HALF-SWEEP GAUSS-SEIDEL ITERATION APPLIED TO TIME-FRACTIONAL DIFFUSION EQUATIONS

Transcription

1 Jural Karya Asli Loreka Ahli Matematik Vol. 8 No. () Page 06-0 Jural Karya Asli Loreka Ahli Matematik HALF-SWEEP GAUSS-SEIDEL ITERATION APPLIED TO TIME-FRACTIONAL DIFFUSION EQUATIONS A. Suarto J. Sulaima ad A. Saudi 3 School of Sciece ad Techology Uiversiti Malaysia Sabah88400 Kota Kiabalu Sabah Malaysia 3 School of Egieerig ad Iformatio Techology Uiversiti Malaysia Sabah Kota Kiabalu Sabah Malaysia adag99@gmail.com umat@ums.edu.my 3 azali60@gmail.com Abstract : I this study we derive a fiite differece approximatio equatio from the discretizatio of the oedimesioal liear time-fractioal diffusio equatios by usig the Caputo s time fractioal derivative. A liear system will be geerated by the Caputo s fiite differece approximatio equatio. The the resultig of the liear system has bee solved usig Half-Sweep Gauss-Seidel (HSGS) iterative method i which its effectiveess will be compared with the existig Gauss-Seidel method (kow as Full-Sweep Gauss-Seidel (FSGS)). A example of the problem is preseted to test the effectiveess the proposed method. The fidigs of this study show that the proposed iterative method is superior compared with the FSGS method. Keywords: Caputo s fractioal derivative; Implicit scheme; HSGS method..0 Itroductio Accordig to previous studies [345] the use of fractioal partial differetial equatios (FPDEs) have attracted may researchers i mathematics physics egieerig ad chemistry to obtai a umerical ad/or aalytical solutios of the fractioal problems. For Istat a fractioal derivative replaces the first-order space partial derivative i a diffusio model ad lead to slower diffusio [5]. So to solve a oe-dimesioal diffusio model with costat coefficiets aalytical solutios are available usig iterative methods. Based o umerical techiques applied to the time fractioal diffusio equatios (TFDE) may proposed methods have bee iitiated such as trasform methods [6] fiite elemets together with the method of lies [3] explicit ad implicit fiite differece methods [37]. Although the explicit methods are coditioally stable this fiite differece schemes are available i the literature [8]. Implicit study The time fractioal diffusio equatios (TFDE) problem will be discretised.by imposig the implicit fiite differece scheme ad Caputo fractioal operator the approximatio equatios ca be used to costruct a liear system at each time level. To solve the liear systems the cocept of the iterative methods have bee discussed by may researchers such as Youg [9] Hackbusch [0] ad Saad [] ad it ca be observed that there are several families of iterative methods. I additio to these iterative methods the cocept of block iteratio has also bee itroduced by Evas [3] Ibrahim ad Abdullah [] Yousif ad Evas [43] to show the efficiecy of its computatio cost. The mai obective of this paper is to study the effectiveess of the Half-Sweep Gauss-Seidel (HSGS) iterative method for solvig time-fractioal parabolic partial differetial equatios (TPPDE s) by usig the Caputo s implicit fiite differece approximatio equatio. To show the capability of the HSGS method we also implemet the Full-Sweep Gauss Seidel (FSGS) iterative methods beig used as a cotrol method. To idicate the efficiecy of this HSGS method let us cosider time-fractioal diffusio equatio (TFDE s) be defied as Ux t U x t U x t a x b x cxu x t () x x Jural Karya Asli Loreka Ahli Matematik Published by Pustaka Ama Press Sd. Bhd.

