1] heat conduction process [2] thermoelasticity [3] plasmaphysics [4] semiconductor modeling [5] biotechnology[6] control theory and inverse problems [7] The analysisdevelopment and implementation of numerical methods forthe solution of such diffusion problems have received wideattention in the literature
Consider an insulated rod of length 119871 located on the 119909-axis of the interval [0 119871] Let the rod have a source of heatLet 119906(119909 119905) denote the temperature in the insulated rod withends held at constant temperatures 119879
1and 119879
2 and the initial
temperature distribution along the rod is 1198921(119909) The problem
is to study the flow of heat in the rod and in this paper thepartial differential equation governing the flow of heat in the
rod is given by the diffusion equation with specification ofenergy
120597119906
120597119905(119909 119905) = 120572
2 1205972119906
1205971199092(119909 119905) + 119902 (119909 119905) 0 le 119909 le 119871 0 le 119905 le 119879
(1)
with the initial constraint
119906 (119909 119905 = 0) = 1198921(119909) 0 le 119909 le 119871 (2)
and the nonlocal boundary constraints
1205851119906 (119909 = 0 119905) + 120585
2119906119909(119909 = 0 119905)
= int
119871
0
1198922(119909) 119906 (119909 119905) 119889119909 + ℎ
1(119905) = 119879
1
1205853119906 (119909 = 119871 119905) + 120585
4119906119909(119909 = 119871 119905)
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 849682 11 pageshttpdxdoiorg1011552014849682
2 Abstract and Applied Analysis
= int
119871
0
1198923(119909) 119906 (119909 119905) 119889119909 + ℎ
2(119905) = 119879
2
0 lt 119905 le 119879
(3)
where 120585119894 119894 = 1 2 3 4 are known constants 119902 119892
119894(119894 = 1 2 3)
are known continuous functions 120572 is the thermal diffusivityof the rod and ℎ
119894(119894 = 1 2) are prescribed functions This
problem has been studied by Dehghan [8] Martın-Vaqueroand Vigo-Aguiar [9] Li and Wu [10] and Golbabai andJavidi [11] Several physical circumstances might be modeledby equation and constraints (1)ndash(3) and several examples ofapplication in physics with comprehensive derivations of theabove mentioned problem can be found in [3 12ndash14]
There are several numerical methods in the literaturethat have been developed for solving the proposed problem(1) subject to initial and nonlocal boundary constraints (2)-(3) The methods were based for instance on the for-ward Euler method the backward Euler approach or theCrank-Nicolson scheme [15 16] Laplace transformation [17]and so forth Dehghan [8] presented four finite differenceapproaches namely the BTCS (backward time centred space)scheme the implicit (3 3) Crandallrsquos formula the 3-pointFTCS (forward time centred space) two-level scheme andthe Dufort-Frankel three-level approach for the numericalsolution of parabolic equation with nonlocal specificationMartın-Vaquero and Vigo-Aguiar [9] improved the order ofconvergence of the implicit (3 3) Crandallrsquos formula pro-posed by Dehghan [8] and also improved the accuracy of themethod Martın-Vaquero and Vigo-Aguiar [18] developed analgorithm for the solution of the heat conduction equationswith nonlocal constraints which reduced the CPU time andenhanced the accuracy of (3 3) Crandallrsquos formula proposedin [8] Li and Wu [10] proposed an algorithm which wasbased on the transverse method of lines (TMOL) whichcan reduce a nonclassical diffusion equation to a series ofordinary differential equations (ODEs) Subsequently theauthors in [10] used an analytic reproducing kernel techniqueto solve ODEs with integral boundary constraints Dehghan[19 20] jointly with Tatari and Dehghan [21] has proposedseveral efficient techniques for the numerical solutions ofpartial differential equations subject to nonlocal boundaryconstraints Golbabai and Javidi [11] introduced a Chebyshevspectral collocation method (CSCM) based on Chebyshevpolynomials for solving a parabolic problem subject tononlocal boundary constraints For more details on othernumerical methods for the solution of a one-dimensionalheat equation subject to nonlocal boundary constraints in theliterature see [22ndash31]
The study of B-spline functions is a key elementin computer-aided geometric design [32ndash35] It has alsoattracted attention in the literature [36ndash51] to the numeri-cal solution of various differential equations [38ndash40] Thisis because they have important geometric properties andfeatures that make them amenable to more detailed analysisNumerical methods based on B-spline functions of variousdegrees have been utilized for solving initial and boundaryvalue problems As examples a cubic B-spline collocation
method was used to solve a nonlinear diffusion equationsubject to certain initial and Dirichlet boundary constraints[41] a finite element method based on bivariate splines hasbeen used for solving parabolic partial differential equation[42] and the combination of finite difference approach andcubic B-spline method was applied for the solution of aone-dimensional heat equation subject to local boundaryconstraints [43 44] Goh et al [45] presented a comparisonof cubic B-spline and extended cubic uniform B-spline basedcollocation methods for solving a one-dimensional heatequationwith a nonlocal initial constraint and concluded thatextended cubic uniform B-spline with an appropriate valueof parameters gives better results than the cubic B-splineA finite difference scheme based on cubic B-spline was alsoused for solving the one-dimensional wave equation [43]advection-diffusion equation [44] one-dimensional coupledviscous Burgersrsquo equation [47] system of strongly cou-pled reaction-diffusion equations [48] and one-dimensionalhyperbolic problems [49]
In our present paper a new two-time level implicittechnique is developed to approximate the solution of thenonclassical diffusion problem (1) subject to initial con-straints in (2) and nonlocal boundary constraints in (1)ndash(3) The technique is based on the cubic trigonometric B-spline functions A finite difference approach and 120579-weightedscheme are applied for the time and space discretizationrespectively Some researchers have considered the ordinaryB-spline collocation method for solving the heat equationsubject to local and nonlocal boundary constraints but sofar as we are aware not with the cubic trigonometric B-spline collocation method Cubic trigonometric B-spline isused as an interpolating function in the space dimensionThe unconditional stability property of the method is provedby von Neumann method The feasibility of the method isshown by test problems with 119896 = 119904ℎ2 119904 = 1 2 4 5 insteadof smaller time step size 119896 = 04ℎ
2 and the approximatedsolutions are found to be in good agreement with the knownexact solutions
The outline of this study is as follows A numericalsolution of nonclassical diffusion problem is presented inSection 2 In Section 3 the cubic trigonometric B-spline isutilized as an interpolating function in the space dimensionThe von Neumann approach is used to prove the stabilityof the method in Section 4 Numerical examples are consid-ered in Section 5 to show the achievability of the proposedmethod Finally the concluding remarks of this study aregiven in Section 6
2 Solution of Nonclassical Diffusion Problem
Consider a uniform mesh Ω with grid points (119909119894 119905119899) to
discretize the grid region Δ = [119886 119887] times [0 119879] with 119909119894= 119886 + 119894ℎ
119894 = 0 1 2 119873 and 119905119899= 119899119896 119899 = 0 1 2 3 119872119872119896 = 119879
Here the quantities ℎ and 119896 are mesh space size and time stepsize respectivelyThe time derivative can be approximated byusing the standard finite difference formula
120597119906119899
120597119905=119906119899+1minus 119906119899
119896 (4)
Abstract and Applied Analysis 3
Using the approximation of (4) (1) becomes
119906119899+1minus 119906119899
119896= 1205722 1205972119906119899
1205971199092+ 119902 (119909
119894 119905119899+1) (5)
Using 120579-weighted technique the space derivatives of (5) canbe written as
119906119899+1minus 119906119899
119896= 120579(120572
2 1205972119906119899+1
1205971199092) + (1 minus 120579) (120572
2 1205972119906119899
1205971199092)
+ 119902 (119909119894 119905119899+1)
(6)
where 0 le 120579 le 1 and the subscripts 119899 and 119899 + 1 are successivetime levels It is noted that the system becomes an explicitschemewhen 120579 = 0 a fully implicit schemewhen 120579 = 1 and aCrank-Nicolson scheme when 120579 = 12 [43 49] In this paperwe use the Crank-Nicolson approach Hence (6) becomes
119906119899+1minus 119906119899
119896=1
2(1205722 1205972119906119899+1
1205971199092) +
1
2(1205722 1205972119906119899
1205971199092) + 119902 (119909
119894 119905119899+1)
(7)
After simplification (7) leads to
2119906119899+1minus 1198961205722119906119899+1
119909119909= 2119906119899+ 1198961205722119906119899
119909119909+ 2119896119902 (119909
119894 119905119899+1) (8)
The space derivatives are approximated by using cubictrigonometric B-spline and are discussed in the next section
3 Cubic Trigonometric B-Spline Technique
In this section we discuss the cubic trigonometric B-splinecollocation method (CuTBSM) for the numerical solution ofthe nonclassical diffusion equation (1) Consider a mesh 119886 le119909 le 119887which is equally divided by knots 119909
119894into119873 subintervals
[119909119894 119909119894+1] 119894 = 0 1 2 119873 minus 1 where 119886 = 119909
0lt 1199091lt sdot sdot sdot lt
119909119873= 119887 Our approach for the nonclassical diffusion equation
using collocation method with cubic trigonometric B-splineis to seek an approximate solution as [37]
119880 (119909 119905) =
119873minus1
sum
119894=minus3
119862119894(119905) 119879119861
119894(119909) (9)
where 119862119894(119905) are to be determined for the approximated
solutions 119880(119909 119905) to the exact solutions 119906(119909 119905) at the point(119909119894 119905119899)119879119861119894(119909) are twice continuously differentiable piecewise
cubic trigonometric B-spline basis functions over the meshdefined by [49ndash51]
119879119861119894(119909)
=1
120596
1199013(119909119894) 119909 isin [119909
119894 119909119894+1]
119901 (119909119894) (119901 (119909
119894) 119902 (119909119894+2)
+119902 (119909119894+3) 119901 (119909
119894+1))
+119902 (119909119894+4) 1199012(119909119894+1) 119909 isin [119909
119894+1 119909119894+2]
119902 (119909119894+4) (119901 (119909
119894+1) 119902 (119909119894+3)
+ 119902 (119909119894+4) 119901 (119909
119894+2))
+119901 (119909119894) 1199022(119909119894+3) 119909 isin [119909
119894+2 119909119894+3]
1199023(119909119894+4) 119909 isin [119909
119894+3 119909119894+4]
(10)
Table 1 Values TB119894(119909) and its derivatives
119909 119909119894
119909119894+1
119909119894+2
119909119894+3
119909119894+4
TB119894
0 1198861
1198862
1198861
0TB1015840119894
0 1198863
0 1198864
0TB10158401015840119894
0 1198865
1198866
1198865
0
where
119901 (119909119894) = sin(
119909 minus 119909119894
2) 119902 (119909
119894) = sin(
119909119894minus 119909
2)
120596 = sin(ℎ2) sin (ℎ) sin(3ℎ
2)
(11)
and where ℎ = (119887 minus 119886)119899 The approximations119880119899119894at the point
(119909119894 119905119899) over subinterval [119909
119894 119909119894+1] can be defined as
119880119899
119894=
119894minus1
sum
119895=119894minus3
