Research Article A New Legendre Collocation Method for...

11
Research Article A New Legendre Collocation Method for Solving a Two-Dimensional Fractional Diffusion Equation A. H. Bhrawy 1,2 1 Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia 2 Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef 62511, Egypt Correspondence should be addressed to A. H. Bhrawy; [email protected] Received 11 March 2014; Accepted 5 April 2014; Published 4 May 2014 Academic Editor: Dumitru Baleanu Copyright © 2014 A. H. Bhrawy. 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 spectral shiſted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two- dimensional initial-boundary fractional diffusion equations with variable coefficients. e method depends basically on the fact that an expansion in a series of shiſted Legendre polynomials , () , (), for the function and its space-fractional derivatives occurring in the partial fractional differential equation (PFDE), is assumed; the expansion coefficients are then determined by reducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for these coefficients. is system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method. is method, in contrast to common finite-difference and finite-element methods, has the exponential rate of convergence for the two spatial discretizations. Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtained by other methods and with the exact solutions more easier. 1. Introduction In recent years there has been a high level of interest of employing spectral methods for numerically solving many types of integral and differential equations, due to their ease of applying them for finite and infinite domains [1–8]. e speed of convergence is one of the great advantages of spectral method. Besides, spectral methods have exponential rates of convergence; they also have high level of accuracy. From the overview of spectral approximation to differential equations, the spectral methods have been divided to four types, namely, collocation [9–11], tau [12, 13], Galerkin [14, 15], and Petrov Galerkin [16, 17] methods. e main idea of all versions of spectral methods is to express the spectral solution of the problem as a finite sum of certain basis functions (combination of orthogonal polynomials or functions) and then to choose the coefficients in order to minimize the difference between the exact and numerical solutions as possible. e spectral collocation method is a specific type of spectral methods, which is more applicable and widely used to solve almost types of differential equations [18–21]. Fractional differential equations [22–31] model many phenomena in several fields such as biology, viscoelasticity, finance, fluid mechanics, physics, chemistry, and engineer- ing. Most fractional differential equations do not have exact solutions, so approximation and numerical techniques must be used. Here, we focus on the application of SL-GL-C scheme for the numerical solution of the time dependent frac- tional diffusion equation in two dimensions. is problem models phenomena exhibiting [32–34] anomalous diffusion that cannot be modeled accurately by the classical two- dimensional diffusion equations with variable coefficients. For the previous numerical methods used for time dependent fractional diffusion equation, see, for example, the papers in [35–40]. e solution of this equation is approximated as , which can be expressed as a finite expansion of shiſted Leg- endre polynomials for the space variables and then evaluate the spatial fractional partial derivatives of the finite expansion of , at the shiſted Legendre-Gauss-Lobatto (SL-GL) quadrature points. Substituting these approximations in the underlined PFDE provides a SODEs in time. is system Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2014, Article ID 636191, 10 pages http://dx.doi.org/10.1155/2014/636191

Transcript of Research Article A New Legendre Collocation Method for...

Page 1: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Research ArticleA New Legendre Collocation Method for Solving aTwo-Dimensional Fractional Diffusion Equation

A H Bhrawy12

1 Department of Mathematics Faculty of Science King Abdulaziz University Jeddah 21589 Saudi Arabia2Department of Mathematics Faculty of Science Beni-Suef University Beni-Suef 62511 Egypt

Correspondence should be addressed to A H Bhrawy alibhrawyyahoocouk

Received 11 March 2014 Accepted 5 April 2014 Published 4 May 2014

Academic Editor Dumitru Baleanu

Copyright copy 2014 A H BhrawyThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

A new spectral shifted Legendre Gauss-Lobatto collocation (SL-GL-C) method is developed and analyzed to solve a class of two-dimensional initial-boundary fractional diffusion equations with variable coefficients The method depends basically on the factthat an expansion in a series of shifted Legendre polynomials 119875

119871119899(119909)119875119871119898

(119910) for the function and its space-fractional derivativesoccurring in the partial fractional differential equation (PFDE) is assumed the expansion coefficients are then determined byreducing the PFDE with its boundary and initial conditions to a system of ordinary differential equations (SODEs) for thesecoefficients This system may be solved numerically by using the fourth-order implicit Runge-Kutta (IRK) method This methodin contrast to common finite-difference and finite-element methods has the exponential rate of convergence for the two spatialdiscretizations Numerical examples are presented in the form of tables and graphs to make comparisons with the results obtainedby other methods and with the exact solutions more easier

1 Introduction

In recent years there has been a high level of interest ofemploying spectral methods for numerically solving manytypes of integral and differential equations due to theirease of applying them for finite and infinite domains [1ndash8]The speed of convergence is one of the great advantages ofspectral method Besides spectral methods have exponentialrates of convergence they also have high level of accuracyFrom the overview of spectral approximation to differentialequations the spectral methods have been divided to fourtypes namely collocation [9ndash11] tau [12 13] Galerkin [14 15]and Petrov Galerkin [16 17] methods The main idea of allversions of spectralmethods is to express the spectral solutionof the problem as a finite sum of certain basis functions(combination of orthogonal polynomials or functions) andthen to choose the coefficients in order to minimize thedifference between the exact and numerical solutions aspossible The spectral collocation method is a specific type ofspectral methods which is more applicable and widely usedto solve almost types of differential equations [18ndash21]

Fractional differential equations [22ndash31] model manyphenomena in several fields such as biology viscoelasticityfinance fluid mechanics physics chemistry and engineer-ing Most fractional differential equations do not have exactsolutions so approximation and numerical techniques mustbe usedHere we focus on the application of SL-GL-C schemefor the numerical solution of the time dependent frac-tional diffusion equation in two dimensions This problemmodels phenomena exhibiting [32ndash34] anomalous diffusionthat cannot be modeled accurately by the classical two-dimensional diffusion equations with variable coefficientsFor the previous numericalmethods used for time dependentfractional diffusion equation see for example the papers in[35ndash40]

The solution 119906 of this equation is approximated as 119906119873119872

which can be expressed as a finite expansion of shifted Leg-endre polynomials for the space variables and then evaluatethe spatial fractional partial derivatives of the finite expansionof 119906119873119872

at the shifted Legendre-Gauss-Lobatto (SL-GL)quadrature points Substituting these approximations in theunderlined PFDE provides a SODEs in time This system

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 636191 10 pageshttpdxdoiorg1011552014636191

2 Abstract and Applied Analysis

may be solved by IRK scheme This scheme is one of thesuitable schemes for solving SODEs of first order Since theIRK scheme has excellent stability properties then it allowsus to reduce computational costs Indeed this is the first workfor employing SL-GL-C scheme to solve two-dimensionalfractional diffusion equations

This paper is organized as follows We present fewrelevant properties of fractional integration in the Riemann-Liouville and shifted Legendre polynomials in the comingsection Section 3 presents the numerical solution of thetwo-dimensional fractional diffusion equations with initial-boundary conditions Section 4 is devoted to solve two testexamples Finally some concluding remarks are given in thelast section

2 Preliminaries and Notation

21 The Fractional Integration in the Riemann-Liouville SenseThere are several definitions of the fractional integration oforder ] gt 0 and not necessarily equivalent to each othersee [41 42] The most used definition is due to Riemann-Liouville which is defined as

119869]119891 (119909) =

1

Γ (])int

119909

0

(119909 minus 119905)]minus1

119891 (119905) 119889119905 ] gt 0 119909 gt 0

1198690119891 (119909) = 119891 (119909)

(1)

One of the basic properties of the operator 119869] is

119869]119909120573=

Γ (120573 + 1)

Γ (120573 + 1 + ])119909120573+]

(2)

The Riemann-Liouville fractional derivative of order ] willbe denoted by 119863

] The next relation defines the Riemann-Liouville fractional derivative of order ] as

119863]119891 (119909) =

119889119898

119889119909119898(119869119898minus]

119891 (119909)) (3)

where 119898 minus 1 lt ] le 119898 119898 isin 119873 and 119898 is the smallest integergreater than ]

Lemma 1 If119898 minus 1 lt ] le 119898 119898 isin 119873 then

119863]119869]119891 (119909) = 119891 (119909)

119869]119863

]119891 (119909) = 119891 (119909) minus

119898minus1

sum

119894=0

119891(119894)(0+)119909119894

119894 119909 gt 0

(4)

22 Properties of Shifted Legendre Polynomials The well-known Legendre polynomials 119875

119894(119909) are defined on the inter-

val (minus1 1) Some properties about the standard Legendrepolynomials will be recalled in this section The Legendrepolynomials 119875

119896(119909) (119896 = 0 1 ) satisfy the following

Rodriguersquos formula

119875119896 (119909) =

(minus1)119896

2119896119896119863119896((1 minus 119909

2)119896

) (5)

We recall also that 119875119896(119909) is a polynomial of degree 119896 and

therefore 119875(119902)119896(119909) (the 119902th derivative of 119875

119896(119909)) is

119875(119902)

119896(119909) =

119896minus119902

sum

119894=0(119896+119894=even)119862119902 (119896 119894) 119875119894 (119909) (6)

where

119862119902 (119896 119894) = 2

119902minus1(2119894 + 1) Γ (

119902 + 119896 minus 119894

2) Γ(

119902 + 119896 + 119894 + 1

2)

times (Γ (119902) Γ (2 minus 119902 + 119896 minus 119894

2) Γ(

3 minus 119902 + 119896 + 119894

2))

minus1

(7)

The Legendre polynomials satisfy the following relations

1198750 (119909) = 1 119875

1 (119909) = 119909

119875119896+2 (119909) =

2119896 + 3

119896 + 2119909119875119896+1 (119909) minus

119896 + 1

119896 + 2119875119896 (119909)