2 7 A. Suarto et. al. a(x) b(x) ad c(x) are kow fuctios or costats as α is a parameter which refers to the fractioal order of time derivative. The orgaizatio of the paper is as follows: I Sectio ad 3 we approximate the formula of the Caputo s fractioal derivative operator ad umerical procedure for solvig time fractioal diffusio equatio () by meas of the implicit fiite differece method. I Sectio 4 formulatio of the HSGS iterative method is itroduced. I Sectio 5 we give a umerical example ad the results ad coclusio are give i Sectio Prelimiaries Before developig the discrete equatio of Problem (). We itroduce some basic defiitios Defiitio. [8] The Riema-Liouville fractioal itegral operator J f x x 0 x t f t dt J of order- is defied as 0 x 0 () Defiitio.[8] The Caputo s fractioal partial derivative operator x m f t D f x dt m m 0 x t with m m mn x 0 D of order - is defied as 0 (3) The purpose of this paper is examie the Half-Sweep Gauss-Seidel (HSGS) iterative method which is compared with the Full-Sweep Gauss-Seidel (FSGS) iterative method for solvig Problem () with variable coefficiets. I solvig umerical of Problem () we derive umerical approximatios based o the Caputo s derivative defiitio with Dirichlet boudary coditios ad cosider the o-local fractioal derivative operator. This approximatio equatio ca be categorized as ucoditioally stable scheme. O stregth of Problem () the solutio domai of the problem has bee restricted to the fiite space domai 0 x with 0 as the parameter refers to the fractioal order of space derivative. To solve of Problem () let us cosider the iitial boudary coditios of Problem () U0 t g0t ad t Ux 0 f x g0t gt ad f x are give fuctios. For a discretize approximatio to the time fractioal derivative i Eq. () we cosider Caputo s fractioal partial derivative of order defied by [89] u( x t) t 0 x s t s ds t 0 0 u t 3.0 Caputo s Implicit Fiite Differece Approximatio Based o Eq. (4) the formulatio of Caputo s fractioal partial derivative of the first order approximatio method is give as D t U k U U (5) k k ad (4)

3 Jural KALAM Vol. 8 No. Page Before discretizig Problem () we assume that the solutio domai of the problem be partitioed uiformly. To do this we cosider some positive itegers m ad i which the grid sizes i space ad 0 time directios for the fiite differece algorithm are defied as h x ad m T k t respectively. Based o these grid sizes we costruct the uiformly grid etwork of the solutio domai the grid poits i the space iterval 0 are idicated as the umbers x i ih i 0... m 0 T are labeled t k U U ad the grid poits i the time iterval The the values of the fuctio x t x t 8 U at the grid poits are deoted as i. By usig Eq. (5) ad the implicit fiite differece discretizatio scheme the Caputo s implicit fiite differece approximatio equatio of Problem () to the grid poit cetered at xi t ihk is give as k ai for i=...m-. U U i i (6) U U U b U U c U i Accordig to Eq. (6) this approximatio equatio is kow as the fully implicit fiite differece approximatio equatio which is cosistet first order accuracy i time ad secod order i space. Basically the approximatio equatio (6) ca be rewritte based o the specified time level. For istace we have for : ai bi ai ai bi k U U U i ci U i U i (7a) h k piu qiu riu U U ai bi pi ai qi ci h a b r i i i Also for = p i U qi U ru i f0 i... m (7b) q f i q k i U. 0 k 0

4 A. Suarto et. al. Accordig to Eq. (7b) it ca be see that the tridiagoal liear system ca be costructed i matrix form as AU f (8) q * r p q * r p3 q * 3 r A 3 p * m q m rm p * m q m mxm U U U U3 Um Um T f U pu 0 U U3 Um Um pmu m T. 4.0 Formulatio of Half-Sweep Gauss-Seidel As metioed i sectio 4 ad see the coefficiet matrix A of Eq.(8) has a large scale ad sparse. The cocept of various iterative methods has bee iitiated ad coducted by may researchers such as Youg [9] Hackbusch [0] Saad [] Evas [3] Yousif ad Evas [43] ad Othma ad Abdullah []. To Solve the tridiagoal liear system Abdullah [4] iitiated Half-Swep iteratio which is the most kow ad widely used iterative techiques to solve liear systems.i additio to that the iteratio has bee extesively used by may researchers; see Ibrahim ad Abdulah [] Othma ad Abdullah [] Suliama et al.[0] Aruchua ad Sulaima [789] Muthuvalu ad Sulaima [6] ad Yousif ad Evas [43].The mai advatage of the Half-Sweep iteratio is to reduce the computatioal complexities durig iteratio process. As a result of this cocept ad usig HSGS method the coefficiet matrix of the liear system (8) ca be expressed as summatio of the three matrices A D L V () D L ad V are diagoal lower triagular ad upper triagular matrices respectively. Thus Half-Sweep Gauss-Seidel iterative method ca be defied geerally as k ( k ) U D L V U f () k U represets a ukow vector at k th iteratio. The implemetatio of the HSGS iterative method ca be described i Algorithm. Algorithm : HSGS method i. Iitializig all the parameters. Set k 0. ii. For i p p p p p Calculate i p ( k ) ( k) U i fi M U M i p p ad i p p p i p i p M ( k ) U 9