119862119899
119896119879119861119895(119909) (12)
In order to obtain the approximations to the solutions thevalues of 119879119861
119894(119909) and its derivatives at nodal points are
required and these derivatives are tabulated in Table 1 where
1198861=
sin2 (ℎ2)sin (ℎ) sin (3ℎ2)
1198862=
2
1 + 2 cos (ℎ)
1198863= minus
3
4 sin (3ℎ2)
1198864=
3
4 sin (3ℎ2)
1198865=
3 (1 + 3 cos (ℎ))16 sin2 (ℎ2) (2 cos (ℎ2) + cos (3ℎ2))
1198866= minus
3 cos2 (ℎ2)sin2 (ℎ2) (2 + 4 cos (ℎ))
(13)
Using approximate functions (10) and (12) and followingMittal and Arora [46] the values at the knots of119880119899
119894and their
derivatives up to second order are determined in terms oftime parameters 119862119899
119895as
119906119899
119894= 1198861119862119899
119894minus3+ 1198862119862119899
119894minus2+ 1198861119862119899
119894minus1
(119906119909)119899
119894= 1198863119862119899
119894minus3+ 1198864119862119899
119894minus1
(119906119909119909)119899
119894= 1198865119862119899
119894minus3+ 1198866119862119899
119894minus2+ 1198865119862119899
119894minus1
(14)
4 Abstract and Applied Analysis
Substituting (12) into (8) gives the following equation
2
119894minus1
sum
119895=119894minus3
119862119899+1
119895119879119861119895(119909119894) minus 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899+1
11989511987911986110158401015840
119895(119909119894)
= 2
119894minus1
sum
119895=119894minus3
119862119899
119895119879119861119895(119909119894) + 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899
11989511987911986110158401015840
119895(119909119894)
+ 2119896119902 (119909119894 119905119899+1)
(15)
The system thus obtained on simplifying (15) consists of119873 + 1 linear equations in 119873 + 3 unknowns 119862119899+1 =
(119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1) at the time level 119905 = 119905
119899+1 Equa-
tion (9) is applied to the boundary constraints (2) and (3) fortwo additional linear equations to obtain a unique solution ofthe resulting system
1205851119880 (0 119905
119899+1) + 1205852119880119909(0 119905119899+1)
= int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
(16)
1205853119880 (119871 119905
119899+1) + 1205854119880119909(119871 119905119899+1)
= int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(17)
From (15) (16) and (17) the system can be written in thematrix vector form as follows
119872119862119899+1
= 119873119862119899+ 119887 (18)
where
119862119899+1
= [119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1]119879
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119887 = [1205721(119905119899+1) 2119896119902 (119909
0 119905119899+1) 2119896119902 (119909
1 119905119899+1)
2119896119902 (119909119873 119905119899+1) 1205731(119905119899+1)]119879
119899 = 0 1 2 119872
(19)
and119872 and119873 are119873 + 3-dimensional matrix given by
119872 =
[[[[[[[[[[[[[[
[
1199010119902011990100
0 sdot sdot sdot 0 0
1199011119902111990110 sdot sdot sdot 0 0
0 1199011119902111990110 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
111990211199011
0 0 sdot sdot sdot sdot sdot sdot 1199010119902011990100
]]]]]]]]]]]]]]
]
119873 =
[[[[[[[[
[
0 0 0 0 sdot sdot sdot 0 0
1199012119902211990120 sdot sdot sdot 0 0
0 1199012119902211990120 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
211990221199012
0 0 sdot sdot sdot sdot sdot sdot 0 0 0
]]]]]]]]
]
(20)
where1199010= 12058511198861+ 12058521198863 119902
0= 12058511198861
11990100= 12058511198861+ 12058521198864 119901
1= 21198861minus 11989612057221198865
1199021= 21198862minus 11989612057221198866 119901
2= 21198861+ 11989612057221198865
1199022= 21198862+ 11989612057221198866
1205721(119905119899+1) = int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
1205731(119905119899+1) = int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(21)
Thus the system (18) becomes a matrix system of dimension(119873 + 3) times (119873 + 3) which is a tridiagonal system that can besolved by theThomas Algorithm [16]
31 Initial State Vector 1198620 After the initial vectors 1198620 havebeen computed from the initial constraints the approximatesolutions 119880119899+1
119894at a particular time level can be calculated
repeatedly by solving the recurrence relation (15) [40]The initial vectors 1198620 can be obtained from the initial
condition and boundary values of the derivatives of the initialcondition as follows [40 49]
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 0
1198800
119894= 1198921(119909119894) 119894 = 0 1 2 119873
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 119873
(22)
Thus (22) yields a (119873+3)times(119873+3)matrix system of the form
119860119862119899= 119889 (23)
where
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119889 = [1198921015840
1(1199090) 1198921(1199090) 119892
1(119909119873) 1198921015840
1(119909119873)]119879
119899 = 0
119860 =
[[[[[[[[[[[[[[
[
11988630 11988640 sdot sdot sdot 0 0
1198861119886211988610 sdot sdot sdot 0 0
0 1198861119886211988610 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119886
111988621198861
0 0 sdot sdot sdot sdot sdot sdot 11988630 1198864
]]]]]]]]]]]]]]
]
(24)
The solution of this system can be found by the use of theThomas Algorithm
4 Stability
In this section the von Neumann stability method is appliedfor investigating the stability of the proposed scheme This
Abstract and Applied Analysis 5
approach has been used by many researchers [18 40 4445 47ndash49] Substituting the approximate solution 119880 andits derivatives at knots with 119902(119909 119905) = 0 into (6) yields adifference equation with variables 119862
119898given by
(1198861minus 119896120579120572
21198865) 119862119899+1
119898minus3+ (1198862minus 119896120579120572
21198866) 119862119899+1
119898minus2
+ (1198861minus 119896120579120572
21198865) 119862119899+1
119898minus1
= (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus3+ (1198862+ 119896 (1 minus 120579) 120572
21198866) 119862119899
119898minus2
+ (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus1
(25)
Substituting the values of 119886119894 119894 = 1 2 5 6 into (25) we obtain
(1199011minus 119896212057912057221199012) 119862119899+1
119898minus3+ (1199013+ 119896212057912057221199014) 119862119899+1
119898minus2
+ (1199011minus 119896212057912057221199012) 119862119899+1
119898minus1
= (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus3
+ (1199013minus 1198962(1 minus 120579) 120572
21199014) 119862119899
119898minus2
+ (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus1
(26)
where
1199011= 16 sin2 (ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
times (1 + 2 cos(ℎ2))
1199012= 3 sin (ℎ) sin(3ℎ
2)(1 + 3 cos(ℎ
2)) cos ec2 (ℎ
2)
times (1 + 2 cos(ℎ2))
1199013= 32 sin (ℎ) sin(3ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
1199014= 24 cot2 (ℎ
2) sin ℎ sin(3ℎ
2)(2 cos(ℎ
2)+cos(3ℎ
2))
(27)
Simplifying it leads to
1199081119862119899+1
119898minus3+ 1199082119862119899+1
119898minus2+ 1199081119862119899+1
119898minus1
= 1199083119862119899
119898minus3+ 1199084119862119899
119898minus2+ 1199083119862119899
119898minus1
(28)
where
1199081= (1199011minus 119896120579120572
21199012)
1199082= (1199013+ 119896120579120572
21199014)
1199083= (1199011+ 119896 (1 minus 120579) 120572
21199012)
1199084= (1199013minus 119896 (1 minus 120579) 120572
21199014)
(29)
Now on inserting the trial solutions (one Fourier mode outof the full solution) at a given point 119909
119898is 119862119899119898= 120575119899 exp(119894119898120578ℎ)
into (28) and rearranging the equations 120578 is the modenumber ℎ is the element size and 119894 = radicminus1 we get
1199081120575119899+1119890119894120578(119898minus3)ℎ
+ 1199082120575119899+1119890119894120578(119898minus2)ℎ
+ 1199081120575119899+1119890119894120578(119898minus1)ℎ
= 1199083120575119899119890119894120578(119898minus3)ℎ
+ 1199084120575119899119890119894120578(119898minus2)ℎ
+ 1199083120575119899119890119894120578(119898minus1)ℎ
(30)
Dividing (30) by 120575119899119890119894120578(119898minus2)ℎ and rearranging the equation weget
120575 (1199082+ 1199081cos (120578ℎ)) = (119908
4+ 1199083cos (120578ℎ)) (31)
Let
119860 = 1199013+ 1199011cos (120578ℎ)
119861 = 1198961205722(1199014minus 1199012cos (120578ℎ))
(32)
Therefore (31) can be written as
120575 (119860 + 120579119861) minus (119860 minus (1 minus 120579) 119861) = 0 (33)
Equation (33) can be rewritten as
120575 [1198832+ 119894119884] minus [119883
1+ 119894119884] = 0 (34)
where
1198831= (119860 minus (1 minus 120579) 119861)
1198832= (119860 + 120579119861)
119884 = 0
(35)
For stability the maximummodulus of the eigenvalues of thematrix has to be less than or equal to one [47] Since 119860 gt 0119861 gt 0 and 0 le 120579 le 1 we always have
10038161003816100381610038161205851003816100381610038161003816
2
=1198832
1+ 1198842
1198832
2+ 1198842
le 1 (36)
Thus from (36) the proposed scheme for nonclassical dif-fusion equation (with term 119902(119909 119905) = 0) is unconditionallystable and it is also unconditionally stable with a general term119902(119909 119905) since themodulus of the eigenvaluesmust be less thanone [47] We recall Duhamelrsquos principle ([15] chapter 9) ascheme is stable for equation 119875
119896ℎV = 119891 if it is stable for the
equation 119875119896ℎV = 0 This means that there are no constraints
on grid sizeℎ and step size in time level 119896 butwe should preferthose values of ℎ and 119896 for which we obtain the best accuracyof the scheme
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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2 Abstract and Applied Analysis
= int
119871
0
1198923(119909) 119906 (119909 119905) 119889119909 + ℎ
2(119905) = 119879
2
0 lt 119905 le 119879
(3)
where 120585119894 119894 = 1 2 3 4 are known constants 119902 119892
119894(119894 = 1 2 3)
are known continuous functions 120572 is the thermal diffusivityof the rod and ℎ
119894(119894 = 1 2) are prescribed functions This
problem has been studied by Dehghan [8] Martın-Vaqueroand Vigo-Aguiar [9] Li and Wu [10] and Golbabai andJavidi [11] Several physical circumstances might be modeledby equation and constraints (1)ndash(3) and several examples ofapplication in physics with comprehensive derivations of theabove mentioned problem can be found in [3 12ndash14]
There are several numerical methods in the literaturethat have been developed for solving the proposed problem(1) subject to initial and nonlocal boundary constraints (2)-(3) The methods were based for instance on the for-ward Euler method the backward Euler approach or theCrank-Nicolson scheme [15 16] Laplace transformation [17]and so forth Dehghan [8] presented four finite differenceapproaches namely the BTCS (backward time centred space)scheme the implicit (3 3) Crandallrsquos formula the 3-pointFTCS (forward time centred space) two-level scheme andthe Dufort-Frankel three-level approach for the numericalsolution of parabolic equation with nonlocal specificationMartın-Vaquero and Vigo-Aguiar [9] improved the order ofconvergence of the implicit (3 3) Crandallrsquos formula pro-posed by Dehghan [8] and also improved the accuracy of themethod Martın-Vaquero and Vigo-Aguiar [18] developed analgorithm for the solution of the heat conduction equationswith nonlocal constraints which reduced the CPU time andenhanced the accuracy of (3 3) Crandallrsquos formula proposedin [8] Li and Wu [10] proposed an algorithm which wasbased on the transverse method of lines (TMOL) whichcan reduce a nonclassical diffusion equation to a series ofordinary differential equations (ODEs) Subsequently theauthors in [10] used an analytic reproducing kernel techniqueto solve ODEs with integral boundary constraints Dehghan[19 20] jointly with Tatari and Dehghan [21] has proposedseveral efficient techniques for the numerical solutions ofpartial differential equations subject to nonlocal boundaryconstraints Golbabai and Javidi [11] introduced a Chebyshevspectral collocation method (CSCM) based on Chebyshevpolynomials for solving a parabolic problem subject tononlocal boundary constraints For more details on othernumerical methods for the solution of a one-dimensionalheat equation subject to nonlocal boundary constraints in theliterature see [22ndash31]