(8)

and the orthogonality relation

(119875119896 (119909) 119875119897 (119909)) = int

1

minus1

119875119896 (119909) 119875119897 (119909)119908 (119909) 119889119909 = ℎ

119896120575119897119896 (9)

where 119908(119909) equiv 1 ℎ119896= 2(2119896 + 1) and 120575

119897119896is the Kronecker

delta function The Legendre-Gauss-Lobatto quadrature isused to evaluate the previous integral accurately For any 120601 isin

1198782119873minus1

[minus1 1] we get

int

1

minus1

120601 (119909) 119889119909 =

119873

sum

119895=0

120603119873119895

120601 (119909119873119895

) (10)

We introduce the following discrete inner product

(119906 V)119873 =119873

sum

119895=0

119906 (119909119873119895

) V (119909119873119895

) 120603119873119895

(11)

where 119909119873119895

(0 le 119895 le 119873) and 120603119873119895

(0 le 119895 le 119873) are usedas usual nodes and the corresponding Christoffel numbersin the interval [minus1 1] respectively Let the shifted Legendrepolynomials 119875

119894((2119909119871) minus 1) be denoted by 119875

119871119894(119909) 119909 isin (0 119871)

Then 119875119871119894(119909) can be obtained with the aid of the following

recurrence formula

(119894 + 1) 119875119871119894+1 (119909) = (2119894 + 1) (2119909

119871minus 1)119875

119871119894 (119909)

minus 119894119875119871119894minus1 (119909) 119894 = 1 2

(12)

The analytic form of the shifted Legendre polynomials 119875119871119894(119909)

of degree 119894 is given by

119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119909119896 (13)

Abstract and Applied Analysis 3

The Riemann-Liouville fractional integration of shifted Leg-endre polynomials 119875

119871119894(119909) of degree 119894 is given by

119869]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)119896

(119894 minus 119896)(119896)2119871119896Γ (119896 + ] + 1)

119909119896+]

119894 = 0 1 119873

119863]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)Γ (119896 + 1)

(119894 minus 119896)(119896)2119871119896Γ (119896 + 1 minus ])

119909119896minus]

119894 = 0 1 119873

(14)

where 119875119871119894(0) = (minus1)

119894 The orthogonality condition is

int

119871

0

119875119871119895 (119909) 119875119871119896 (119909) 119908119871 (119909) 119889119909 = ℎ

119871119896120575119895119896 (15)

where 119908119871(119909) equiv 1 and ℎ

119871119896= 119871(2119896 + 1)

The function 119906(119909) square integrable in (0 119871) may beexpressed in terms of shifted Legendre polynomials as

119906 (119909) =

infin

sum

119895=0

119888119895119875119871119895 (119909) (16)

where the coefficients 119888119895are given by

119888119895=

1

ℎ119871119895

int

119871

0

119906 (119909) 119875119871119895 (119909) 119889119909 119895 = 0 1 2 (17)

In practice only the first (119873 + 1)-terms shifted Legendrepolynomials are considered Hence 119906(119909) can be expressed inthe form

119906119873 (119909) ≃

119873

sum

119895=0

119888119895119875119871119895 (119909) (18)

3 Shifted Legendre SpectralCollocation Method

In this section we derive the SL-GL-C method for solvingtwo-dimensional fractional diffusion equation of the form

120597V (119909 119910 119905)120597119905

= 1198921(119909 119910)

120597]1V (119909 119910 119905)120597119909]1

+ 1198922(119909 119910)

120597]2V (119909 119910 119905)120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1198711] times [0 119871

2] times (0 119879]

(19)

which is subject to the initial condition

V (119909 119910 0) = 1198923(119909 119910) (119909 119910) isin [0 119871

1] times [0 119871

2] (20)

and boundary conditions

V (0 119910 119905) = 1198924(119910 119905) V (119871

1 119910 119905) = 119892

5(119910 119905)

(119910 119905) isin [0 1198712] times [0 119879]

V (119909 0 119905) = 1198926 (119909 119905) V (119909 119871

2 119905) = 119892

7 (119909 119905)

(119909 119905) isin [0 1198711] times [0 119879]

(21)

Denote by 119909119873119895

(119909119871119873119895

) 0 ⩽ 119895 ⩽ 119873 and120603119873119895

(120603119871119873119895

) 0 le

119894 le 119873 the nodes and Christoffel numbers of the standard(shifted) Legendre-Gauss-Lobatto quadrature on the inter-vals (minus1 1) (0 119871) respectively Then one can clearly deducethat

119909119871119873119895

=119871

2(119909119873119895

+ 1) 0 le 119895 le 119873

120603119871119873119895

= (119871

2)

120572+120573+1

120603119873119895

0 le 119895 le 119873

(22)

and if 119878119873(0 119871) denotes the set of all polynomials of degree at

most119873 then it follows for any 120601 isin 1198782119873minus1

(0 119871) that

int

119871

0

119908119871 (119909) 120601 (119909) 119889119909 = (

119871

2)int

1

minus1

120601(119871

2(119909 + 1)) 119889119909

= (119871

2)

119873

sum

119895=0

120603119873119895

120601(119871

2(119909119873119895

+ 1))

=

119873

sum

119895=0

120603119871119873119895

120601 (119909119871119873119895

)

(23)

Here the SL-GL-Cmethodwill be employed to transformthe previous two-dimensional fractional diffusion equationinto SODEs Let us expand the dependent variables in theform

V (119909 119910 119905) =119873

sum

119894=0

119872

sum

119895=0

119886119894119895 (119905) 119875119871

1119894 (119909) 119875119871

2119895(119910) (24)

We observe from (15) that the coefficients 119886119894119895(119905) can be

approximated by

119886119894119895 (119905) =

1

ℎ1198711119894ℎ1198712119895

times

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

times V (1199091198711119873119897

1199101198712119872119896

119905)

(25)

4 Abstract and Applied Analysis

Table 1 Maximum absolute errors for problem (40) for 119879 = 1

119873 = 119872SL-GL-C method [43]

119872119905

119864119872119864

Δ119909 = Δ119910 = Δ119905 119872119864

4 341785 times 10minus3

911427 times 10minus3 mdash mdash

6 213706 times 10minus5

683861 times 10minus5 1

5105564 times 10

minus3

8 132866 times 10minus6

485909 times 10minus6 1

10261694 times 10

minus4

10 174023 times 10minus7

707142 times 10minus7 1

20657067 times 10

minus5

12 343827 times 10minus8

152415 times 10minus7 1

40155866 times 10

minus5

Table 2 Maximum absolute errors for problem (40) for 119879 = 1 and119873 = 119872 = 12

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 274515 times 10minus11 01 05 1 917923 times 10

minus12 01 1 1 441521 times 10minus11

02 213241 times 10minus10 02 757278 times 10

minus11 02 358585 times 10minus10

03 722081 times 10minus10 03 261927 times 10

minus10 03 120622 times 10minus9

04 171192 times 10minus9 04 643406 times 10

minus10 04 293613 times 10minus9

05 335365 times 10minus9 05 129893 times 10

minus9 05 58445 times 10minus9

06 580344 times 10minus9 06 233485 times 10

minus9 06 102404 times 10minus8

07 920721 times 10minus9 07 382976 times 10

minus9 07 165193 times 10minus8

08 139884 times 10minus8 08 584963 times 10

minus9 08 250569 times 10minus8

09 206847 times 10minus8 09 85766 times 10

minus9 09 364012 times 10minus8

1 275218 times 10minus8 1 126484 times 10

minus8 1 560815 times 10minus8

Table 3 Maximum absolute errors for problem (46) for 119879 = 1 and]1= ]2= 15

119873 = 119872 119872119905

119864119872119864

6 257399 times 10minus8

102853 times 10minus7

8 151431 times 10minus8

121098 times 10minus7

10 824341 times 10minus14

413336 times 10minus13

The expression (24) can be rewritten as

V (119909 119910 119905)

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909) 119875119871

2119895(119910) V119897119896 (119905)

(26)

where V(1199091198711119873119897

1199101198712119872119896

119905) = V119897119896(119905) In what follows the

approximation of the fractional spatial partial derivative

120597]1V(119909 119910 119905)120597119909]1 with respect to 119909 for V(119909 119910 119905) can be

computed as

120597]1V (119909 119910 119905)120597119909]1

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 119863(]1)(1198751198711119894 (119909)) 119875119871

2119895(119910) V119897119896 (119905)

(27)

Also the fractional spatial partial derivative of fractionalorder 120597]2V(119909 119910 119905)120597119910]

2 with respect to 119910 of the approximatesolution is

120597]2V (119909 119910 119905)120597119910]2

=

119873

sum

119897=0

119872

sum

119896=0

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909)119863

(]2)(1198751198712119895(119910)) V

119897119896 (119905)

(28)

Abstract and Applied Analysis 5

Table 4 Maximum absolute errors for problem (46) for 119879 = 1 and ]1= ]2= 15

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 406522 times 10minus16 01 05 1 705989 times 10

minus15 01 1 1 235055 times 10minus15

02 312066 times 10minus15 02 7445 times 10

minus15 02 1 663792 times 10minus15

03 422817 times 10minus15 03 173386 times 10

minus15 03 1 692588 times 10minus15

04 87365 times 10minus15 04 425354 times 10

minus15 04 1 170948 times 10minus14

05 140339 times 10minus14 05 156351 times 10

minus14 05 1 283974 times 10minus14

06 181075 times 10minus14 06 114631 times 10

minus14 06 1 344343 times 10minus14

07 269246 times 10minus14 07 18454 times 10

minus14 07 1 544946 times 10minus14

08 342964 times 10minus14 08 314054 times 10

minus14 08 1 678416 times 10minus14

09 43951 times 10minus14 09 278319 times 10

minus14 09 1 903513 times 10minus14

1 31379 times 10minus15 1 120095 times 10

minus14 1 1 10103 times 10minus14

The previous fractional derivatives can be computed at theSL-GL interpolation nodes as