5 Jural KALAM Vol. 8 No. Page 06-0 iii. Covergece test. If. U k k U 0 to Step (iv). Otherwise go back to Step (ii). 0 is satisfied go iv. Display approximate solutios. 5.0 Numerical Experimet I this sectio a example of the time fractioal diffusio equatio is give to illustrate the accuracy ad effectiveess properties of the Full-Sweep Gauss-Seidel (GS) ad Half-Sweep Gauss-Seidel (HSGS) iterative methods. For compariso purposes three criteria have bee cosidered such as umber of iteratios executio time (i secods) ad maximum absolute error at three differet values of α = ad For implemetatio of three iterative schemes the covergece test 0 cosidered the tolerace error which is fixed as = 0. Let us cosider the time fractioal iitial boudary value problem be give as [5] Ux t Ux t 0 0 x t 0 (3) t x the boudary coditios are stated i fractioal terms kt kt U 0 t U t (4) ad the iitial coditio Ux0 x. (5) it ca be see that Eq. (3) ca be reduced to the stadard From Problem (3) as takig diffusio equatio U xt Uxt t x 0 x t 0 (6) subected to the iitial coditio Ux 0 x ad boudary coditios U0 t kt U t kt The the aalytical solutio of Problem (6) is obtaied as follows Ux t x kt. Now by applyig the series m m mi i U x0 t Ux0 t U x t mi 0 t! i0 t i U x t for 0 it ca be show that the aalytical solutio of Problem (3) is give as to t U x t x k. All results of umerical experimets for Problem (3) which were obtaied from implemetatio of FSGS ad HSGS iterative methods have bee recorded i Table at differet values of mesh sizes m = ad 8. 0

6 A. Suarto et. al. TABLE. Compariso of umber iteratios the executio time ( secods) ad maximum errors for the iterative methods usig example at α = 0.5 α = 0.50 α = 0.75 M Method K Time Max Error K Time Max Error K Time Max Error 8 FSGS e e e- HSGS e e e- 56 FSGS e e e- HSGS e e e- 5 FSGS e e e0-4 HSGS e e e- FSGS e e e- HSGS e e e- 8 FSGS e e e- HSGS e e e- 6.0 Coclusio For the umerical solutio of the time fractioal diffusio problems the paper presets the derivatio of the Caputo s implicit fiite differece approximatio equatios i which this approximatio equatio leads to a tridiagoal liear system. From observatio of all experimetal results by imposig the FSGS ad HSGS iterative methods it is obvious at 0. 5that umber of iteratios have declied approximately by % correspods to the HSGS iterative method compared with the FSGS methods. Agai i terms of executio time implemetatios of HSGS method are much faster about % tha the FSGS methods. It meas that the HSGS method requires smaller umber of iteratios ad computatioal time at 0. 5 as compared with FSGS iterative methods. Based o the accuracy of both iterative methods it ca be cocluded that their umerical solutios are i good agreemet. Refereces [] Maiard F 997 Fractals ad Fractioal Calculus Cotiuum Mechaics. Heidelberg: Spriger-Verlag [] Diethelm K. ad Freed A.D 999 O the solutio of oliear fractioal order differetial equatio used i the modelig of viscoelasticity i Scietific Computig i Chemical egieerig II Computatioal Fluid Dyamic Reactio Egieerig ad Molecular Properties Heidelberg: Spriger-Verlag 7-4. [3] Meerschaert M.M. ad Tadera C 0 Fiite differece approximatio for fractioal advectio-dispersio flow equatios JCAM 7(0) [4] Chaves A 998 Fractioal diffusio equatio to describe Levy flight Phys.Lett A