The study of B-spline functions is a key elementin computer-aided geometric design [32ndash35] It has alsoattracted attention in the literature [36ndash51] to the numeri-cal solution of various differential equations [38ndash40] Thisis because they have important geometric properties andfeatures that make them amenable to more detailed analysisNumerical methods based on B-spline functions of variousdegrees have been utilized for solving initial and boundaryvalue problems As examples a cubic B-spline collocation
method was used to solve a nonlinear diffusion equationsubject to certain initial and Dirichlet boundary constraints[41] a finite element method based on bivariate splines hasbeen used for solving parabolic partial differential equation[42] and the combination of finite difference approach andcubic B-spline method was applied for the solution of aone-dimensional heat equation subject to local boundaryconstraints [43 44] Goh et al [45] presented a comparisonof cubic B-spline and extended cubic uniform B-spline basedcollocation methods for solving a one-dimensional heatequationwith a nonlocal initial constraint and concluded thatextended cubic uniform B-spline with an appropriate valueof parameters gives better results than the cubic B-splineA finite difference scheme based on cubic B-spline was alsoused for solving the one-dimensional wave equation [43]advection-diffusion equation [44] one-dimensional coupledviscous Burgersrsquo equation [47] system of strongly cou-pled reaction-diffusion equations [48] and one-dimensionalhyperbolic problems [49]
In our present paper a new two-time level implicittechnique is developed to approximate the solution of thenonclassical diffusion problem (1) subject to initial con-straints in (2) and nonlocal boundary constraints in (1)ndash(3) The technique is based on the cubic trigonometric B-spline functions A finite difference approach and 120579-weightedscheme are applied for the time and space discretizationrespectively Some researchers have considered the ordinaryB-spline collocation method for solving the heat equationsubject to local and nonlocal boundary constraints but sofar as we are aware not with the cubic trigonometric B-spline collocation method Cubic trigonometric B-spline isused as an interpolating function in the space dimensionThe unconditional stability property of the method is provedby von Neumann method The feasibility of the method isshown by test problems with 119896 = 119904ℎ2 119904 = 1 2 4 5 insteadof smaller time step size 119896 = 04ℎ
2 and the approximatedsolutions are found to be in good agreement with the knownexact solutions
The outline of this study is as follows A numericalsolution of nonclassical diffusion problem is presented inSection 2 In Section 3 the cubic trigonometric B-spline isutilized as an interpolating function in the space dimensionThe von Neumann approach is used to prove the stabilityof the method in Section 4 Numerical examples are consid-ered in Section 5 to show the achievability of the proposedmethod Finally the concluding remarks of this study aregiven in Section 6
2 Solution of Nonclassical Diffusion Problem
Consider a uniform mesh Ω with grid points (119909119894 119905119899) to
discretize the grid region Δ = [119886 119887] times [0 119879] with 119909119894= 119886 + 119894ℎ
119894 = 0 1 2 119873 and 119905119899= 119899119896 119899 = 0 1 2 3 119872119872119896 = 119879
Here the quantities ℎ and 119896 are mesh space size and time stepsize respectivelyThe time derivative can be approximated byusing the standard finite difference formula
120597119906119899
120597119905=119906119899+1minus 119906119899
119896 (4)
Abstract and Applied Analysis 3
Using the approximation of (4) (1) becomes
119906119899+1minus 119906119899
119896= 1205722 1205972119906119899
1205971199092+ 119902 (119909
119894 119905119899+1) (5)
Using 120579-weighted technique the space derivatives of (5) canbe written as
119906119899+1minus 119906119899
119896= 120579(120572
2 1205972119906119899+1
1205971199092) + (1 minus 120579) (120572
2 1205972119906119899
1205971199092)
+ 119902 (119909119894 119905119899+1)
(6)
where 0 le 120579 le 1 and the subscripts 119899 and 119899 + 1 are successivetime levels It is noted that the system becomes an explicitschemewhen 120579 = 0 a fully implicit schemewhen 120579 = 1 and aCrank-Nicolson scheme when 120579 = 12 [43 49] In this paperwe use the Crank-Nicolson approach Hence (6) becomes
119906119899+1minus 119906119899
119896=1
2(1205722 1205972119906119899+1
1205971199092) +
1
2(1205722 1205972119906119899
1205971199092) + 119902 (119909
119894 119905119899+1)
(7)
After simplification (7) leads to
2119906119899+1minus 1198961205722119906119899+1
119909119909= 2119906119899+ 1198961205722119906119899
119909119909+ 2119896119902 (119909
119894 119905119899+1) (8)
The space derivatives are approximated by using cubictrigonometric B-spline and are discussed in the next section
3 Cubic Trigonometric B-Spline Technique
In this section we discuss the cubic trigonometric B-splinecollocation method (CuTBSM) for the numerical solution ofthe nonclassical diffusion equation (1) Consider a mesh 119886 le119909 le 119887which is equally divided by knots 119909
119894into119873 subintervals
[119909119894 119909119894+1] 119894 = 0 1 2 119873 minus 1 where 119886 = 119909
0lt 1199091lt sdot sdot sdot lt
119909119873= 119887 Our approach for the nonclassical diffusion equation
using collocation method with cubic trigonometric B-splineis to seek an approximate solution as [37]
119880 (119909 119905) =
119873minus1
sum
119894=minus3
119862119894(119905) 119879119861
119894(119909) (9)
where 119862119894(119905) are to be determined for the approximated
solutions 119880(119909 119905) to the exact solutions 119906(119909 119905) at the point(119909119894 119905119899)119879119861119894(119909) are twice continuously differentiable piecewise
cubic trigonometric B-spline basis functions over the meshdefined by [49ndash51]
119879119861119894(119909)
=1
120596
1199013(119909119894) 119909 isin [119909
119894 119909119894+1]
119901 (119909119894) (119901 (119909
119894) 119902 (119909119894+2)
+119902 (119909119894+3) 119901 (119909
119894+1))
+119902 (119909119894+4) 1199012(119909119894+1) 119909 isin [119909
119894+1 119909119894+2]
119902 (119909119894+4) (119901 (119909
119894+1) 119902 (119909119894+3)
+ 119902 (119909119894+4) 119901 (119909
119894+2))
+119901 (119909119894) 1199022(119909119894+3) 119909 isin [119909
119894+2 119909119894+3]
1199023(119909119894+4) 119909 isin [119909
119894+3 119909119894+4]
(10)
Table 1 Values TB119894(119909) and its derivatives
119909 119909119894
119909119894+1
119909119894+2
119909119894+3
119909119894+4
TB119894
0 1198861
1198862
1198861
0TB1015840119894
0 1198863
0 1198864
0TB10158401015840119894
0 1198865
1198866
1198865
0
where
119901 (119909119894) = sin(
119909 minus 119909119894
2) 119902 (119909
119894) = sin(
119909119894minus 119909
2)
120596 = sin(ℎ2) sin (ℎ) sin(3ℎ
2)
(11)
and where ℎ = (119887 minus 119886)119899 The approximations119880119899119894at the point
(119909119894 119905119899) over subinterval [119909
119894 119909119894+1] can be defined as
119880119899
119894=
119894minus1
sum
119895=119894minus3
119862119899
119896119879119861119895(119909) (12)
In order to obtain the approximations to the solutions thevalues of 119879119861
119894(119909) and its derivatives at nodal points are
required and these derivatives are tabulated in Table 1 where
1198861=
sin2 (ℎ2)sin (ℎ) sin (3ℎ2)
1198862=
2
1 + 2 cos (ℎ)
1198863= minus
3
4 sin (3ℎ2)
1198864=
3
4 sin (3ℎ2)
1198865=
3 (1 + 3 cos (ℎ))16 sin2 (ℎ2) (2 cos (ℎ2) + cos (3ℎ2))
1198866= minus
3 cos2 (ℎ2)sin2 (ℎ2) (2 + 4 cos (ℎ))
(13)
Using approximate functions (10) and (12) and followingMittal and Arora [46] the values at the knots of119880119899
119894and their
derivatives up to second order are determined in terms oftime parameters 119862119899
119895as
119906119899
119894= 1198861119862119899
119894minus3+ 1198862119862119899
119894minus2+ 1198861119862119899
119894minus1
(119906119909)119899
119894= 1198863119862119899
119894minus3+ 1198864119862119899
119894minus1
(119906119909119909)119899
119894= 1198865119862119899
119894minus3+ 1198866119862119899
119894minus2+ 1198865119862119899
119894minus1
(14)
4 Abstract and Applied Analysis
Substituting (12) into (8) gives the following equation
2
119894minus1
sum
119895=119894minus3
119862119899+1
119895119879119861119895(119909119894) minus 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899+1
11989511987911986110158401015840
119895(119909119894)
= 2
119894minus1
sum
119895=119894minus3
119862119899
119895119879119861119895(119909119894) + 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899
11989511987911986110158401015840
119895(119909119894)
+ 2119896119902 (119909119894 119905119899+1)
(15)
The system thus obtained on simplifying (15) consists of119873 + 1 linear equations in 119873 + 3 unknowns 119862119899+1 =
(119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1) at the time level 119905 = 119905
119899+1 Equa-
tion (9) is applied to the boundary constraints (2) and (3) fortwo additional linear equations to obtain a unique solution ofthe resulting system
1205851119880 (0 119905
119899+1) + 1205852119880119909(0 119905119899+1)
= int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
(16)
1205853119880 (119871 119905
119899+1) + 1205854119880119909(119871 119905119899+1)
= int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(17)
From (15) (16) and (17) the system can be written in thematrix vector form as follows
119872119862119899+1
= 119873119862119899+ 119887 (18)
where
119862119899+1
= [119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1]119879
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119887 = [1205721(119905119899+1) 2119896119902 (119909
0 119905119899+1) 2119896119902 (119909
1 119905119899+1)
2119896119902 (119909119873 119905119899+1) 1205731(119905119899+1)]119879
119899 = 0 1 2 119872
(19)
and119872 and119873 are119873 + 3-dimensional matrix given by
119872 =
[[[[[[[[[[[[[[
[
1199010119902011990100
0 sdot sdot sdot 0 0
1199011119902111990110 sdot sdot sdot 0 0
0 1199011119902111990110 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
111990211199011
0 0 sdot sdot sdot sdot sdot sdot 1199010119902011990100
]]]]]]]]]]]]]]
]
119873 =
[[[[[[[[
[
0 0 0 0 sdot sdot sdot 0 0
1199012119902211990120 sdot sdot sdot 0 0
0 1199012119902211990120 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
211990221199012
0 0 sdot sdot sdot sdot sdot sdot 0 0 0
]]]]]]]]
]