(120597]1V (119909 119910 119905)120597119909]1

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(120597]2V (119909 119910 119905)120597119910]2

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(29)

where

120588119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]1)1198751198711119894(1199091198711119873119899

) 1198751198712119895(1199101198712119872119898

)

120582119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]2)1198751198712119895(1199101198712119872119898

) 1198751198711119894(1199091198711119873119899

)

(30)

In the proposed SL-GL-C method the residual of (19) is setto zero at (119873 minus 1) times (119872 minus 1) of SL-GL points Moreover thevalues of V

0119896(119905) V119873119896

(119905) V1198970(119905) and V

119897119872(119905) are given by

V0119896 (119905) = 119892

4(1199101198712119872119896

119905) V119873119896 (119905) = 119892

5(1199101198712119872119896

119905)

119896 = 0 119872

V1198970 (119905) = 119892

6(1199091198711119873119897

119905) V119897119872 (119905) = 119892

7(1199091198711119873119897

119905)

119897 = 0 119873

(31)

Therefore adapting (26)ndash(31) enables one to write (19)ndash(21)as a (119873 minus 1) times (119872 minus 1) SODEs

V119899119898 (119905) = 119892

119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(32)

which is subject to the initial conditions

V119899119898 (0) = 119892

119899119898

3 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(33)

where

1198921(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

1

1198922(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

2

1198923(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

3

119891 (1199091198711119873119899

1199101198712119872119898

119905) = 119891119899119898

(119905)

(34)

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 2: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

2 Abstract and Applied Analysis

may be solved by IRK scheme This scheme is one of thesuitable schemes for solving SODEs of first order Since theIRK scheme has excellent stability properties then it allowsus to reduce computational costs Indeed this is the first workfor employing SL-GL-C scheme to solve two-dimensionalfractional diffusion equations

This paper is organized as follows We present fewrelevant properties of fractional integration in the Riemann-Liouville and shifted Legendre polynomials in the comingsection Section 3 presents the numerical solution of thetwo-dimensional fractional diffusion equations with initial-boundary conditions Section 4 is devoted to solve two testexamples Finally some concluding remarks are given in thelast section

2 Preliminaries and Notation

21 The Fractional Integration in the Riemann-Liouville SenseThere are several definitions of the fractional integration oforder ] gt 0 and not necessarily equivalent to each othersee [41 42] The most used definition is due to Riemann-Liouville which is defined as

119869]119891 (119909) =

1

Γ (])int

119909

0

(119909 minus 119905)]minus1

119891 (119905) 119889119905 ] gt 0 119909 gt 0

1198690119891 (119909) = 119891 (119909)

(1)

One of the basic properties of the operator 119869] is

119869]119909120573=

Γ (120573 + 1)

Γ (120573 + 1 + ])119909120573+]

(2)

The Riemann-Liouville fractional derivative of order ] willbe denoted by 119863

] The next relation defines the Riemann-Liouville fractional derivative of order ] as

119863]119891 (119909) =

119889119898

119889119909119898(119869119898minus]

119891 (119909)) (3)

where 119898 minus 1 lt ] le 119898 119898 isin 119873 and 119898 is the smallest integergreater than ]

Lemma 1 If119898 minus 1 lt ] le 119898 119898 isin 119873 then

119863]119869]119891 (119909) = 119891 (119909)

119869]119863

]119891 (119909) = 119891 (119909) minus

119898minus1

sum

119894=0

119891(119894)(0+)119909119894

119894 119909 gt 0

(4)

22 Properties of Shifted Legendre Polynomials The well-known Legendre polynomials 119875

119894(119909) are defined on the inter-

val (minus1 1) Some properties about the standard Legendrepolynomials will be recalled in this section The Legendrepolynomials 119875

119896(119909) (119896 = 0 1 ) satisfy the following

Rodriguersquos formula

119875119896 (119909) =

(minus1)119896

2119896119896119863119896((1 minus 119909

2)119896

) (5)

We recall also that 119875119896(119909) is a polynomial of degree 119896 and

therefore 119875(119902)119896(119909) (the 119902th derivative of 119875

119896(119909)) is

119875(119902)

119896(119909) =

119896minus119902

sum

119894=0(119896+119894=even)119862119902 (119896 119894) 119875119894 (119909) (6)

where

119862119902 (119896 119894) = 2

119902minus1(2119894 + 1) Γ (

119902 + 119896 minus 119894

2) Γ(

119902 + 119896 + 119894 + 1

2)

times (Γ (119902) Γ (2 minus 119902 + 119896 minus 119894

2) Γ(

3 minus 119902 + 119896 + 119894

2))

minus1

(7)

The Legendre polynomials satisfy the following relations

1198750 (119909) = 1 119875

1 (119909) = 119909

119875119896+2 (119909) =

2119896 + 3

119896 + 2119909119875119896+1 (119909) minus

119896 + 1

119896 + 2119875119896 (119909)

(8)

and the orthogonality relation

(119875119896 (119909) 119875119897 (119909)) = int

1

minus1

119875119896 (119909) 119875119897 (119909)119908 (119909) 119889119909 = ℎ

119896120575119897119896 (9)

where 119908(119909) equiv 1 ℎ119896= 2(2119896 + 1) and 120575

119897119896is the Kronecker

delta function The Legendre-Gauss-Lobatto quadrature isused to evaluate the previous integral accurately For any 120601 isin

1198782119873minus1

[minus1 1] we get

int

1

minus1

120601 (119909) 119889119909 =

119873

sum

119895=0

120603119873119895

120601 (119909119873119895

) (10)

We introduce the following discrete inner product

(119906 V)119873 =119873

sum

119895=0

119906 (119909119873119895

) V (119909119873119895

) 120603119873119895

(11)

where 119909119873119895

(0 le 119895 le 119873) and 120603119873119895

(0 le 119895 le 119873) are usedas usual nodes and the corresponding Christoffel numbersin the interval [minus1 1] respectively Let the shifted Legendrepolynomials 119875

119894((2119909119871) minus 1) be denoted by 119875

119871119894(119909) 119909 isin (0 119871)

Then 119875119871119894(119909) can be obtained with the aid of the following

recurrence formula

(119894 + 1) 119875119871119894+1 (119909) = (2119894 + 1) (2119909

119871minus 1)119875

119871119894 (119909)

minus 119894119875119871119894minus1 (119909) 119894 = 1 2

(12)

The analytic form of the shifted Legendre polynomials 119875119871119894(119909)

of degree 119894 is given by

119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119909119896 (13)

Abstract and Applied Analysis 3

The Riemann-Liouville fractional integration of shifted Leg-endre polynomials 119875

119871119894(119909) of degree 119894 is given by

119869]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)119896

(119894 minus 119896)(119896)2119871119896Γ (119896 + ] + 1)

119909119896+]

119894 = 0 1 119873

119863]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)Γ (119896 + 1)

(119894 minus 119896)(119896)2119871119896Γ (119896 + 1 minus ])

119909119896minus]

119894 = 0 1 119873

(14)

where 119875119871119894(0) = (minus1)

119894 The orthogonality condition is

int

119871

0

119875119871119895 (119909) 119875119871119896 (119909) 119908119871 (119909) 119889119909 = ℎ

119871119896120575119895119896 (15)

where 119908119871(119909) equiv 1 and ℎ

119871119896= 119871(2119896 + 1)

The function 119906(119909) square integrable in (0 119871) may beexpressed in terms of shifted Legendre polynomials as

119906 (119909) =

infin

sum

119895=0

119888119895119875119871119895 (119909) (16)

where the coefficients 119888119895are given by

119888119895=

1

ℎ119871119895

int

119871

0

119906 (119909) 119875119871119895 (119909) 119889119909 119895 = 0 1 2 (17)

In practice only the first (119873 + 1)-terms shifted Legendrepolynomials are considered Hence 119906(119909) can be expressed inthe form

119906119873 (119909) ≃

119873

sum

119895=0

119888119895119875119871119895 (119909) (18)

3 Shifted Legendre SpectralCollocation Method

In this section we derive the SL-GL-C method for solvingtwo-dimensional fractional diffusion equation of the form

120597V (119909 119910 119905)120597119905

= 1198921(119909 119910)

120597]1V (119909 119910 119905)120597119909]1

+ 1198922(119909 119910)

120597]2V (119909 119910 119905)120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1198711] times [0 119871

2] times (0 119879]

(19)

which is subject to the initial condition

V (119909 119910 0) = 1198923(119909 119910) (119909 119910) isin [0 119871

1] times [0 119871

2] (20)

and boundary conditions

V (0 119910 119905) = 1198924(119910 119905) V (119871

1 119910 119905) = 119892

5(119910 119905)

(119910 119905) isin [0 1198712] times [0 119879]

V (119909 0 119905) = 1198926 (119909 119905) V (119909 119871

2 119905) = 119892

7 (119909 119905)

(119909 119905) isin [0 1198711] times [0 119879]

(21)

Denote by 119909119873119895

(119909119871119873119895

) 0 ⩽ 119895 ⩽ 119873 and120603119873119895

(120603119871119873119895

) 0 le

119894 le 119873 the nodes and Christoffel numbers of the standard(shifted) Legendre-Gauss-Lobatto quadrature on the inter-vals (minus1 1) (0 119871) respectively Then one can clearly deducethat

119909119871119873119895

=119871

2(119909119873119895

+ 1) 0 le 119895 le 119873

120603119871119873119895

= (119871

2)