7 Jural KALAM Vol. 8 No. Page 06-0 [5] Agrawal O.P 00 Solutio for a fractioal diffusio-wave equatio defied i a bouded doma Noliear Dyamic 9(00) [6] Yuste S.B. adacedo L 0 A explicit fiite differece method ad a ew vo Neuma-type stability aalysis for fractioal diffusio equatios SIAM J. Numer. Aal.4 (5) (0) [7] Yuste S.B 006 Weighted average fiite differece method for fractioal diffusio equatios J.Comp.Phys.6 (006)64-74 [8] Zhag Y 009 A Fiite Differece Method For Fractioal Partial Differetial Equatio Applied Mathematics Ad Computatio 5 (009) [9] Youg D.M 97 Iterative solutio of large liear systems. Lodo: Academic Press. [0] Hackbusch W 995 Iterative solutio of large sparse systems of equatios. New York: Spriger-Verlag. [] Saad Y 996 Iterative method for sparse liear systems. Bosto: Iteratioal Thomas Publishig. [] Ibrahim A ad Abdullah A.R 995 Solvig the two dimesioal diffusio equatio by the Four Poit Explicit Decoupled Group (EDG) iterative method. Iteratioal Joural Computer Mathematics. 58 (995) [3] D.J. Evas. 985 Group Explicit Iterative methods for solvig large liear systems It. J. Computer Maths. 7:8-08. [4] Yousif W. S ad Evas D.J Explicit Group Iterative Methods for solvig elliptic partial differetial equatios i 3- space dimesios. It. J. Computer Maths.8 (986) [5] Al S E Ozgur B ad Korkmaz E Aalysis Of Fractioal Partial Differetial Equatios By Taylor Series Expasio. Boudary Value Problem A Spriger Ope (03)68. [6] Muthuvalu M.S ad Sulaima J Computatioal Methods Based o Complexity Reductio Approach for First id Liear Fredholm Itegral Equatios with Semi-Smooth Kerel Jural Of Moder i Numeric Mathematics (0) [7] Aruchua E ad Sulaima J Half-Sweep Cougate Gardiet method for Solvig First Order Liear Fredholm Itegro-differetial Equatios Australia Joural of Basic ad applied Scieces (0) [8] Aruchua E ad Sulama J. Half Sweep Quadrature-Differece Schemes with Iterative Method I Solvig Liear Fredholm Itegro-Differetial Equatios Progress i Applied Mathematics (03) -. [9] Aruchua E ad Sulaima J. Applicatios of the Cetral-Differece with Half-Sweep Gauss-Seidel Method for Solvig First Order Liear Fredholm Itegro-Differetial Equatios World Academy of Sciece Egieerig ad Techology (0) [0] Sulaima J Othma M ad Hasa M.K A New Half-Sweep Algebraic Multigrid (HSAMG) Algorithm for Two-Poit Boudary Problems Proceedigs of the Iteratioal Cofrece o Statistics ad Mathematics ad Its Applicatios i the Developmet of Sciece ad Techology (0) [] Othma M ad Abdullah A.R The Half-Sweep Multigrid Method as a fast Multigrid Poisso Solver It J.Comp Math (998) [] Sulaima J Hasa M.K ad Othma M The Half-Sweep Iterative Alteratig Decompositio Explicit (HSIADE) Method for Diffusio Equatios I Comp.l ad Ifo Sciece (0). [3] Yousif W ad Evas D.J Explicit De-coupled Group Iterative Methods ad Their Implemetatios Parallel Algorithm ad Applicatios (995) 53-7 [4] Abdulah A.R The Four Poit Explicit Decoupled group (EDG) Method: A Fast Poisso Solver It.J Comp.Math (99)6-70.