(20)
where1199010= 12058511198861+ 12058521198863 119902
0= 12058511198861
11990100= 12058511198861+ 12058521198864 119901
1= 21198861minus 11989612057221198865
1199021= 21198862minus 11989612057221198866 119901
2= 21198861+ 11989612057221198865
1199022= 21198862+ 11989612057221198866
1205721(119905119899+1) = int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
1205731(119905119899+1) = int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(21)
Thus the system (18) becomes a matrix system of dimension(119873 + 3) times (119873 + 3) which is a tridiagonal system that can besolved by theThomas Algorithm [16]
31 Initial State Vector 1198620 After the initial vectors 1198620 havebeen computed from the initial constraints the approximatesolutions 119880119899+1
119894at a particular time level can be calculated
repeatedly by solving the recurrence relation (15) [40]The initial vectors 1198620 can be obtained from the initial
condition and boundary values of the derivatives of the initialcondition as follows [40 49]
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 0
1198800
119894= 1198921(119909119894) 119894 = 0 1 2 119873
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 119873
(22)
Thus (22) yields a (119873+3)times(119873+3)matrix system of the form
119860119862119899= 119889 (23)
where
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119889 = [1198921015840
1(1199090) 1198921(1199090) 119892
1(119909119873) 1198921015840
1(119909119873)]119879
119899 = 0
119860 =
[[[[[[[[[[[[[[
[
11988630 11988640 sdot sdot sdot 0 0
1198861119886211988610 sdot sdot sdot 0 0
0 1198861119886211988610 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119886
111988621198861
0 0 sdot sdot sdot sdot sdot sdot 11988630 1198864
]]]]]]]]]]]]]]
]
(24)
The solution of this system can be found by the use of theThomas Algorithm
4 Stability
In this section the von Neumann stability method is appliedfor investigating the stability of the proposed scheme This
Abstract and Applied Analysis 5
approach has been used by many researchers [18 40 4445 47ndash49] Substituting the approximate solution 119880 andits derivatives at knots with 119902(119909 119905) = 0 into (6) yields adifference equation with variables 119862
119898given by
(1198861minus 119896120579120572
21198865) 119862119899+1
119898minus3+ (1198862minus 119896120579120572
21198866) 119862119899+1
119898minus2
+ (1198861minus 119896120579120572
21198865) 119862119899+1
119898minus1
= (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus3+ (1198862+ 119896 (1 minus 120579) 120572
21198866) 119862119899
119898minus2
+ (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus1
(25)
Substituting the values of 119886119894 119894 = 1 2 5 6 into (25) we obtain
(1199011minus 119896212057912057221199012) 119862119899+1
119898minus3+ (1199013+ 119896212057912057221199014) 119862119899+1
119898minus2
+ (1199011minus 119896212057912057221199012) 119862119899+1
119898minus1
= (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus3
+ (1199013minus 1198962(1 minus 120579) 120572
21199014) 119862119899
119898minus2
+ (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus1
(26)
where
1199011= 16 sin2 (ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
times (1 + 2 cos(ℎ2))
1199012= 3 sin (ℎ) sin(3ℎ
2)(1 + 3 cos(ℎ
2)) cos ec2 (ℎ
2)
times (1 + 2 cos(ℎ2))
1199013= 32 sin (ℎ) sin(3ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
1199014= 24 cot2 (ℎ
2) sin ℎ sin(3ℎ
2)(2 cos(ℎ
2)+cos(3ℎ
2))
(27)
Simplifying it leads to
1199081119862119899+1
119898minus3+ 1199082119862119899+1
119898minus2+ 1199081119862119899+1
119898minus1
= 1199083119862119899
119898minus3+ 1199084119862119899
119898minus2+ 1199083119862119899
119898minus1
(28)
where
1199081= (1199011minus 119896120579120572
21199012)
1199082= (1199013+ 119896120579120572
21199014)
1199083= (1199011+ 119896 (1 minus 120579) 120572
21199012)
1199084= (1199013minus 119896 (1 minus 120579) 120572
21199014)
(29)
Now on inserting the trial solutions (one Fourier mode outof the full solution) at a given point 119909
119898is 119862119899119898= 120575119899 exp(119894119898120578ℎ)
into (28) and rearranging the equations 120578 is the modenumber ℎ is the element size and 119894 = radicminus1 we get
1199081120575119899+1119890119894120578(119898minus3)ℎ
+ 1199082120575119899+1119890119894120578(119898minus2)ℎ
+ 1199081120575119899+1119890119894120578(119898minus1)ℎ
= 1199083120575119899119890119894120578(119898minus3)ℎ
+ 1199084120575119899119890119894120578(119898minus2)ℎ
+ 1199083120575119899119890119894120578(119898minus1)ℎ
(30)
Dividing (30) by 120575119899119890119894120578(119898minus2)ℎ and rearranging the equation weget
120575 (1199082+ 1199081cos (120578ℎ)) = (119908
4+ 1199083cos (120578ℎ)) (31)
Let
119860 = 1199013+ 1199011cos (120578ℎ)
119861 = 1198961205722(1199014minus 1199012cos (120578ℎ))
(32)
Therefore (31) can be written as
120575 (119860 + 120579119861) minus (119860 minus (1 minus 120579) 119861) = 0 (33)
Equation (33) can be rewritten as
120575 [1198832+ 119894119884] minus [119883
1+ 119894119884] = 0 (34)
where
1198831= (119860 minus (1 minus 120579) 119861)
1198832= (119860 + 120579119861)
119884 = 0
(35)
For stability the maximummodulus of the eigenvalues of thematrix has to be less than or equal to one [47] Since 119860 gt 0119861 gt 0 and 0 le 120579 le 1 we always have
10038161003816100381610038161205851003816100381610038161003816
2
=1198832
1+ 1198842
1198832
2+ 1198842
le 1 (36)
Thus from (36) the proposed scheme for nonclassical dif-fusion equation (with term 119902(119909 119905) = 0) is unconditionallystable and it is also unconditionally stable with a general term119902(119909 119905) since themodulus of the eigenvaluesmust be less thanone [47] We recall Duhamelrsquos principle ([15] chapter 9) ascheme is stable for equation 119875
119896ℎV = 119891 if it is stable for the
equation 119875119896ℎV = 0 This means that there are no constraints
on grid sizeℎ and step size in time level 119896 butwe should preferthose values of ℎ and 119896 for which we obtain the best accuracyof the scheme
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
Submit your manuscripts athttpwwwhindawicom
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
Using the approximation of (4) (1) becomes
119906119899+1minus 119906119899
119896= 1205722 1205972119906119899
1205971199092+ 119902 (119909
119894 119905119899+1) (5)
Using 120579-weighted technique the space derivatives of (5) canbe written as
119906119899+1minus 119906119899
119896= 120579(120572
2 1205972119906119899+1
1205971199092) + (1 minus 120579) (120572
2 1205972119906119899
1205971199092)
+ 119902 (119909119894 119905119899+1)
(6)
where 0 le 120579 le 1 and the subscripts 119899 and 119899 + 1 are successivetime levels It is noted that the system becomes an explicitschemewhen 120579 = 0 a fully implicit schemewhen 120579 = 1 and aCrank-Nicolson scheme when 120579 = 12 [43 49] In this paperwe use the Crank-Nicolson approach Hence (6) becomes
119906119899+1minus 119906119899
119896=1
2(1205722 1205972119906119899+1
1205971199092) +
1
2(1205722 1205972119906119899
1205971199092) + 119902 (119909
119894 119905119899+1)
(7)
After simplification (7) leads to
2119906119899+1minus 1198961205722119906119899+1
119909119909= 2119906119899+ 1198961205722119906119899
119909119909+ 2119896119902 (119909
119894 119905119899+1) (8)
The space derivatives are approximated by using cubictrigonometric B-spline and are discussed in the next section
3 Cubic Trigonometric B-Spline Technique
In this section we discuss the cubic trigonometric B-splinecollocation method (CuTBSM) for the numerical solution ofthe nonclassical diffusion equation (1) Consider a mesh 119886 le119909 le 119887which is equally divided by knots 119909
119894into119873 subintervals
[119909119894 119909119894+1] 119894 = 0 1 2 119873 minus 1 where 119886 = 119909
0lt 1199091lt sdot sdot sdot lt
119909119873= 119887 Our approach for the nonclassical diffusion equation
using collocation method with cubic trigonometric B-splineis to seek an approximate solution as [37]
119880 (119909 119905) =
119873minus1
sum
119894=minus3
119862119894(119905) 119879119861
119894(119909) (9)
where 119862119894(119905) are to be determined for the approximated
solutions 119880(119909 119905) to the exact solutions 119906(119909 119905) at the point(119909119894 119905119899)119879119861119894(119909) are twice continuously differentiable piecewise
cubic trigonometric B-spline basis functions over the meshdefined by [49ndash51]
119879119861119894(119909)
=1
120596
1199013(119909119894) 119909 isin [119909
119894 119909119894+1]
119901 (119909119894) (119901 (119909
119894) 119902 (119909119894+2)
+119902 (119909119894+3) 119901 (119909
119894+1))
+119902 (119909119894+4) 1199012(119909119894+1) 119909 isin [119909
119894+1 119909119894+2]
119902 (119909119894+4) (119901 (119909
119894+1) 119902 (119909119894+3)
+ 119902 (119909119894+4) 119901 (119909
119894+2))
+119901 (119909119894) 1199022(119909119894+3) 119909 isin [119909
119894+2 119909119894+3]
1199023(119909119894+4) 119909 isin [119909
119894+3 119909119894+4]
(10)
Table 1 Values TB119894(119909) and its derivatives
119909 119909119894
119909119894+1
119909119894+2
119909119894+3
119909119894+4
TB119894
0 1198861
1198862
1198861
0TB1015840119894
0 1198863
0 1198864
0TB10158401015840119894
0 1198865
1198866
1198865
0
where
119901 (119909119894) = sin(
119909 minus 119909119894
2) 119902 (119909
119894) = sin(
119909119894minus 119909
2)
120596 = sin(ℎ2) sin (ℎ) sin(3ℎ
2)
(11)
and where ℎ = (119887 minus 119886)119899 The approximations119880119899119894at the point
(119909119894 119905119899) over subinterval [119909
119894 119909119894+1] can be defined as
119880119899
119894=
119894minus1
sum
119895=119894minus3
119862119899
119896119879119861119895(119909) (12)
In order to obtain the approximations to the solutions thevalues of 119879119861
119894(119909) and its derivatives at nodal points are
required and these derivatives are tabulated in Table 1 where
1198861=
sin2 (ℎ2)sin (ℎ) sin (3ℎ2)
1198862=
2
1 + 2 cos (ℎ)
1198863= minus
3
4 sin (3ℎ2)
1198864=
3
4 sin (3ℎ2)
1198865=
3 (1 + 3 cos (ℎ))16 sin2 (ℎ2) (2 cos (ℎ2) + cos (3ℎ2))
1198866= minus
3 cos2 (ℎ2)sin2 (ℎ2) (2 + 4 cos (ℎ))
(13)
Using approximate functions (10) and (12) and followingMittal and Arora [46] the values at the knots of119880119899
119894and their
derivatives up to second order are determined in terms oftime parameters 119862119899
119895as
119906119899
119894= 1198861119862119899
119894minus3+ 1198862119862119899
119894minus2+ 1198861119862119899
119894minus1
(119906119909)119899
119894= 1198863119862119899
119894minus3+ 1198864119862119899
119894minus1
(119906119909119909)119899
119894= 1198865119862119899
119894minus3+ 1198866119862119899
119894minus2+ 1198865119862119899
119894minus1
(14)
4 Abstract and Applied Analysis
Substituting (12) into (8) gives the following equation
2
119894minus1
sum
119895=119894minus3
119862119899+1
119895119879119861119895(119909119894) minus 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899+1
11989511987911986110158401015840
119895(119909119894)
= 2
119894minus1
sum
119895=119894minus3
119862119899
119895119879119861119895(119909119894) + 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899
11989511987911986110158401015840
119895(119909119894)
+ 2119896119902 (119909119894 119905119899+1)