120572+120573+1

120603119873119895

0 le 119895 le 119873

(22)

and if 119878119873(0 119871) denotes the set of all polynomials of degree at

most119873 then it follows for any 120601 isin 1198782119873minus1

(0 119871) that

int

119871

0

119908119871 (119909) 120601 (119909) 119889119909 = (

119871

2)int

1

minus1

120601(119871

2(119909 + 1)) 119889119909

= (119871

2)

119873

sum

119895=0

120603119873119895

120601(119871

2(119909119873119895

+ 1))

=

119873

sum

119895=0

120603119871119873119895

120601 (119909119871119873119895

)

(23)

Here the SL-GL-Cmethodwill be employed to transformthe previous two-dimensional fractional diffusion equationinto SODEs Let us expand the dependent variables in theform

V (119909 119910 119905) =119873

sum

119894=0

119872

sum

119895=0

119886119894119895 (119905) 119875119871

1119894 (119909) 119875119871

2119895(119910) (24)

We observe from (15) that the coefficients 119886119894119895(119905) can be

approximated by

119886119894119895 (119905) =

1

ℎ1198711119894ℎ1198712119895

times

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

times V (1199091198711119873119897

1199101198712119872119896

119905)

(25)

4 Abstract and Applied Analysis

Table 1 Maximum absolute errors for problem (40) for 119879 = 1

119873 = 119872SL-GL-C method [43]

119872119905

119864119872119864

Δ119909 = Δ119910 = Δ119905 119872119864

4 341785 times 10minus3

911427 times 10minus3 mdash mdash

6 213706 times 10minus5

683861 times 10minus5 1

5105564 times 10

minus3

8 132866 times 10minus6

485909 times 10minus6 1

10261694 times 10

minus4

10 174023 times 10minus7

707142 times 10minus7 1

20657067 times 10

minus5

12 343827 times 10minus8

152415 times 10minus7 1

40155866 times 10

minus5

Table 2 Maximum absolute errors for problem (40) for 119879 = 1 and119873 = 119872 = 12

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 274515 times 10minus11 01 05 1 917923 times 10

minus12 01 1 1 441521 times 10minus11

02 213241 times 10minus10 02 757278 times 10

minus11 02 358585 times 10minus10

03 722081 times 10minus10 03 261927 times 10

minus10 03 120622 times 10minus9

04 171192 times 10minus9 04 643406 times 10

minus10 04 293613 times 10minus9

05 335365 times 10minus9 05 129893 times 10

minus9 05 58445 times 10minus9

06 580344 times 10minus9 06 233485 times 10

minus9 06 102404 times 10minus8

07 920721 times 10minus9 07 382976 times 10

minus9 07 165193 times 10minus8

08 139884 times 10minus8 08 584963 times 10

minus9 08 250569 times 10minus8

09 206847 times 10minus8 09 85766 times 10

minus9 09 364012 times 10minus8

1 275218 times 10minus8 1 126484 times 10

minus8 1 560815 times 10minus8

Table 3 Maximum absolute errors for problem (46) for 119879 = 1 and]1= ]2= 15

119873 = 119872 119872119905

119864119872119864

6 257399 times 10minus8

102853 times 10minus7

8 151431 times 10minus8

121098 times 10minus7

10 824341 times 10minus14

413336 times 10minus13

The expression (24) can be rewritten as

V (119909 119910 119905)

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909) 119875119871

2119895(119910) V119897119896 (119905)

(26)

where V(1199091198711119873119897

1199101198712119872119896

119905) = V119897119896(119905) In what follows the

approximation of the fractional spatial partial derivative

120597]1V(119909 119910 119905)120597119909]1 with respect to 119909 for V(119909 119910 119905) can be

computed as

120597]1V (119909 119910 119905)120597119909]1

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 119863(]1)(1198751198711119894 (119909)) 119875119871

2119895(119910) V119897119896 (119905)

(27)

Also the fractional spatial partial derivative of fractionalorder 120597]2V(119909 119910 119905)120597119910]

2 with respect to 119910 of the approximatesolution is

120597]2V (119909 119910 119905)120597119910]2

=

119873

sum

119897=0

119872

sum

119896=0

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909)119863

(]2)(1198751198712119895(119910)) V

119897119896 (119905)

(28)

Abstract and Applied Analysis 5

Table 4 Maximum absolute errors for problem (46) for 119879 = 1 and ]1= ]2= 15

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 406522 times 10minus16 01 05 1 705989 times 10

minus15 01 1 1 235055 times 10minus15

02 312066 times 10minus15 02 7445 times 10

minus15 02 1 663792 times 10minus15

03 422817 times 10minus15 03 173386 times 10

minus15 03 1 692588 times 10minus15

04 87365 times 10minus15 04 425354 times 10

minus15 04 1 170948 times 10minus14

05 140339 times 10minus14 05 156351 times 10

minus14 05 1 283974 times 10minus14

06 181075 times 10minus14 06 114631 times 10

minus14 06 1 344343 times 10minus14

07 269246 times 10minus14 07 18454 times 10

minus14 07 1 544946 times 10minus14

08 342964 times 10minus14 08 314054 times 10

minus14 08 1 678416 times 10minus14

09 43951 times 10minus14 09 278319 times 10

minus14 09 1 903513 times 10minus14

1 31379 times 10minus15 1 120095 times 10

minus14 1 1 10103 times 10minus14

The previous fractional derivatives can be computed at theSL-GL interpolation nodes as

(120597]1V (119909 119910 119905)120597119909]1

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(120597]2V (119909 119910 119905)120597119910]2

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(29)

where

120588119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]1)1198751198711119894(1199091198711119873119899

) 1198751198712119895(1199101198712119872119898

)

120582119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]2)1198751198712119895(1199101198712119872119898

) 1198751198711119894(1199091198711119873119899

)

(30)

In the proposed SL-GL-C method the residual of (19) is setto zero at (119873 minus 1) times (119872 minus 1) of SL-GL points Moreover thevalues of V

0119896(119905) V119873119896

(119905) V1198970(119905) and V

119897119872(119905) are given by

V0119896 (119905) = 119892

4(1199101198712119872119896

119905) V119873119896 (119905) = 119892

5(1199101198712119872119896

119905)

119896 = 0 119872

V1198970 (119905) = 119892

6(1199091198711119873119897

119905) V119897119872 (119905) = 119892

7(1199091198711119873119897

119905)

119897 = 0 119873

(31)

Therefore adapting (26)ndash(31) enables one to write (19)ndash(21)as a (119873 minus 1) times (119872 minus 1) SODEs

V119899119898 (119905) = 119892

119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(32)

which is subject to the initial conditions

V119899119898 (0) = 119892

119899119898

3 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(33)

where

1198921(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

1

1198922(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

2

1198923(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

3

119891 (1199091198711119873119899

1199101198712119872119898

119905) = 119891119899119898

(119905)

(34)

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 3: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Abstract and Applied Analysis 3

The Riemann-Liouville fractional integration of shifted Leg-endre polynomials 119875

119871119894(119909) of degree 119894 is given by

119869]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)119896

(119894 minus 119896)(119896)2119871119896Γ (119896 + ] + 1)

119909119896+]

119894 = 0 1 119873

119863]119875119871119894 (119909) =

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)

(119894 minus 119896)(119896)2119871119896

119869]119909119896

=

119894

sum

119896=0

(minus1)119894+119896 (119894 + 119896)Γ (119896 + 1)

(119894 minus 119896)(119896)2119871119896Γ (119896 + 1 minus ])

119909119896minus]

119894 = 0 1 119873

(14)

where 119875119871119894(0) = (minus1)

119894 The orthogonality condition is

int

119871

0

119875119871119895 (119909) 119875119871119896 (119909) 119908119871 (119909) 119889119909 = ℎ

119871119896120575119895119896 (15)

where 119908119871(119909) equiv 1 and ℎ

119871119896= 119871(2119896 + 1)

The function 119906(119909) square integrable in (0 119871) may beexpressed in terms of shifted Legendre polynomials as

119906 (119909) =

infin

sum

119895=0

119888119895119875119871119895 (119909) (16)

where the coefficients 119888119895are given by

119888119895=

1

ℎ119871119895

int

119871

0

119906 (119909) 119875119871119895 (119909) 119889119909 119895 = 0 1 2 (17)

In practice only the first (119873 + 1)-terms shifted Legendrepolynomials are considered Hence 119906(119909) can be expressed inthe form

119906119873 (119909) ≃

119873

sum

119895=0

119888119895119875119871119895 (119909) (18)

3 Shifted Legendre SpectralCollocation Method

In this section we derive the SL-GL-C method for solvingtwo-dimensional fractional diffusion equation of the form

120597V (119909 119910 119905)120597119905

= 1198921(119909 119910)

120597]1V (119909 119910 119905)120597119909]1

+ 1198922(119909 119910)

120597]2V (119909 119910 119905)120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1198711] times [0 119871

2] times (0 119879]

(19)

which is subject to the initial condition

V (119909 119910 0) = 1198923(119909 119910) (119909 119910) isin [0 119871

1] times [0 119871

2] (20)

and boundary conditions

V (0 119910 119905) = 1198924(119910 119905) V (119871

1 119910 119905) = 119892

5(119910 119905)

(119910 119905) isin [0 1198712] times [0 119879]

V (119909 0 119905) = 1198926 (119909 119905) V (119909 119871

2 119905) = 119892

7 (119909 119905)

(119909 119905) isin [0 1198711] times [0 119879]

(21)

Denote by 119909119873119895

(119909119871119873119895

) 0 ⩽ 119895 ⩽ 119873 and120603119873119895

(120603119871119873119895

) 0 le

119894 le 119873 the nodes and Christoffel numbers of the standard(shifted) Legendre-Gauss-Lobatto quadrature on the inter-vals (minus1 1) (0 119871) respectively Then one can clearly deducethat

119909119871119873119895

=119871

2(119909119873119895

+ 1) 0 le 119895 le 119873

120603119871119873119895

= (119871

2)

120572+120573+1

120603119873119895

0 le 119895 le 119873

(22)

and if 119878119873(0 119871) denotes the set of all polynomials of degree at

most119873 then it follows for any 120601 isin 1198782119873minus1

(0 119871) that

int

119871

0

119908119871 (119909) 120601 (119909) 119889119909 = (

119871

2)int

1

minus1

120601(119871

2(119909 + 1)) 119889119909

= (119871

2)

119873

sum

119895=0

120603119873119895

120601(119871

2(119909119873119895

+ 1))

=

119873

sum

119895=0

120603119871119873119895

120601 (119909119871119873119895

)

(23)

Here the SL-GL-Cmethodwill be employed to transformthe previous two-dimensional fractional diffusion equationinto SODEs Let us expand the dependent variables in theform

V (119909 119910 119905) =119873

sum

119894=0

119872

sum

119895=0

119886119894119895 (119905) 119875119871

1119894 (119909) 119875119871

2119895(119910) (24)

We observe from (15) that the coefficients 119886119894119895(119905) can be

approximated by

119886119894119895 (119905) =

1

ℎ1198711119894ℎ1198712119895

times

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

times V (1199091198711119873119897

1199101198712119872119896

119905)

(25)

4 Abstract and Applied Analysis

Table 1 Maximum absolute errors for problem (40) for 119879 = 1

119873 = 119872SL-GL-C method [43]

119872119905

119864119872119864

Δ119909 = Δ119910 = Δ119905 119872119864

4 341785 times 10minus3

911427 times 10minus3 mdash mdash

6 213706 times 10minus5

683861 times 10minus5 1

5105564 times 10

minus3

8 132866 times 10minus6

485909 times 10minus6 1

10261694 times 10

minus4

10 174023 times 10minus7

707142 times 10minus7 1

20657067 times 10

minus5

12 343827 times 10minus8

152415 times 10minus7 1

40155866 times 10

minus5

Table 2 Maximum absolute errors for problem (40) for 119879 = 1 and119873 = 119872 = 12

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 274515 times 10minus11 01 05 1 917923 times 10

minus12 01 1 1 441521 times 10minus11

02 213241 times 10minus10 02 757278 times 10

minus11 02 358585 times 10minus10

03 722081 times 10minus10 03 261927 times 10

minus10 03 120622 times 10minus9

04 171192 times 10minus9 04 643406 times 10

minus10 04 293613 times 10minus9

05 335365 times 10minus9 05 129893 times 10

minus9 05 58445 times 10minus9

06 580344 times 10minus9 06 233485 times 10

minus9 06 102404 times 10minus8

07 920721 times 10minus9 07 382976 times 10

minus9 07 165193 times 10minus8

08 139884 times 10minus8 08 584963 times 10

minus9 08 250569 times 10minus8

09 206847 times 10minus8 09 85766 times 10

minus9 09 364012 times 10minus8

1 275218 times 10minus8 1 126484 times 10

minus8 1 560815 times 10minus8

Table 3 Maximum absolute errors for problem (46) for 119879 = 1 and]1= ]2= 15

119873 = 119872 119872119905

119864119872119864

6 257399 times 10minus8

102853 times 10minus7

8 151431 times 10minus8

121098 times 10minus7

10 824341 times 10minus14

413336 times 10minus13

The expression (24) can be rewritten as

V (119909 119910 119905)

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909) 119875119871

2119895(119910) V119897119896 (119905)

(26)

where V(1199091198711119873119897

1199101198712119872119896

119905) = V119897119896(119905) In what follows the

approximation of the fractional spatial partial derivative

120597]1V(119909 119910 119905)120597119909]1 with respect to 119909 for V(119909 119910 119905) can be

computed as

120597]1V (119909 119910 119905)120597119909]1

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 119863(]1)(1198751198711119894 (119909)) 119875119871

2119895(119910) V119897119896 (119905)

(27)

Also the fractional spatial partial derivative of fractionalorder 120597]2V(119909 119910 119905)120597119910]

2 with respect to 119910 of the approximatesolution is

120597]2V (119909 119910 119905)120597119910]2

=

119873

sum

119897=0

119872

sum

119896=0

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909)119863

(]2)(1198751198712119895(119910)) V

119897119896 (119905)

(28)

Abstract and Applied Analysis 5

Table 4 Maximum absolute errors for problem (46) for 119879 = 1 and ]1= ]2= 15

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 406522 times 10minus16 01 05 1 705989 times 10

minus15 01 1 1 235055 times 10minus15

02 312066 times 10minus15 02 7445 times 10

minus15 02 1 663792 times 10minus15

03 422817 times 10minus15 03 173386 times 10

minus15 03 1 692588 times 10minus15

04 87365 times 10minus15 04 425354 times 10

minus15 04 1 170948 times 10minus14

05 140339 times 10minus14 05 156351 times 10

minus14 05 1 283974 times 10minus14

06 181075 times 10minus14 06 114631 times 10

minus14 06 1 344343 times 10minus14

07 269246 times 10minus14 07 18454 times 10

minus14 07 1 544946 times 10minus14

08 342964 times 10minus14 08 314054 times 10

minus14 08 1 678416 times 10minus14

09 43951 times 10minus14 09 278319 times 10

minus14 09 1 903513 times 10minus14

1 31379 times 10minus15 1 120095 times 10

minus14 1 1 10103 times 10minus14

The previous fractional derivatives can be computed at theSL-GL interpolation nodes as

(120597]1V (119909 119910 119905)120597119909]1

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(120597]2V (119909 119910 119905)120597119910]2

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(29)

where

120588119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]1)1198751198711119894(1199091198711119873119899

) 1198751198712119895(1199101198712119872119898

)

120582119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]2)1198751198712119895(1199101198712119872119898

) 1198751198711119894(1199091198711119873119899

)

(30)

In the proposed SL-GL-C method the residual of (19) is setto zero at (119873 minus 1) times (119872 minus 1) of SL-GL points Moreover thevalues of V

0119896(119905) V119873119896

(119905) V1198970(119905) and V

119897119872(119905) are given by

V0119896 (119905) = 119892

4(1199101198712119872119896

119905) V119873119896 (119905) = 119892

5(1199101198712119872119896

119905)

119896 = 0 119872

V1198970 (119905) = 119892

6(1199091198711119873119897

119905) V119897119872 (119905) = 119892

7(1199091198711119873119897

119905)

119897 = 0 119873

(31)

Therefore adapting (26)ndash(31) enables one to write (19)ndash(21)as a (119873 minus 1) times (119872 minus 1) SODEs

V119899119898 (119905) = 119892

119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(32)

which is subject to the initial conditions

V119899119898 (0) = 119892

119899119898

3 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(33)

where

1198921(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

1

1198922(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

2

1198923(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

3

119891 (1199091198711119873119899

1199101198712119872119898

119905) = 119891119899119898

(119905)

(34)

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 4: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

4 Abstract and Applied Analysis

Table 1 Maximum absolute errors for problem (40) for 119879 = 1

119873 = 119872SL-GL-C method [43]

119872119905

119864119872119864

Δ119909 = Δ119910 = Δ119905 119872119864

4 341785 times 10minus3

911427 times 10minus3 mdash mdash

6 213706 times 10minus5

683861 times 10minus5 1

5105564 times 10

minus3

8 132866 times 10minus6

485909 times 10minus6 1

10261694 times 10

minus4

10 174023 times 10minus7

707142 times 10minus7 1

20657067 times 10

minus5

12 343827 times 10minus8

152415 times 10minus7 1

40155866 times 10

minus5

Table 2 Maximum absolute errors for problem (40) for 119879 = 1 and119873 = 119872 = 12

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 274515 times 10minus11 01 05 1 917923 times 10

minus12 01 1 1 441521 times 10minus11

02 213241 times 10minus10 02 757278 times 10

minus11 02 358585 times 10minus10

03 722081 times 10minus10 03 261927 times 10

minus10 03 120622 times 10minus9

04 171192 times 10minus9 04 643406 times 10

minus10 04 293613 times 10minus9

05 335365 times 10minus9 05 129893 times 10

minus9 05 58445 times 10minus9

06 580344 times 10minus9 06 233485 times 10

minus9 06 102404 times 10minus8

07 920721 times 10minus9 07 382976 times 10

minus9 07 165193 times 10minus8

08 139884 times 10minus8 08 584963 times 10

minus9 08 250569 times 10minus8

09 206847 times 10minus8 09 85766 times 10

minus9 09 364012 times 10minus8

1 275218 times 10minus8 1 126484 times 10

minus8 1 560815 times 10minus8

Table 3 Maximum absolute errors for problem (46) for 119879 = 1 and]1= ]2= 15

119873 = 119872 119872119905

119864119872119864

6 257399 times 10minus8

102853 times 10minus7

8 151431 times 10minus8

121098 times 10minus7

10 824341 times 10minus14

413336 times 10minus13

The expression (24) can be rewritten as

V (119909 119910 119905)

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909) 119875119871

2119895(119910) V119897119896 (119905)

(26)

where V(1199091198711119873119897

1199101198712119872119896

119905) = V119897119896(119905) In what follows the

approximation of the fractional spatial partial derivative

120597]1V(119909 119910 119905)120597119909]1 with respect to 119909 for V(119909 119910 119905) can be

computed as

120597]1V (119909 119910 119905)120597119909]1

=

119873

sum

119894=0

119872

sum

119895=0

119873

sum

119897=0

119872

sum

119896=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 119863(]1)(1198751198711119894 (119909)) 119875119871