(15)
The system thus obtained on simplifying (15) consists of119873 + 1 linear equations in 119873 + 3 unknowns 119862119899+1 =
(119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1) at the time level 119905 = 119905
119899+1 Equa-
tion (9) is applied to the boundary constraints (2) and (3) fortwo additional linear equations to obtain a unique solution ofthe resulting system
1205851119880 (0 119905
119899+1) + 1205852119880119909(0 119905119899+1)
= int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
(16)
1205853119880 (119871 119905
119899+1) + 1205854119880119909(119871 119905119899+1)
= int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(17)
From (15) (16) and (17) the system can be written in thematrix vector form as follows
119872119862119899+1
= 119873119862119899+ 119887 (18)
where
119862119899+1
= [119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1]119879
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119887 = [1205721(119905119899+1) 2119896119902 (119909
0 119905119899+1) 2119896119902 (119909
1 119905119899+1)
2119896119902 (119909119873 119905119899+1) 1205731(119905119899+1)]119879
119899 = 0 1 2 119872
(19)
and119872 and119873 are119873 + 3-dimensional matrix given by
119872 =
[[[[[[[[[[[[[[
[
1199010119902011990100
0 sdot sdot sdot 0 0
1199011119902111990110 sdot sdot sdot 0 0
0 1199011119902111990110 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
111990211199011
0 0 sdot sdot sdot sdot sdot sdot 1199010119902011990100
]]]]]]]]]]]]]]
]
119873 =
[[[[[[[[
[
0 0 0 0 sdot sdot sdot 0 0
1199012119902211990120 sdot sdot sdot 0 0
0 1199012119902211990120 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
211990221199012
0 0 sdot sdot sdot sdot sdot sdot 0 0 0
]]]]]]]]
]
(20)
where1199010= 12058511198861+ 12058521198863 119902
0= 12058511198861
11990100= 12058511198861+ 12058521198864 119901
1= 21198861minus 11989612057221198865
1199021= 21198862minus 11989612057221198866 119901
2= 21198861+ 11989612057221198865
1199022= 21198862+ 11989612057221198866
1205721(119905119899+1) = int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
1205731(119905119899+1) = int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(21)
Thus the system (18) becomes a matrix system of dimension(119873 + 3) times (119873 + 3) which is a tridiagonal system that can besolved by theThomas Algorithm [16]
31 Initial State Vector 1198620 After the initial vectors 1198620 havebeen computed from the initial constraints the approximatesolutions 119880119899+1
119894at a particular time level can be calculated
repeatedly by solving the recurrence relation (15) [40]The initial vectors 1198620 can be obtained from the initial
condition and boundary values of the derivatives of the initialcondition as follows [40 49]
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 0
1198800
119894= 1198921(119909119894) 119894 = 0 1 2 119873
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 119873
(22)
Thus (22) yields a (119873+3)times(119873+3)matrix system of the form
119860119862119899= 119889 (23)
where
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119889 = [1198921015840
1(1199090) 1198921(1199090) 119892
1(119909119873) 1198921015840
1(119909119873)]119879
119899 = 0
119860 =
[[[[[[[[[[[[[[
[
11988630 11988640 sdot sdot sdot 0 0
1198861119886211988610 sdot sdot sdot 0 0
0 1198861119886211988610 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119886
111988621198861
0 0 sdot sdot sdot sdot sdot sdot 11988630 1198864
]]]]]]]]]]]]]]
]
(24)
The solution of this system can be found by the use of theThomas Algorithm
4 Stability
In this section the von Neumann stability method is appliedfor investigating the stability of the proposed scheme This
Abstract and Applied Analysis 5
approach has been used by many researchers [18 40 4445 47ndash49] Substituting the approximate solution 119880 andits derivatives at knots with 119902(119909 119905) = 0 into (6) yields adifference equation with variables 119862
119898given by
(1198861minus 119896120579120572
21198865) 119862119899+1
119898minus3+ (1198862minus 119896120579120572
21198866) 119862119899+1
119898minus2
+ (1198861minus 119896120579120572
21198865) 119862119899+1
119898minus1
= (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus3+ (1198862+ 119896 (1 minus 120579) 120572
21198866) 119862119899
119898minus2
+ (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus1
(25)
Substituting the values of 119886119894 119894 = 1 2 5 6 into (25) we obtain
(1199011minus 119896212057912057221199012) 119862119899+1
119898minus3+ (1199013+ 119896212057912057221199014) 119862119899+1
119898minus2
+ (1199011minus 119896212057912057221199012) 119862119899+1
119898minus1
= (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus3
+ (1199013minus 1198962(1 minus 120579) 120572
21199014) 119862119899
119898minus2
+ (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus1
(26)
where
1199011= 16 sin2 (ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
times (1 + 2 cos(ℎ2))
1199012= 3 sin (ℎ) sin(3ℎ
2)(1 + 3 cos(ℎ
2)) cos ec2 (ℎ
2)
times (1 + 2 cos(ℎ2))
1199013= 32 sin (ℎ) sin(3ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
1199014= 24 cot2 (ℎ
2) sin ℎ sin(3ℎ
2)(2 cos(ℎ
2)+cos(3ℎ
2))
(27)
Simplifying it leads to
1199081119862119899+1
119898minus3+ 1199082119862119899+1
119898minus2+ 1199081119862119899+1
119898minus1
= 1199083119862119899
119898minus3+ 1199084119862119899
119898minus2+ 1199083119862119899
119898minus1
(28)
where
1199081= (1199011minus 119896120579120572
21199012)
1199082= (1199013+ 119896120579120572
21199014)
1199083= (1199011+ 119896 (1 minus 120579) 120572
21199012)
1199084= (1199013minus 119896 (1 minus 120579) 120572
21199014)
(29)
Now on inserting the trial solutions (one Fourier mode outof the full solution) at a given point 119909
119898is 119862119899119898= 120575119899 exp(119894119898120578ℎ)
into (28) and rearranging the equations 120578 is the modenumber ℎ is the element size and 119894 = radicminus1 we get
1199081120575119899+1119890119894120578(119898minus3)ℎ
+ 1199082120575119899+1119890119894120578(119898minus2)ℎ
+ 1199081120575119899+1119890119894120578(119898minus1)ℎ
= 1199083120575119899119890119894120578(119898minus3)ℎ
+ 1199084120575119899119890119894120578(119898minus2)ℎ
+ 1199083120575119899119890119894120578(119898minus1)ℎ
(30)
Dividing (30) by 120575119899119890119894120578(119898minus2)ℎ and rearranging the equation weget
120575 (1199082+ 1199081cos (120578ℎ)) = (119908
4+ 1199083cos (120578ℎ)) (31)
Let
119860 = 1199013+ 1199011cos (120578ℎ)
119861 = 1198961205722(1199014minus 1199012cos (120578ℎ))
(32)
Therefore (31) can be written as
120575 (119860 + 120579119861) minus (119860 minus (1 minus 120579) 119861) = 0 (33)
Equation (33) can be rewritten as
120575 [1198832+ 119894119884] minus [119883
1+ 119894119884] = 0 (34)
where
1198831= (119860 minus (1 minus 120579) 119861)
1198832= (119860 + 120579119861)
119884 = 0
(35)
For stability the maximummodulus of the eigenvalues of thematrix has to be less than or equal to one [47] Since 119860 gt 0119861 gt 0 and 0 le 120579 le 1 we always have
10038161003816100381610038161205851003816100381610038161003816
2
=1198832
1+ 1198842
1198832
2+ 1198842
le 1 (36)
Thus from (36) the proposed scheme for nonclassical dif-fusion equation (with term 119902(119909 119905) = 0) is unconditionallystable and it is also unconditionally stable with a general term119902(119909 119905) since themodulus of the eigenvaluesmust be less thanone [47] We recall Duhamelrsquos principle ([15] chapter 9) ascheme is stable for equation 119875
119896ℎV = 119891 if it is stable for the
equation 119875119896ℎV = 0 This means that there are no constraints
on grid sizeℎ and step size in time level 119896 butwe should preferthose values of ℎ and 119896 for which we obtain the best accuracyof the scheme
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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
4 Abstract and Applied Analysis
Substituting (12) into (8) gives the following equation
2
119894minus1
sum
119895=119894minus3
119862119899+1
119895119879119861119895(119909119894) minus 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899+1
11989511987911986110158401015840
119895(119909119894)
= 2
119894minus1
sum
119895=119894minus3
119862119899
119895119879119861119895(119909119894) + 119896120572
2
119894minus1
sum
119895=119894minus3
119862119899
11989511987911986110158401015840
119895(119909119894)
+ 2119896119902 (119909119894 119905119899+1)
(15)
The system thus obtained on simplifying (15) consists of119873 + 1 linear equations in 119873 + 3 unknowns 119862119899+1 =
(119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1) at the time level 119905 = 119905
119899+1 Equa-
tion (9) is applied to the boundary constraints (2) and (3) fortwo additional linear equations to obtain a unique solution ofthe resulting system
1205851119880 (0 119905
119899+1) + 1205852119880119909(0 119905119899+1)
= int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
(16)
1205853119880 (119871 119905
119899+1) + 1205854119880119909(119871 119905119899+1)
= int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(17)
From (15) (16) and (17) the system can be written in thematrix vector form as follows
119872119862119899+1
= 119873119862119899+ 119887 (18)
where
119862119899+1
= [119862119899+1
minus3 119862119899+1
minus2 119862119899+1
minus1 119862
119899+1
119873minus1]119879
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119887 = [1205721(119905119899+1) 2119896119902 (119909
0 119905119899+1) 2119896119902 (119909
1 119905119899+1)
2119896119902 (119909119873 119905119899+1) 1205731(119905119899+1)]119879
119899 = 0 1 2 119872
(19)
and119872 and119873 are119873 + 3-dimensional matrix given by
119872 =
[[[[[[[[[[[[[[
[
1199010119902011990100
0 sdot sdot sdot 0 0
1199011119902111990110 sdot sdot sdot 0 0
0 1199011119902111990110 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
111990211199011
0 0 sdot sdot sdot sdot sdot sdot 1199010119902011990100
]]]]]]]]]]]]]]
]
119873 =
[[[[[[[[
[
0 0 0 0 sdot sdot sdot 0 0
1199012119902211990120 sdot sdot sdot 0 0
0 1199012119902211990120 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119901
211990221199012
0 0 sdot sdot sdot sdot sdot sdot 0 0 0
]]]]]]]]
]
(20)
where1199010= 12058511198861+ 12058521198863 119902
0= 12058511198861
11990100= 12058511198861+ 12058521198864 119901
1= 21198861minus 11989612057221198865
1199021= 21198862minus 11989612057221198866 119901
2= 21198861+ 11989612057221198865
1199022= 21198862+ 11989612057221198866
1205721(119905119899+1) = int
119871
0
1198922(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
1(119905119899+1)
1205731(119905119899+1) = int
119871
0
1198923(119909) 119906 (119909 119905
119899+1) 119889119909 + ℎ
2(119905119899+1)
(21)
Thus the system (18) becomes a matrix system of dimension(119873 + 3) times (119873 + 3) which is a tridiagonal system that can besolved by theThomas Algorithm [16]
31 Initial State Vector 1198620 After the initial vectors 1198620 havebeen computed from the initial constraints the approximatesolutions 119880119899+1
119894at a particular time level can be calculated
repeatedly by solving the recurrence relation (15) [40]The initial vectors 1198620 can be obtained from the initial
condition and boundary values of the derivatives of the initialcondition as follows [40 49]