2119895(119910) V119897119896 (119905)

(27)

Also the fractional spatial partial derivative of fractionalorder 120597]2V(119909 119910 119905)120597119910]

2 with respect to 119910 of the approximatesolution is

120597]2V (119909 119910 119905)120597119910]2

=

119873

sum

119897=0

119872

sum

119896=0

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ1198711119894ℎ1198712119895

times 1198751198711119894 (119909)119863

(]2)(1198751198712119895(119910)) V

119897119896 (119905)

(28)

Abstract and Applied Analysis 5

Table 4 Maximum absolute errors for problem (46) for 119879 = 1 and ]1= ]2= 15

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 406522 times 10minus16 01 05 1 705989 times 10

minus15 01 1 1 235055 times 10minus15

02 312066 times 10minus15 02 7445 times 10

minus15 02 1 663792 times 10minus15

03 422817 times 10minus15 03 173386 times 10

minus15 03 1 692588 times 10minus15

04 87365 times 10minus15 04 425354 times 10

minus15 04 1 170948 times 10minus14

05 140339 times 10minus14 05 156351 times 10

minus14 05 1 283974 times 10minus14

06 181075 times 10minus14 06 114631 times 10

minus14 06 1 344343 times 10minus14

07 269246 times 10minus14 07 18454 times 10

minus14 07 1 544946 times 10minus14

08 342964 times 10minus14 08 314054 times 10

minus14 08 1 678416 times 10minus14

09 43951 times 10minus14 09 278319 times 10

minus14 09 1 903513 times 10minus14

1 31379 times 10minus15 1 120095 times 10

minus14 1 1 10103 times 10minus14

The previous fractional derivatives can be computed at theSL-GL interpolation nodes as

(120597]1V (119909 119910 119905)120597119909]1

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(120597]2V (119909 119910 119905)120597119910]2

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(29)

where

120588119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]1)1198751198711119894(1199091198711119873119899

) 1198751198712119895(1199101198712119872119898

)

120582119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]2)1198751198712119895(1199101198712119872119898

) 1198751198711119894(1199091198711119873119899

)

(30)

In the proposed SL-GL-C method the residual of (19) is setto zero at (119873 minus 1) times (119872 minus 1) of SL-GL points Moreover thevalues of V

0119896(119905) V119873119896

(119905) V1198970(119905) and V

119897119872(119905) are given by

V0119896 (119905) = 119892

4(1199101198712119872119896

119905) V119873119896 (119905) = 119892

5(1199101198712119872119896

119905)

119896 = 0 119872

V1198970 (119905) = 119892

6(1199091198711119873119897

119905) V119897119872 (119905) = 119892

7(1199091198711119873119897

119905)

119897 = 0 119873

(31)

Therefore adapting (26)ndash(31) enables one to write (19)ndash(21)as a (119873 minus 1) times (119872 minus 1) SODEs

V119899119898 (119905) = 119892

119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(32)

which is subject to the initial conditions

V119899119898 (0) = 119892

119899119898

3 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(33)

where

1198921(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

1

1198922(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

2

1198923(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

3

119891 (1199091198711119873119899

1199101198712119872119898

119905) = 119891119899119898

(119905)

(34)

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 5: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Abstract and Applied Analysis 5

Table 4 Maximum absolute errors for problem (46) for 119879 = 1 and ]1= ]2= 15

119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905) 119909 119910 119905 119864(119909 119910 119905)

01 01 1 406522 times 10minus16 01 05 1 705989 times 10

minus15 01 1 1 235055 times 10minus15

02 312066 times 10minus15 02 7445 times 10

minus15 02 1 663792 times 10minus15

03 422817 times 10minus15 03 173386 times 10

minus15 03 1 692588 times 10minus15

04 87365 times 10minus15 04 425354 times 10

minus15 04 1 170948 times 10minus14

05 140339 times 10minus14 05 156351 times 10

minus14 05 1 283974 times 10minus14

06 181075 times 10minus14 06 114631 times 10

minus14 06 1 344343 times 10minus14

07 269246 times 10minus14 07 18454 times 10

minus14 07 1 544946 times 10minus14

08 342964 times 10minus14 08 314054 times 10

minus14 08 1 678416 times 10minus14

09 43951 times 10minus14 09 278319 times 10

minus14 09 1 903513 times 10minus14

1 31379 times 10minus15 1 120095 times 10

minus14 1 1 10103 times 10minus14

The previous fractional derivatives can be computed at theSL-GL interpolation nodes as

(120597]1V (119909 119910 119905)120597119909]1

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(120597]2V (119909 119910 119905)120597119910]2

)

119909=1199091198711119873119899

119910=1199101198712119872119898

=

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

119899 = 0 1 119873 119898 = 0 1 119872

(29)

where

120588119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]1)1198751198711119894(1199091198711119873119899

) 1198751198712119895(1199101198712119872119898

)

120582119899119898

119897119896=

119873

sum

119894=0

119872

sum

119895=0

(1198751198712119895(1199101198712119872119896

) 120603119872119896

1198751198711119894(1199091198711119873119897

) 120603119873119897)

ℎ119894ℎ119895

times 119863(]2)1198751198712119895(1199101198712119872119898

) 1198751198711119894(1199091198711119873119899

)

(30)

In the proposed SL-GL-C method the residual of (19) is setto zero at (119873 minus 1) times (119872 minus 1) of SL-GL points Moreover thevalues of V

0119896(119905) V119873119896

(119905) V1198970(119905) and V

119897119872(119905) are given by

V0119896 (119905) = 119892

4(1199101198712119872119896

119905) V119873119896 (119905) = 119892

5(1199101198712119872119896

119905)

119896 = 0 119872

V1198970 (119905) = 119892

6(1199091198711119873119897

119905) V119897119872 (119905) = 119892

7(1199091198711119873119897

119905)

119897 = 0 119873

(31)

Therefore adapting (26)ndash(31) enables one to write (19)ndash(21)as a (119873 minus 1) times (119872 minus 1) SODEs

V119899119898 (119905) = 119892

119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(32)

which is subject to the initial conditions

V119899119898 (0) = 119892

119899119898

3 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(33)

where

1198921(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

1

1198922(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

2

1198923(1199091198711119873119899

1199101198712119872119898

) = 119892119899119898

3

119891 (1199091198711119873119899

1199101198712119872119898

119905) = 119891119899119898

(119905)

(34)

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 6: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

6 Abstract and Applied Analysis

Finally we can arrange (32) and (33) to their matrix formula-tion which can be solved by IRK scheme

(((((

(

V11 (119905) sdot sdot sdot V

1119872minus1 (119905)

V21 (119905)

V2119872minus1 (119905)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (119905)

V119873minus2119872minus1 (119905)

V119873minus11 (119905) sdot sdot sdot V

119873minus1119872minus1 (119905)

)))))

)

=((

(

11986711 (119905 V1 V119873minus1) sdot sdot sdot 1198671119872minus1 (119905 V1 V119873minus1)

11986721 (119905 V1 V119873minus1) 1198672119872minus1 (119905 V1 V119873minus1)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119867119873minus21 (119905 V1 V119873minus1) 119867119873minus2119872minus1 (119905 V1 V119873minus1)

119867119873minus11 (119905 V1 V119873minus1) sdot sdot sdot 119867119873minus1119872minus1 (119905 V1 V119873minus1)

))

)

(((((

(

V11 (0) sdot sdot sdot V

1119872minus1 (0)

V21 (0)

V2119872minus1 (0)

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

V119873minus21 (0)

V119873minus2119872minus1 (0)

V119873minus11 (0) sdot sdot sdot V

119873minus1119872minus1 (0)

)))))

)

=

(((((((

(

11989211

3sdot sdot sdot 119892

1119872minus1

3

11989221

3

1198922119872minus1

3

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

sdot sdot sdot d sdot sdot sdot

119892119873minus21

3

119892119873minus2119872minus1

3

119892119873minus11

3sdot sdot sdot 119892119873minus1119872minus1

3

)))))))

)

119905 isin [minus120591 0]

(35)where

119867119899119898

(119905 V11 V

119873minus1119872minus1)

= 119892119899119898

1

119873

sum

119897=0

119872

sum

119896=0

120588119899119898

119897119896V119897119896 (119905) + 119892

119899119898

2

119873

sum

119897=0

119872

sum

119896=0

120582119899119898

119897119896V119897119896 (119905)

+ 119891119899119898

(119905) 119899 = 1 119873 minus 1 119898 = 1 119872 minus 1

(36)

The Runge-Kutta method can be expressed as one ofpowerful numerical integrations tools used for initial valueSODEs of first order The IRK method presents a subclassof the well-known family of Runge-Kutta methods and hasmany applications in the numerical solution of systems ofordinary differential equations

4 Numerical Results

In this section we present two numerical examples to showthe accuracy robustness and applicability of the proposed

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 1 Space graph of numerical solution of problem (40)

00

05

10x

00

05

10

y

0E(xy1) 40 times 10

minus8

20 times 10minus8

Figure 2 Space graph of absolute error of problem (40)

method We compare the results obtained by our methodwith those obtained in [43]

The difference between themeasured value of the approx-imate solution and the exact solution (absolute error) is givenby

119864 (x y 119905) = 1003816100381610038161003816119906 (x y 119905) minus 119906119873(x y 119905)1003816100381610038161003816 (37)

where x y 119905 119906(x y 119905) and 119906119873(x y 119905) are the space vectors

time exact and numerical solutions respectively Moreoverthe maximum absolute error (MAEs) is given by

119872119864= Max 119864 (x y 119905) over all domains (38)

Also we define the norm infinity as

119872119905

119864= Max 119864 (x y 119905

119896) over all space domains (39)

Example 2 Consider the two-dimensional problem [43] is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

12059718119906 (119909 119910 119905)

12059711990918+ 1198922 (119909 119905)

12059716119906 (119909 119910 119905)

12059711990916

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(40)

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 7: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Abstract and Applied Analysis 7

00 02 04 06 08 10000

001

002

003

004

005

006

007

x

uanduN

u(x 05 00)

u(x 05 05)

u(x 05 10)uN(x 05 00)

uN(x 05 05)

uN(x 05 10)

Figure 3 119909-direction curves of numerical and exact solutions ofproblem (40)

00 02 04 06 08 10y

u(05 y 00)

u(05 y 05)

u(05 y 10)uN(05 y 00)

uN(05 y 05)

uN(05 y 10)

000

002

004

006

008

010

uanduN

Figure 4 119910-direction curves of numerical and exact solutions ofproblem (40)

with the diffusion coefficients

1198921 (119909 119905) =

Γ (22) 11990928119910

6 119892

2 (119909 119905) =211990911991026

Γ (46) (41)

and consider the forcing function is

119891 (119909 119910 119905) = minus (1 + 2119909119910) 119890minus119905119909311991036 (42)

with the initial condition

119906 (119909 119910 0) = 119909311991036 (43)

and Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus11990511991036

119906 (119909 1 119905) = 119890minus1199051199093

forall119905 ge 0

(44)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus119905119909311991036 (45)

From the first look on Table 1 we see high accurate resultsbased on MAEs at different choices of 119873 and 119872 comparedwith the results given in [43] Moreover results listed inTable 2 confirm that the proposed method is very accurate

In Figure 1 we display the numerical solution of problem(40) computed by the present method at 119873 = 119872 = 12while we plot the space graph of absolute error of problem(40) computed by the present method at 119873 = 119872 = 12 inFigure 2 As shown in Figures 3 and 4 we see the agreementof numerical and exact solutions curves

Example 3 Consider the two-dimensional fractional diffu-sion equation is

120597119906

120597119905(119909 119910 119905)

= 1198921 (119909 119905)

120597]1119906 (119909 119910 119905)

120597119909]1+ 1198922 (119909 119905)

120597]2119906 (119909 119910 119905)

120597119910]2

+ 119891 (119909 119910 119905) (119909 119910 119905) isin [0 1] times [0 1] times [0 119879]

(46)

with the diffusion coefficients

1198921(119909 119910) =

(3 minus 2119909) Γ (3 minus ]1)

2

1198922(119909 119910) =

(4 minus 119910) Γ (4 minus ]2)

6

(47)

and consider the forcing function is

119891 (119909 119910 119905) = 119890minus119905(1199092(minus11991032

+ 119910 minus 4) 11991032

+ radic119909 (2119909 minus 3) 1199103)

(48)

with the initial condition

119906 (119909 119910 0) = 11990921199103 (49)

and the Dirichlet boundary conditions

119906 (0 119910 119905) = 119906 (119909 0 119905) = 0 119906 (1 119910 119905) = 119890minus1199051199103

119906 (119909 1 119905) = 119890minus1199051199092

forall119905 ge 0

(50)

The exact solution of this two-dimensional fractional diffu-sion equation is given by

119906 (119909 119910 119905) = 119890minus11990511990921199103 (51)

More accurate results formaximumabsolute and absoluteerrors of (46) have been obtained in Tables 3 and 4 respec-tively with values of parameters listed in their captions theapproximate solution 119906

119873119872at ]1= ]2= 15 and119873 = 119872 = 10

is plotted in Figure 5 Moreover we plot the absolute error forthis problem at ]

1= ]2= 15 and 119873 = 119872 = 10 in Figure 6

Finally the 119909- and 119910-directions curves of absolute error at]1= ]2= 15 and 119873 = 119872 = 10 are displayed in Figures 7

and 8

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 8: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

8 Abstract and Applied Analysis

00

05

10 00

05

10

000204

06

uN(xy05)

x

y

Figure 5 Space graph of numerical solution of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00

05

10x

00

05

10

y

0E(xy1)

30 times 10minus13

20 times 10minus13

10 times 10minus13

Figure 6 Space graph of absolute error of problem (46) at ]1= ]2=

15 and119873 = 119872 = 10

5 Concluding Remarks and Future Work

The SL-GL-C method was investigated successfully in spatialdiscretizations to get accurate approximate solution of thetwo-dimensional fractional diffusion equations with variablecoefficients This problem was transformed into SODEs intime variable which greatly simplifies the problem The IRKscheme was then applied to the resulting system From thenumerical experiments the obtained results demonstratedthe effectiveness and the high accuracy of SL-GL-C methodfor solving the mentioned problem

The technique can be extended to more sophisticatedproblems in two- and three-dimensional spaces In princi-ple this method may be extended to related problems inmathematical physics It is possible to use other orthogonalpolynomials say Chebyshev polynomials or Jacobi polyno-mials to solve the mentioned problem in this paper Fur-thermore the proposed spectral method might be developedby considering the Legendre pseudospectral approximationin both temporal and spatial discretizations We should notethat as a numerical method we are restricted to solvingproblem over a finite domain Also the pseudospectralapproximation might be employed based on generalized

00 02 04 06 08 100

x

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(x051)

Figure 7 119909-direction curve of absolute error of problem (46) at ]1=

]2= 15 and119873 = 119872 = 10

00 02 04 06 08 100

y

60 times 10minus14

50 times 10minus14

40 times 10minus14

30 times 10minus14

20 times 10minus14

10 times 10minus14

E(05y1)

Figure 8 119910-direction curves of absolute error of problem (46) at]1= ]2= 15 and119873 = 119872 = 10

Laguerre or modified generalized Laguerre polynomials [44]to solve similar problems in a semi-infinite spatial intervals

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This paper was funded by the Deanship of Scientific ResearchDSR King Abdulaziz University Jeddah Grant no 130-53-D1435The author therefore acknowledges with thanks DSRfor the technical and financial support

References

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

[2] H Schamel and K Elsaesser ldquoThe application of the spectralmethod to nonlinear wave propagationrdquo Journal of Computa-tional Physics vol 22 no 4 pp 501ndash516 1976

[3] W M Abd-Elhameed E H Doha and Y H Youssri ldquoEfficientspectral-Petrov-Galerkin methods for third- and fifth-orderdifferential equations using general parameters generalized

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 9: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

Abstract and Applied Analysis 9

Jacobi polynomialsrdquoQuaestiones Mathematicae vol 36 pp 15ndash38 2013

[4] E H Doha A H Bhrawy M A Abdelkawy and R M HafezldquoA Jacobi collocation approximation for nonlinear coupledviscous Burgersrsquo equationrdquo Central European Journal of Physicsvol 12 pp 111ndash122 2014

[5] H Adibi and A M Rismani ldquoOn using a modified Legendre-spectralmethod for solving singular IVPs of Lane-Emden typerdquoComputers andMathematics with Applications vol 60 no 7 pp2126ndash2130 2010

[6] E H Doha A H Bhrawy M A Abdelkawy and R Avan Gorder ldquoJacobi-Gauss-Lobatto collocation method for thenumerical solution of 1 + 1 nonlinear Schrodinger equationsrdquoJournal of Computational Physics vol 261 pp 244ndash255 2014

[7] E H Doha A H Bhrawy D Baleanu and R M Hafez ldquoA newJacobi rational-Gauss collocation method for numerical solu-tion of generalized Pantograph equationsrdquo Applied NumericalMathematics vol 77 pp 43ndash54 2014

[8] FMMahfouz ldquoNumerical simulation of free convectionwithinan eccentric annulus filled with micropolar fluid using spectralmethodrdquo Applied Mathematics and Computation vol 219 pp5397ndash5409 2013

[9] J Ma B-W Li and J R Howell ldquoThermal radiation heattransfer in one- and two-dimensional enclosures using thespectral collocation method with full spectrum k-distributionmodelrdquo International Journal of Heat and Mass Transfer vol 71pp 35ndash43 2014

[10] X Ma and C Huang ldquoSpectral collocation method for linearfractional integro-differential equationsrdquoAppliedMathematicalModelling vol 38 pp 1434ndash1448 2014

[11] W M Abd-Elhameed E H Doha and Y H Youssri ldquoNewwavelets collocation method for solving second-order multi-point boundary value problems using Chebyshev polynomialsof third and fourthrdquo Abstract and Applied Analysis vol 2013Article ID 542839 9 pages 2013

[12] S R Lau and R H Price ldquoSparse spectral-tau method forthe three-dimensional helically reduced wave equation on two-center domainsrdquo Journal of Computational Physics vol 231 pp7695ndash7714 2012

[13] F Ghoreishi and S Yazdani ldquoAn extension of the spectralTau method for numerical solution of multi-order fractionaldifferential equations with convergence analysisrdquo ComputersandMathematics withApplications vol 61 no 1 pp 30ndash43 2011

[14] E H Doha and A H Bhrawy ldquoAn efficient direct solverfor multidimensional elliptic Robin boundary value problemsusing a Legendre spectral-Galerkin methodrdquo Computers andMathematics with Applications vol 64 pp 558ndash571 2012

[15] T Boaca and I Boaca ldquoSpectral galerkinmethod in the study ofmass transfer in laminar and turbulent flowsrdquo Computer AidedChemical Engineering vol 24 pp 99ndash104 2007