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 0
1198800
119894= 1198921(119909119894) 119894 = 0 1 2 119873
(1198800
119894)119909= 1198921015840
1(119909119894) 119894 = 119873
(22)
Thus (22) yields a (119873+3)times(119873+3)matrix system of the form
119860119862119899= 119889 (23)
where
119862119899= [119862119899
minus3 119862119899
minus2 119862119899
minus1 119862
119899
119873minus1]119879
119889 = [1198921015840
1(1199090) 1198921(1199090) 119892
1(119909119873) 1198921015840
1(119909119873)]119879
119899 = 0
119860 =
[[[[[[[[[[[[[[
[
11988630 11988640 sdot sdot sdot 0 0
1198861119886211988610 sdot sdot sdot 0 0
0 1198861119886211988610 sdot sdot sdot 0
0 0 sdot sdot sdot sdot sdot sdot 119886
111988621198861
0 0 sdot sdot sdot sdot sdot sdot 11988630 1198864
]]]]]]]]]]]]]]
]
(24)
The solution of this system can be found by the use of theThomas Algorithm
4 Stability
In this section the von Neumann stability method is appliedfor investigating the stability of the proposed scheme This
Abstract and Applied Analysis 5
approach has been used by many researchers [18 40 4445 47ndash49] Substituting the approximate solution 119880 andits derivatives at knots with 119902(119909 119905) = 0 into (6) yields adifference equation with variables 119862
119898given by
(1198861minus 119896120579120572
21198865) 119862119899+1
119898minus3+ (1198862minus 119896120579120572
21198866) 119862119899+1
119898minus2
+ (1198861minus 119896120579120572
21198865) 119862119899+1
119898minus1
= (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus3+ (1198862+ 119896 (1 minus 120579) 120572
21198866) 119862119899
119898minus2
+ (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus1
(25)
Substituting the values of 119886119894 119894 = 1 2 5 6 into (25) we obtain
(1199011minus 119896212057912057221199012) 119862119899+1
119898minus3+ (1199013+ 119896212057912057221199014) 119862119899+1
119898minus2
+ (1199011minus 119896212057912057221199012) 119862119899+1
119898minus1
= (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus3
+ (1199013minus 1198962(1 minus 120579) 120572
21199014) 119862119899
119898minus2
+ (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus1
(26)
where
1199011= 16 sin2 (ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
times (1 + 2 cos(ℎ2))
1199012= 3 sin (ℎ) sin(3ℎ
2)(1 + 3 cos(ℎ
2)) cos ec2 (ℎ
2)
times (1 + 2 cos(ℎ2))
1199013= 32 sin (ℎ) sin(3ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
1199014= 24 cot2 (ℎ
2) sin ℎ sin(3ℎ
2)(2 cos(ℎ
2)+cos(3ℎ
2))
(27)
Simplifying it leads to
1199081119862119899+1
119898minus3+ 1199082119862119899+1
119898minus2+ 1199081119862119899+1
119898minus1
= 1199083119862119899
119898minus3+ 1199084119862119899
119898minus2+ 1199083119862119899
119898minus1
(28)
where
1199081= (1199011minus 119896120579120572
21199012)
1199082= (1199013+ 119896120579120572
21199014)
1199083= (1199011+ 119896 (1 minus 120579) 120572
21199012)
1199084= (1199013minus 119896 (1 minus 120579) 120572
21199014)
(29)
Now on inserting the trial solutions (one Fourier mode outof the full solution) at a given point 119909
119898is 119862119899119898= 120575119899 exp(119894119898120578ℎ)
into (28) and rearranging the equations 120578 is the modenumber ℎ is the element size and 119894 = radicminus1 we get
1199081120575119899+1119890119894120578(119898minus3)ℎ
+ 1199082120575119899+1119890119894120578(119898minus2)ℎ
+ 1199081120575119899+1119890119894120578(119898minus1)ℎ
= 1199083120575119899119890119894120578(119898minus3)ℎ
+ 1199084120575119899119890119894120578(119898minus2)ℎ
+ 1199083120575119899119890119894120578(119898minus1)ℎ
(30)
Dividing (30) by 120575119899119890119894120578(119898minus2)ℎ and rearranging the equation weget
120575 (1199082+ 1199081cos (120578ℎ)) = (119908
4+ 1199083cos (120578ℎ)) (31)
Let
119860 = 1199013+ 1199011cos (120578ℎ)
119861 = 1198961205722(1199014minus 1199012cos (120578ℎ))
(32)
Therefore (31) can be written as
120575 (119860 + 120579119861) minus (119860 minus (1 minus 120579) 119861) = 0 (33)
Equation (33) can be rewritten as
120575 [1198832+ 119894119884] minus [119883
1+ 119894119884] = 0 (34)
where
1198831= (119860 minus (1 minus 120579) 119861)
1198832= (119860 + 120579119861)
119884 = 0
(35)
For stability the maximummodulus of the eigenvalues of thematrix has to be less than or equal to one [47] Since 119860 gt 0119861 gt 0 and 0 le 120579 le 1 we always have
10038161003816100381610038161205851003816100381610038161003816
2
=1198832
1+ 1198842
1198832
2+ 1198842
le 1 (36)
Thus from (36) the proposed scheme for nonclassical dif-fusion equation (with term 119902(119909 119905) = 0) is unconditionallystable and it is also unconditionally stable with a general term119902(119909 119905) since themodulus of the eigenvaluesmust be less thanone [47] We recall Duhamelrsquos principle ([15] chapter 9) ascheme is stable for equation 119875
119896ℎV = 119891 if it is stable for the
equation 119875119896ℎV = 0 This means that there are no constraints
on grid sizeℎ and step size in time level 119896 butwe should preferthose values of ℎ and 119896 for which we obtain the best accuracyof the scheme
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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Differential EquationsInternational Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
approach has been used by many researchers [18 40 4445 47ndash49] Substituting the approximate solution 119880 andits derivatives at knots with 119902(119909 119905) = 0 into (6) yields adifference equation with variables 119862
119898given by
(1198861minus 119896120579120572
21198865) 119862119899+1
119898minus3+ (1198862minus 119896120579120572
21198866) 119862119899+1
119898minus2
+ (1198861minus 119896120579120572
21198865) 119862119899+1
119898minus1
= (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus3+ (1198862+ 119896 (1 minus 120579) 120572
21198866) 119862119899
119898minus2
+ (1198861+ 119896 (1 minus 120579) 120572
21198865) 119862119899
119898minus1
(25)
Substituting the values of 119886119894 119894 = 1 2 5 6 into (25) we obtain
(1199011minus 119896212057912057221199012) 119862119899+1
119898minus3+ (1199013+ 119896212057912057221199014) 119862119899+1
119898minus2
+ (1199011minus 119896212057912057221199012) 119862119899+1
119898minus1
= (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus3
+ (1199013minus 1198962(1 minus 120579) 120572
21199014) 119862119899
119898minus2
+ (1199011+ 1198962(1 minus 120579) 120572
21199012) 119862119899
119898minus1
(26)
where
1199011= 16 sin2 (ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
times (1 + 2 cos(ℎ2))
1199012= 3 sin (ℎ) sin(3ℎ
2)(1 + 3 cos(ℎ
2)) cos ec2 (ℎ
2)
times (1 + 2 cos(ℎ2))
1199013= 32 sin (ℎ) sin(3ℎ
2)(2 cos(ℎ
2) + cos(3ℎ
2))
1199014= 24 cot2 (ℎ
2) sin ℎ sin(3ℎ
2)(2 cos(ℎ
2)+cos(3ℎ
2))
(27)
Simplifying it leads to
1199081119862119899+1
119898minus3+ 1199082119862119899+1
119898minus2+ 1199081119862119899+1
119898minus1
= 1199083119862119899
119898minus3+ 1199084119862119899
119898minus2+ 1199083119862119899
119898minus1
(28)
where
1199081= (1199011minus 119896120579120572
21199012)
1199082= (1199013+ 119896120579120572
21199014)
1199083= (1199011+ 119896 (1 minus 120579) 120572
21199012)
1199084= (1199013minus 119896 (1 minus 120579) 120572
21199014)
(29)
Now on inserting the trial solutions (one Fourier mode outof the full solution) at a given point 119909
119898is 119862119899119898= 120575119899 exp(119894119898120578ℎ)
into (28) and rearranging the equations 120578 is the modenumber ℎ is the element size and 119894 = radicminus1 we get
1199081120575119899+1119890119894120578(119898minus3)ℎ
+ 1199082120575119899+1119890119894120578(119898minus2)ℎ
+ 1199081120575119899+1119890119894120578(119898minus1)ℎ
= 1199083120575119899119890119894120578(119898minus3)ℎ
+ 1199084120575119899119890119894120578(119898minus2)ℎ
+ 1199083120575119899119890119894120578(119898minus1)ℎ
(30)
Dividing (30) by 120575119899119890119894120578(119898minus2)ℎ and rearranging the equation weget
120575 (1199082+ 1199081cos (120578ℎ)) = (119908
4+ 1199083cos (120578ℎ)) (31)
Let
119860 = 1199013+ 1199011cos (120578ℎ)
119861 = 1198961205722(1199014minus 1199012cos (120578ℎ))
(32)
Therefore (31) can be written as
120575 (119860 + 120579119861) minus (119860 minus (1 minus 120579) 119861) = 0 (33)
Equation (33) can be rewritten as
120575 [1198832+ 119894119884] minus [119883
1+ 119894119884] = 0 (34)
where
1198831= (119860 minus (1 minus 120579) 119861)
1198832= (119860 + 120579119861)
119884 = 0
(35)
For stability the maximummodulus of the eigenvalues of thematrix has to be less than or equal to one [47] Since 119860 gt 0119861 gt 0 and 0 le 120579 le 1 we always have
10038161003816100381610038161205851003816100381610038161003816
2
=1198832
1+ 1198842
1198832
2+ 1198842
le 1 (36)
Thus from (36) the proposed scheme for nonclassical dif-fusion equation (with term 119902(119909 119905) = 0) is unconditionallystable and it is also unconditionally stable with a general term119902(119909 119905) since themodulus of the eigenvaluesmust be less thanone [47] We recall Duhamelrsquos principle ([15] chapter 9) ascheme is stable for equation 119875
119896ℎV = 119891 if it is stable for the
equation 119875119896ℎV = 0 This means that there are no constraints
on grid sizeℎ and step size in time level 119896 butwe should preferthose values of ℎ and 119896 for which we obtain the best accuracyof the scheme
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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
6 Abstract and Applied Analysis
Table 2 Absolute errors for Example 1 with ℎ = 005 and several values of time step by present method (CuTBSM) and compare with TMOL[10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method
TMOL [10] TMOL [10] TMOL [10]01 45119864 minus 04 431119864 minus 04 40119864 minus 05 196119864 minus 04 15119864 minus 05 825119864 minus 06
03 14119864 minus 03 588119864 minus 04 14119864 minus 04 269119864 minus 04 25119864 minus 05 133119864 minus 05
05 25119864 minus 03 520119864 minus 04 25119864 minus 04 238119864 minus 04 24119864 minus 05 120119864 minus 05
07 40119864 minus 03 423119864 minus 04 40119864 minus 04 192119864 minus 04 27119864 minus 05 836119864 minus 06
09 55119864 minus 03 332119864 minus 04 55119864 minus 04 150119864 minus 04 60119864 minus 05 403119864 minus 06
10 60119864 minus 04 291119864 minus 04 60119864 minus 04 131119864 minus 04 68119864 minus 05 190119864 minus 06
5 Results and Discussions
In this section the cubic trigonometric B-spline collocationmethod is employed to obtain the numerical solutions forone-dimensional nonclassical diffusion problem with non-local boundary constraints given in (1)ndash(3) Two numericalexamples are discussed in this section to exhibit the capabilityand efficiency of the proposed trigonometric spline methodNumerical results are compared with existing methods in theliterature and with the exact solution at the different nodalpoints 119909
119894for some time levels 119905
119899using some particular space
step size ℎ and time step 119896 In order to calculate themaximumerrors and relative 119871
2error norms of the proposed method
numerically we use the following formulas
119871infin= max119894
1003816100381610038161003816119880num (119909119894 119879) minus 119906exact (119909119894 119879)1003816100381610038161003816
1198712=