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

[17] W M Abd-Elhameed E H Doha and M A BassuonyldquoTwo Legendre-Dual-Petrov-Galerkin algorithms for solvingthe integrated forms of high odd-order boundary value prob-lemsrdquoThe ScientificWorld Journal vol 2013 Article ID 30926411 pages 2013

[18] LWang YMa and ZMeng ldquoHaar wavelet method for solvingfractional partial differential equations numericallyrdquo AppliedMathematics and Computation vol 227 pp 66ndash76 2014

[19] 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

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

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

[22] R Garrappa and M Popolizio ldquoOn the use of matrix functionsfor fractional partial differential equationsrdquo Mathematics andComputers in Simulation vol 81 no 5 pp 1045ndash1056 2011

[23] A A Pedas and E Tamme ldquoNumerical solution of nonlinearfractional differential equations by spline collocation methodsrdquoJournal of Computational and AppliedMathematics vol 255 pp216ndash230 2014

[24] F Gao X Lee F Fei H Tong Y Deng and H Zhao ldquoIden-tification time-delayed fractional order chaos with functionalextrema model via differential evolutionrdquo Expert Systems WithApplications vol 41 pp 1601ndash1608 2014

[25] Y Zhao D F Cheng and X J Yang ldquoApproximation solu-tions for local fractional Schrodinger equation in the one-dimensional Cantorian systemrdquo Advances in MathematicalPhysics vol 2013 Article ID 291386 5 pages 2013

[26] A H Bhrawy and M M Al-Shomrani ldquoA shifted Legendrespectral method for fractional-order multi-point boundaryvalue problemsrdquo Advances in Difference Equations vol 2012article 8 2012

[27] J W Kirchner X Feng and C Neal ldquoFrail chemistry and itsimplications for contaminant transport in catchmentsrdquo Naturevol 403 no 6769 pp 524ndash526 2000

[28] A Pedas and E Tamme ldquoPiecewise polynomial collocationfor linear boundary value problems of fractional differentialequationsrdquo Journal of Computational and Applied Mathematicsvol 236 no 13 pp 3349ndash3359 2012

[29] A Ahmadian M Suleiman S Salahshour and D Baleanu ldquoAJacobi operational matrix for solving a fuzzy linear fractionaldifferential equationrdquo Advances in Difference Equations vol2013 article 104 29 pages 2013

[30] A H Bhrawy and M A Alghamdi ldquoA shifted Jacobi-Gauss-Lobatto collocation method for solving nonlinear fractionalLangevin equation involving two fractional orders in differentintervalsrdquo Boundary Value Problems vol 2012 article 62 2012

[31] R Hilfer Applications of Fractional Calculus in Physics WordScientific Singapore 2000

[32] E Kotomin and V Kuzovkov Modern Aspects of Diffusion-Controlled Reactions Cooperative Phenomena in BimolecularProcesses Comprehensive Chemical Kinetics Elsevier 1996

[33] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods Series on ComplexityNonlinearity and Chaos World Scientific Singapore 2012

[34] R S Cantrell and C Cosner Spatial Ecology via ReactionDiffusion Equations Wiley 2004

[35] HWang andNDu ldquoFast solutionmethods for space-fractionaldiffusion equationsrdquo Journal of Computational and AppliedMathematics vol 255 pp 376ndash383 2014

[36] A H Bhrawy and D Baleanu ldquoA spectral Legendre-Gauss-Lobatto collocation method for a space-fractional advection

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 10: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

10 Abstract and Applied Analysis

diffusion equations with variable coefficientsrdquoReports onMath-ematical Physics vol 72 pp 219ndash233 2013

[37] E H Doha A H Bhrawy and S S Ezz-Eldien ldquoNumericalapproximations for fractional diffusion equations via a Cheby-shev spectral-tau methodrdquo Central European Journal of Physicsvol 11 pp 1494ndash1503 2013

[38] F Liu P Zhuang I Turner K Burrage and V Anh ldquoAnew fractional finite volume method for solving the fractionaldiffusion equationrdquo Applied Mathematical Modelling 2013

[39] S B Yuste and J Quintana-Murillo ldquoA finite difference methodwith non-uniform timesteps for fractional diffusion equationsrdquoComputer Physics Communications vol 183 pp 2594ndash26002012

[40] K Wang and H Wang ldquoA fast characteristic finite differencemethod for fractional advection-diffusion equationsrdquoAdvancesin Water Resources vol 34 no 7 pp 810ndash816 2011

[41] KMiller and B RossAn Introduction to the Fractional Calaulusand Fractional Differential Equations John Wiley amp Sons NewYork NY USA 1993

[42] I Podluny Fractional Differential Equations Academic PressSan Diego Calif USA 1999

[43] C Tadjeran and M M Meerschaert ldquoA second-order accuratenumerical method for the two-dimensional fractional diffusionequationrdquo Journal of Computational Physics vol 220 no 2 pp813ndash823 2007

[44] D Baleanu A H Bhrawy and T M Taha ldquoTwo efficientgeneralized Laguerre spectral algorithms for fractional initialvalue problemsrdquoAbstract andAppliedAnalysis vol 2013 ArticleID 546502 10 pages 2013

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

Page 11: Research Article A New Legendre Collocation Method for ...downloads.hindawi.com/journals/aaa/2014/636191.pdf · Research Article A New Legendre Collocation Method for Solving a Two-Dimensional

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


Research Article New Wavelets Collocation Method …downloads.hindawi.com/journals/aaa/2013/542839.pdfNew Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value

Research Article New Wavelets Collocation Method …downloads.hindawi.com/journals/aaa/2013/542839.pdfNew Wavelets Collocation Method for Solving Second-Order Multipoint Boundary Value

An adaptive wavelet stochastic collocation method for ...

An adaptive wavelet stochastic collocation method for ...

Laguerre collocation method for solving Lane-Emden type ...

Laguerre collocation method for solving Lane-Emden type ...

Hermite Wavelet Collocation Method for the Numerical ...Hermite Wavelet Collocation Method for the Numerical Solution of Integral and Integro- ... is based upon Hermite polynomials

Hermite Wavelet Collocation Method for the Numerical ...Hermite Wavelet Collocation Method for the Numerical Solution of Integral and Integro- ... is based upon Hermite polynomials

RKPM with Augmented Corrected Collocation Method for ...

RKPM with Augmented Corrected Collocation Method for ...

Exponential Convergence of Multiquadric Collocation Method: a Numerical Study

Exponential Convergence of Multiquadric Collocation Method: a Numerical Study

APPLICATIONS OF SPLINE COLLOCATION METHOD TO PARTIAL ...

APPLICATIONS OF SPLINE COLLOCATION METHOD TO PARTIAL ...

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

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

BOUNDARY ELEMENT COLLOCATION METHOD FOR …etna.mcs.kent.edu/vol.39.2012/pp113-143.dir/pp113-143.pdf · boundary element collocation method for solving the exterior neumann problem

BOUNDARY ELEMENT COLLOCATION METHOD FOR …etna.mcs.kent.edu/vol.39.2012/pp113-143.dir/pp113-143.pdf · boundary element collocation method for solving the exterior neumann problem

Quintic B-spline collocation method for numerical solution ...

Quintic B-spline collocation method for numerical solution ...

Radial basis function collocation method for an elliptic ...svajunas/files/.../Sajavicius_2016a_Eng_Anal_Bound_… · tional related references. A meshless method based on RBF collocation

Radial basis function collocation method for an elliptic ...svajunas/files/.../Sajavicius_2016a_Eng_Anal_Bound_… · tional related references. A meshless method based on RBF collocation

BOUNDARY ELEMENT COLLOCATION METHOD FOR SOLVING …

BOUNDARY ELEMENT COLLOCATION METHOD FOR SOLVING …

Spectral Collocation Method for Fractional Differential ...downloads.hindawi.com/journals/ijde/2019/3734617.pdf · ResearchArticle Spectral Collocation Method for Fractional Differential/Integral

Spectral Collocation Method for Fractional Differential ...downloads.hindawi.com/journals/ijde/2019/3734617.pdf · ResearchArticle Spectral Collocation Method for Fractional Differential/Integral

Second-Generation Wavelet Collocation Method for the ...

Second-Generation Wavelet Collocation Method for the ...

An-Najah National University Faculty of Graduate Studies · scaling functions and wavelets, Homotopy perturbation method, Legendre polynomial method and Taylor collocation method.

An-Najah National University Faculty of Graduate Studies · scaling functions and wavelets, Homotopy perturbation method, Legendre polynomial method and Taylor collocation method.

A local RBF method based on a finite collocation approach€¦ · This collocation system is fully populated and non-symmetric. In the RBF-FC method, these collocation systems are

A local RBF method based on a finite collocation approach€¦ · This collocation system is fully populated and non-symmetric. In the RBF-FC method, these collocation systems are

Chebyshev Collocation Method for Differential Equations

Chebyshev Collocation Method for Differential Equations

AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD ...elman/papers/Cho-Elman.pdfAN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES

AN ADAPTIVE REDUCED BASIS COLLOCATION METHOD ...elman/papers/Cho-Elman.pdfAN ADAPTIVE REDUCED BASIS COLLOCATION METHOD BASED ON PCM ANOVA DECOMPOSITION FOR ANISOTROPIC STOCHASTIC PDES

A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra's Population Growth Model

A Multiple-Step Legendre-Gauss Collocation Method for Solving Volterra's Population Growth Model

Collocation Least-squares Polynomial Chaos Methodwillamette.edu/cla/cs/faculty/cheng/papers/collocation-least-square.pdf · Collocation Least-squares Polynomial Chaos Method Haiyan

Collocation Least-squares Polynomial Chaos Methodwillamette.edu/cla/cs/faculty/cheng/papers/collocation-least-square.pdf · Collocation Least-squares Polynomial Chaos Method Haiyan