1003816100381610038161003816119880num (119909 119905) minus 119906exact (119909 119905)1003816100381610038161003816
1003816100381610038161003816119906exact (119909 119905)1003816100381610038161003816
(37)
Example 1 Consider the nonclassical diffusion problem ((1)ndash(3)) with
119902 (119909 119905) = minus119890minus(119909+sin 119905)
(1 + cos 119905) 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
minus119909 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0
1198922(119909) =
119890
119890 minus 2119909 119892
3(119909) =
2
119890 + sin 1 minus cos 1cos119909
119871 = 1
ℎ119894(119905) (119894 = 1 2) = 0 0 lt 119905 le 1
(38)
This test problem is from Dehghan [8] and Li and Wu [10]and the known solution is 119906(119909 119905) = 119890minus(119909+sin 119905)We compare themaximum errors with TMOL [10] when they are consideredwith space size ℎ = 005 and errors are recorded with severalvalues of time step 119896 given in Table 2 and also shown inFigure 1 It is worth noting that the results obtained usingCuTBSM are more accurate as compared to TMOL [10]We also compare the relative errors of numerical value of119906(06 01) with different space step ℎ = 005 0025 001
Abso
lute
erro
rs
00 02 04 06 08 10
2
4
6
8
10
12
14times10
minus6
x
At t = 01
At t = 03
At t = 05
At t = 07
At t = 09
At t = 10
Figure 1 Absolute errors of numerical values at different time levelsfor Example 1 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
03
04
05
06
07
08
09
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 2 A comparison of numerical solution and known solutionat different time levels for Example 1 with ℎ = 005 and 119896 = 001
and with time step size 119896 = 119904ℎ2 119904 = 1 2 4 5 instead of
119896 = 04ℎ2 which was used in the BTCS [8] the implicit
(3 3) Crandallrsquos formula [8] the 3-point FTCS [8] andthe Dufort-Frankel three-level approach [8] and TMOL[10] and these results are tabulated in Table 3 It is clearlyshown from this table that the obtained results by usingCuTBSM aremore precise as compared tomethods in [8 10]Figure 2 shows the approximate solution and exact solutionfor this example at different time levels with ℎ = 005 and119896 = 001
Abstract and Applied Analysis 7
Table 3 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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 Relative errors of numerical method at 119906 (06 01) for Example 1 with 119896 = 04 ℎ2 and various values of space step
ℎPresent method
(119896 = 5ℎ2)Present method
(119896 = 4ℎ2)Present method
(119896 = 2ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04ℎ2)
005 494119864 minus 04 388119864 minus 04 176119864 minus 04 708119864 minus 05 739119864 minus 06
0025 123119864 minus 04 969119864 minus 05 441119864 minus 05 177119864 minus 05 187119864 minus 06
0010 197119864 minus 05 155119864 minus 05 708119864 minus 06 286119864 minus 06 326119864 minus 07
ℎBTCS [8](119896 = 04ℎ2)
Crandall [8](119896 = 04ℎ2)
FTCS [8](119896 = 04ℎ2)
Dufort-Frankel[8]
(119896 = 04ℎ2)
TMOL [10](119896 = 04ℎ2)
005 63119864 minus 02 39119864 minus 03 64119864 minus 02 68119864 minus 02 28119864 minus 04
0025 15119864 minus 02 24119864 minus 04 16119864 minus 02 17119864 minus 02 71119864 minus 05
0010 40119864 minus 03 15119864 minus 05 41119864 minus 03 41119864 minus 03 mdash
Table 4 Absolute errors for Example 2 with ℎ = 005 and several values of time step by present method and TMOL [10]
119905119896 = 001 Present method 119896 = 0005 Present method 119896 = 0001 Present method 119896 = 00005 Present method
TMOL [10] TMOL [10] TMOL [10] TMOL [10]01 70119864 minus 04 180119864 minus 03 16119864 minus 04 866119864 minus 04 80119864 minus 05 115119864 minus 04 40119864 minus 05 231119864 minus 05
03 90119864 minus 04 177119864 minus 03 18119864 minus 04 844119864 minus 04 10119864 minus 05 995119864 minus 05 50119864 minus 05 809119864 minus 06
05 70119864 minus 04 116119864 minus 03 14119864 minus 04 548119864 minus 04 70119864 minus 05 549119864 minus 05 35119864 minus 05 827119864 minus 06
07 45119864 minus 04 756119864 minus 04 90119864 minus 05 351119864 minus 04 45119864 minus 05 279119864 minus 05 24119864 minus 05 131119864 minus 06
09 30119864 minus 04 509119864 minus 04 60119864 minus 05 233119864 minus 04 30119864 minus 05 133119864 minus 05 14119864 minus 05 146119864 minus 06
10 24119864 minus 04 425119864 minus 04 45119864 minus 05 193119864 minus 04 24119864 minus 05 884119864 minus 06 12119864 minus 05 146119864 minus 06
Table 5 Absolute errors of numerical value of 119906 for Example 2 with different space step size 119896 = 00001 at selected time levels and comparewith CSCM [11]
119905ℎ = 025 Present method ℎ = 0125 Present method ℎ = 00625 Present methodCSCM [11] CSCM [11] CSCM [11]
01 107119864 minus 02 191119864 minus 03 180119864 minus 03 450119864 minus 04 322119864 minus 04 977119864 minus 05
05 860119864 minus 03 173119864 minus 03 160119864 minus 03 421119864 minus 04 277119864 minus 04 958119864 minus 05
06 790119864 minus 03 152119864 minus 03 140119864 minus 03 370119864 minus 04 253119864 minus 04 849119864 minus 05
09 570119864 minus 03 106119864 minus 03 100119864 minus 03 258119864 minus 04 184119864 minus 04 604119864 minus 05
10 510119864 minus 03 950119864 minus 04 926119864 minus 04 232119864 minus 04 164119864 minus 04 544119864 minus 05
Example 2 We consider another numerical test problemwith
119902 (119909 119905) =
minus2 (1199092+ 119905 + 1)
(119905 + 1)3
0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119909
2 0 le 119909 le 1
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) = 119909
1198923(119909) = 119909 119871 = 1 0 le 119909 le 1
ℎ1(119905) =
minus1
4(119905 + 1)2
ℎ2(119905) =
3
4(119905 + 1)2
0 lt 119905 le 1
(39)
The exact solution of this equation is 119906(119909 119905) = 1199092(119905 + 1)
2
and this test problem has been taken from [8 10 11] Thisproblem is tested using different values of ℎ and 119896 to showthe capability of the present method for solving nonclassicaldiffusion equation ((1)ndash(3)) The final time is taken 119879 = 10Themaximum errors of the numerical method are calculatedat different time levels with different time step size and it isobserved that they are more accurate as compared to TMOL[10] and Chebyshev spectral collocation method (CSCM)based on Chebyshev polynomials [11] The numerical errorsare tabulated in Tables 4 and 5 and are also depictedgraphically in Figure 3The relative errors of numerical value119906(06 10) with different space step ℎ = 005 0025 001 andwith time step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 instead of 119896 =04ℎ2 which was used in the BTCS [8] the implicit (3 3)
Crandallrsquos formula [8] the 3-point FTCS [8] and the Dufort-Frankel three-level approach [8] and TMOL [10] and they arerecorded in Table 6 It is worth noting that numerical resultsare much better than the methods in [8 10 11] A comparison
8 Abstract and Applied Analysis
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
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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
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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
Table 6 Relative errors of numerical method at 119906 (06 10) for Example 2 with various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 528119864 minus 04 415119864 minus 04 189119864 minus 04 781119864 minus 05 884119864 minus 06
0025 114119864 minus 04 896119864 minus 05 474119864 minus 05 191119864 minus 05 221119864 minus 06
0010 211119864 minus 05 166119864 minus 05 759119864 minus 06 306119864 minus 06 353119864 minus 07
ℎBTCS [8](119896 = 04 ℎ2)
Crandall [8](119896 = 04 ℎ2)
FTCS [8](119896 = 04 ℎ2)
Dufort-Frankel [8](119896 = 04 ℎ2)
TMOL [10](119896 = 04 ℎ2)
005 73119864 minus 02 38119864 minus 03 75119864 minus 02 78119864 minus 02 16119864 minus 03
0025 18119864 minus 02 21119864 minus 04 19119864 minus 02 19119864 minus 02 40119864 minus 04
0010 44119864 minus 03 12119864 minus 05 40119864 minus 03 39119864 minus 03 mdash
02 04 06 08 10
000002
000004
000006
000008
00001At t = 01
At t = 03
At t = 05
At t = 07
At t = 09At t = 10
x
Abso
lute
erro
rs
Figure 3 Absolute errors of numerical values at different time levelsfor Example 2 by using ℎ = 005 and 119896 = 0001
02 04 06 08 10
02
04
06
08
u(xt)
x
Exact and Approx Sol at t = 01
Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10
Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07
Exact and Approx Sol at t = 05
Figure 4 A comparison of numerical solution and known solutionat different time levels for Example 2 with ℎ = 005 and 119896 = 001
of numerical solutions at different time levels with knownsolution is presented graphically in the Figure 4
Example 3 Finally we consider nonclassical diffusion prob-lem ((1)-(3)) with
119902 (119909 119905) =minus119890119909(1 + 119905)
2
(1 + 1199052)2 0 le 119909 le 1 0 lt 119905 le 1
1198921(119909) = 119890
119909 0 le 119909 le 1
02 04 06 08 10
000001
000002
000003
000004
000005
Abso
lute
erro
rs
x
k = 5h2
k = 4h2
k = 2h2
k = h2
k = 04h2
Figure 5 Absolute errors of numerical values at119879 = 1 for Example 3with ℎ = 001 and different time step size
120585119894(119894 = 1 3) = 1 120585
119894(119894 = 2 4) = 0 119892
2(119909) =
(119909 + 1)
119890
1198923(119909) = 0 119871 = 1 0 le 119909 le 1
ℎ1(119905) = 0 ℎ
2(119905) =
119890
(1 + 1199052)0 lt 119905 le 1
(40)
This test problem has been taken from [15] and its exactsolution is 119906(119909 119905) = 119890
119909(1 + 119905
2) The final time is taken
as 119879 = 1 The maximum errors of the proposed schemeare considered at 119879 = 1 with different time step sizes thatare depicted graphically in Figure 5 The relative errors ofnumerical value 119906(06 10) are calculated with different spacesize step with 119896 = 119904ℎ
2 119904 = 1 2 4 5 and they are given inTable 7 It is worth noting that the numerical results are foundto be in good agreement with exact solutions A comparisonof numerical solutions at different time levels with knownsolution is presented graphically in Figure 6
6 Concluding Remarks
In this paper a new two-time level implicit scheme basedon cubic trigonometric B-spline has been used to solve thenonclassical diffusion problem with known initial and withnonlocal boundary constraints instead of the usual boundary
Abstract and Applied Analysis 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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 9
Table 7 Relative errors of numerical method at 119906 (06 10) for Example 3 with different time step size and various values of space step
ℎPresent method
(119896 = 5 ℎ2)Present method
(119896 = 4 ℎ2)Present method
(119896 = 2 ℎ2)
Present method(119896 = ℎ
2)Present method(119896 = 04 ℎ2)
005 121119864 minus 03 945119864 minus 04 423119864 minus 04 162119864 minus 04 607119864 minus 06
0025 301119864 minus 04 235119864 minus 04 105119864 minus 04 405119864 minus 05 151119864 minus 06
0010 481119864 minus 05 377119864 minus 05 169119864 minus 05 649119864 minus 06 241119864 minus 07
02 04 06 08 10
10
15
20
25
u(xt)
x
Exact and Approx Sol at t = 01Exact and Approx Sol at t = 03
Exact and Approx Sol at t = 10Exact and Approx Sol at t = 09
Exact and Approx Sol at t = 07Exact and Approx Sol at t = 05
Figure 6 A comparison of numerical solution and known solutionat different time levels for Example 3 with ℎ = 005 and 119896 = 001
constraints A usual finite difference discretization is usedfor time derivatives and cubic trigonometric B-spline isapplied for space derivatives It is noted that the accuracyof solution may reduce as time increases due to the timetruncation errors of time derivative term [47] The cubictrigonometric B-spline method used in this paper is simpleand straightforward to apply An advantage of using the cubictrigonometric B-spline method outlined in this paper is thatit produces a spline function on each new time line whichcan be used to obtain the solutions at any intermediate pointin the space direction whereas the finite difference approachyields the solution only at the selected points The CuTBSMhas approximated the solution with more accurate results fortime step size 119896 = 119904ℎ
2 119904 = 1 2 4 5 as compared to somefinite difference schemes with smaller time step size 119896 =
04ℎ2 such as BTCS Crandallrsquos formula FTCS the Dufort-
Frankel scheme and TMOL based on reproducing kernelThe proposed method is shown to be unconditionally stableIt is also evident from the examples that the approximatesolution is very close to the exact solution
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the anonymous reviewers fortheir helpful valuable comments and suggestions to improvethe paper This study was fully supported by FRGS Grant
no 203PMATHS6711324 from the School of Mathemat-ical Sciences Universiti Sains Malaysia Penang MalaysiaThe first author was supported by Post Doctorate Fellow-ship from School of Mathematical Sciences Universiti SainsMalaysia Penang Malaysia during a part of the time inwhich the research was carried out
References
[1] Y S Choi and K-Y Chan ldquoA parabolic equation with nonlocalboundary conditions arising from electrochemistryrdquo NonlinearAnalysis Theory Methods amp Applications vol 18 no 4 pp 317ndash331 1992
[2] A Bouziani ldquoOn a class of parabolic equations with a nonlocalboundary conditionrdquo Academie Royale de Belgique Bulletin dela Classe des Sciences 6e Serie vol 10 no 1ndash6 pp 61ndash77 1999
[3] W A Day ldquoParabolic equations and thermodynamicsrdquo Quar-terly of Applied Mathematics vol 50 no 3 pp 523ndash533 1992
[4] A A Samarskiı ldquoSome problems of the theory of differentialequationsrdquo Differential Equations vol 16 no 11 pp 1925ndash19351980
[5] A R Bahadır ldquoApplication of cubic B-spline finite elementtechnique to the termistor problemrdquo Applied Mathematics andComputation vol 149 no 2 pp 379ndash387 2004
[6] J H Cushman B X Hu and F Deng ldquoNonlocal reactivetransport with physical and chemical heterogeneity localiza-tion errorsrdquo Water Resources Research vol 31 no 9 pp 2219ndash2237 1995
[7] F Kanca ldquoThe inverse problem of the heat equation withperiodic boundary and integral overdetermination conditionsrdquoJournal of Inequalities and Applications vol 2013 article 1082013
[8] M Dehghan ldquoEfficient techniques for the second-orderparabolic equation subject to nonlocal specificationsrdquo AppliedNumerical Mathematics vol 52 no 1 pp 39ndash62 2005
[9] J Martın-Vaquero and J Vigo-Aguiar ldquoA note on efficienttechniques for the second-order parabolic equation subject tonon-local conditionsrdquo Applied Numerical Mathematics vol 59no 6 pp 1258ndash1264 2009
[10] X Li and B Wu ldquoNew algorithm for nonclassical parabolicproblems based on the reproducing kernel methodrdquoMathemat-ical Sciences vol 7 article 4 2013
[11] A Golbabai and M Javidi ldquoA numerical solution for non-classical parabolic problem based on Chebyshev spectral col-location methodrdquo Applied Mathematics and Computation vol190 no 1 pp 179ndash185 2007
[12] W A Day ldquoA decreasing property of solutions of parabolicequations with applications to thermoelasticityrdquo Quarterly ofApplied Mathematics vol 40 no 4 pp 468ndash475 1983
10 Abstract and Applied Analysis
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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
[13] W A Day ldquoExtensions of a property of the heat equation tolinear thermoelasticity and other theoriesrdquoQuarterly of AppliedMathematics vol 40 no 3 pp 319ndash330 1982
[14] W A DayHeat Conduction within LinearThermoelasticity vol30 of Springer Tracts in Natural Philosophy Springer New YorkNY USA 1985
[15] J C Strikwerda Finite Difference Schemes and Partial Differen-tial Equations Society for Industrial and Applied MathematicsPhiladelphia Pa USA 2nd edition 2004
[16] Du V Rosenberg Methods for Solution of Partial DifferentialEquations vol 113 American Elsevier New York NY USA1969
[17] W T Ang ldquoA method of solution for the one-dimensionalheat equation subject to nonlocal conditionsrdquo Southeast AsianBulletin of Mathematics vol 26 no 2 pp 197ndash203 2002
[18] J Martın-Vaquero and J Vigo-Aguiar ldquoOn the numericalsolution of the heat conduction equations subject to nonlocalconditionsrdquo Applied Numerical Mathematics vol 59 no 10 pp2507ndash2514 2009
[19] M Dehghan ldquoOn the numerical solution of the diffusionequation with a nonlocal boundary conditionrdquo MathematicalProblems in Engineering vol 2 pp 81ndash92 2003
[20] M Dehghan ldquoA computational study of the one-dimensionalparabolic equation subject to nonclassical boundary specifica-tionsrdquoNumericalMethods for Partial Differential Equations vol22 no 1 pp 220ndash257 2006
[21] M Tatari and M Dehghan ldquoOn the solution of the non-local parabolic partial differential equations via radial basisfunctionsrdquo Applied Mathematical Modelling vol 33 no 3 pp1729ndash1738 2009
[22] J H Cushman ldquoDiffusion in fractal porous mediardquo WaterResources Research vol 27 no 4 pp 643ndash644 1991
[23] V V Shelukhin ldquoA non-local in time model for radionu-clides propagation in Stokes fluidrdquo Siberian Branch of RussianAcademy of Sciences Institute of Hydrodynamics no 107 pp180ndash193 1993
[24] A S Vasudeva Murthy and J G Verwer ldquoSolving parabolicintegro-differential equations by an explicit integrationmethodrdquo Journal of Computational and Applied Mathematicsvol 39 no 1 pp 121ndash132 1992
[25] C V Pao ldquoNumerical methods for nonlinear integro-parabolicequations of Fredholm typerdquo Computers amp Mathematics withApplications vol 41 no 7-8 pp 857ndash877 2001
[26] C V Pao ldquoReaction diffusion equations with nonlocal bound-ary and nonlocal initial conditionsrdquo Journal of MathematicalAnalysis and Applications vol 195 no 3 pp 702ndash718 1995
[27] J R Cannon and A L Matheson ldquoA numerical procedurefor diffusion subject to the specification of massrdquo InternationalJournal of Engineering Science vol 31 no 3 pp 347ndash355 1993
[28] M Dehghan ldquoNumerical solution of a parabolic equation withnon-local boundary specificationsrdquo Applied Mathematics andComputation vol 145 no 1 pp 185ndash194 2003
[29] G Ekolin ldquoFinite difference methods for a nonlocal boundaryvalue problem for the heat equationrdquoBITNumericalMathemat-ics vol 31 no 2 pp 245ndash261 1991
[30] G Fairweather and J C Lopez-Marcos ldquoGalerkin methodsfor a semilinear parabolic problem with nonlocal boundaryconditionsrdquoAdvances in ComputationalMathematics vol 6 no3-4 pp 243ndash262 1996
[31] Y Liu ldquoNumerical solution of the heat equation with nonlocalboundary conditionsrdquo Journal of Computational and AppliedMathematics vol 110 no 1 pp 115ndash127 1999
[32] G E Farin Curves and Surfaces for Computer-Aided GeometricDesign A Practical Code Academic Press 1996
[33] J Hoschek and D Lasser Fundamentals of Computer AidedGeometric Design A K Peters Boston Mass USA 1993
[34] Y S Lai W P Du and R H Wang ldquoThe viro method forconstruction of piecewise algebraic hypersurfacesrdquoAbstract andApplied Analysis vol 2013 Article ID 690341 7 pages 2013
[35] RHWangXQ Shi Z X Luo andZX SuMultivariate Splineand Its Applications Science Press Beijing China KluwerAcademic Publishers New York NY USA 2001
[36] P M Prenter Splines and Variational Methods John Wiley ampSons New York NY USA 1989
[37] C de Boor A Practical Guide to Splines vol 27 of AppliedMathematical Sciences Springer 1978
[38] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of third-order boundary-value problems with fourth-degree 119861-spline functionsrdquo International Journal of ComputerMathematics vol 71 no 3 pp 373ndash381 1999
[39] H N Caglar S H Caglar and E H Twizell ldquoThe numericalsolution of fifth-order boundary value problems with sixth-degree 119861-spline functionsrdquoApplied Mathematics Letters vol 12no 5 pp 25ndash30 1999
[40] I Dag D Irk and B Saka ldquoA numerical solution of theBurgersrsquo equation using cubic B-splinesrdquo Applied Mathematicsand Computation vol 163 no 1 pp 199ndash211 2005
[41] J Rashidinia M Ghasemi and R Jalilian ldquoA collocationmethod for the solution of nonlinear one-dimensional para-bolic equationsrdquo Mathematical Sciences Quarterly Journal vol4 no 1 pp 87ndash104 2010
[42] K Qu ZWang and B Jiang ldquoA finite elementmethod by usingbivariate splines for one dimensional heat equationsrdquo Journal ofInformation amp Computational Science vol 10 no 12 pp 3659ndash3666 2013
[43] J Goh A A Majid and A I M Ismail ldquoNumerical methodusing cubic B-spline for the heat andwave equationrdquoComputersand Mathematics with Applications vol 62 no 12 pp 4492ndash4498 2011
[44] J Goh A A Majid and A I M Ismail ldquoCubic B-spline collo-cation method for one-dimensional heat and advection-diffu-sion equationsrdquo Journal of Applied Mathematics vol 2012Article ID 458701 8 pages 2012
[45] J Goh A A Majid and A I M Ismail ldquoA comparisonof some splines-based methods for the one-dimensional heatequationrdquo Proceedings ofWorld Academy of Science Engineeringand Technology vol 70 pp 858ndash861 2010
[46] R C Mittal and G Arora ldquoQuintic B-spline collocationmethod for numerical solution of the Kuramoto-Sivashinskyequationrdquo Communications in Nonlinear Science and NumericalSimulation vol 15 no 10 pp 2798ndash2808 2010
[47] R C Mittal and G Arora ldquoNumerical solution of the coupledviscous Burgersrsquo equationrdquo Communications in Nonlinear Sci-ence andNumerical Simulation vol 16 no 3 pp 1304ndash1313 2011
[48] M Abbas A A Majid A I M Ismail and A RashidldquoNumerical method using cubic B-spline for a strongly coupledreaction-diffusion systemrdquo PLoS ONE vol 9 no 1 articlee83265 2014
[49] M Abbas A A Majid A I M Ismail and A Rashid ldquoTheapplication of cubic trigonometric B-spline to the numericalsolution of the hyperbolic problemsrdquo Applied Mathematics andComputation vol 239 pp 74ndash88 2014
Abstract and Applied Analysis 11
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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
[50] A Nikolis ldquoNumerical solutions of ordinary differential equa-tions with quadratic trigonometric splinesrdquo Applied Mathemat-ics E-Notes vol 4 pp 142ndash149 1995
[51] N N Abd Hamid A A Majid and A I M Ismail ldquoCubictrigonometric B-spline applied to linear two-point boundaryvalue problems of order twordquoWorldAcademy of Science Engine-ering and Technology vol 70 pp 798ndash803 2010
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
![C Rational Cubic/Linear Trigonometric Interpolation Spline ... · preserving interpolation surfaces developed in [21], [22], [23] were based on the claim given in [24]: bi-cubic partially](https://static.fdocuments.net/doc/165x107/5f1f49d4d22078629c51e4b0/c-rational-cubiclinear-trigonometric-interpolation-spline-preserving-interpolation.jpg)
