Caputo s Implicit Solution of Time-Fractional Diffusion Equation Using Half-Sweep AOR Iteration

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1 Global Joural o Pure ad Applied Mathematics. ISSN Volume Number 4 (06) pp Research Idia Publicatios Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio sig Hal-Sweep AOR Iteratio A. Suarto* J. Sulaiama ad 3 A. Saudi Mathematics with Ecoomic Programme Faculty o Sciece ad Natural Resources iversiti Malaysia Sabah Kota Kiabalu Sabah Malaysia *Correspodig Author: 3 Sotware Egieerig Programme Faculty o Computig ad Iormatics iversiti Malaysia Sabah Kota Kiabalu Sabah Malaysia Abstract The aim o this paper deals with the applicatio o Hal-Sweep Accelerated Over- Relaxatio (HSAOR) iterative method usig a ucoditioally implicit iite dierece approximatio equatio rom the discretizatio o the oedimesioal liear time-ractioal diusio equatios by usig the Caputo s time ractioal derivative. The Caputo s implicit iite dierece approximatio equatio leads a liear system which is solved by usig the proposed HSAOR iterative method. Throughout implemetatios o two umerical experimets coducted it has show that the HSAOR method is superior as compared with FSAOR ad FSOR methods. Keywords: Caputo s ractioal derivative; Implicit iite dierece; HSAOR method.. Itroductio From previous studies [345] may scietiic ad egieerig have proposed to get ractioal partial dieretial equatios (FPDE s) a umerical ad/or aalytical approximate solutios o the ractioal problems. Actually various umerical techiques that ca be used to solve the time ractioal diusio equatios (TFDE s) such as trasorm methods [6] iite elemets together with the method o lies [3] explicit ad implicit iite dierece methods [37]. Based o the iite dierece methods there exist three discretizatio schemes ca be costructed such as explicit semi-implicit ad implicit.

2 3470 A. Suarto et al To get a iite dierece solutio o the time-ractioal diusio equatios (TFDE s) problem eeds to be discretized via iite dierece discretizatio scheme by imposig the implicit iite dierece scheme ad Caputo s ractioal operator. The correspodig approximatio equatios ca be used to costruct a liear system at each time level. To solve the liear system various iterative methods have bee proposed ad discussed by Youg [8] Hackbusch [9] ad Saad [0]. From the previous studies o iteratios there exists several amilies o iterative methods. I additio to these iterative methods the cocept o block iteratio has also bee itroduced by Evas [] ad urthermore explaatio o this block cocept have bee exteded by Ibrahim ad Abdullah [] Yousi ad Evas [34] i which these block iterative methods ca be oe o the eiciet iterative methods. To improve the covergece rate o iterative process the hal-sweep iteratio cocept is itroduced by Abdullah [5] via Explicit Decoupled Group (EDG) iterative method to solve two dimesioal Poisso equatios. Due to the advatage o this cocept this hal-sweep iterative methods have bee extesively studied by may researchers; see Ibrahim ad Abdulah [] Yousi ad Evas [34] Othma ad Abdullah [6] Sulaima Hasa ad Othma [7] Aruchua ad Sulaima [89] Muthuvalu ad Sulaima [0] Saudi ad Sulaima [3] Fauzi ad Sulaima [4] Hasa ad YitHoe [5] Sulaima [6] ad Akhir [7]. As metioed the advatages o AOR method [8] ad hal-sweep iteratio rom previous studies we examie the applicatio o the Hal-Sweep Accelerated Over- Relaxatio (HSAOR) iterative method or solvig time-ractioal diusio equatios (TFDE s) by usig the Caputo s implicit iite dierece approximatio equatio. To test perormace o the HSAOR method we also implemet the Full-Sweep Successive Over-Relaxatio (FSSOR) iterative method beig used as cotrol methods ad Full-Sweep Accelerated Over-Relaxatio (FSAOR) iterative method. To ivestigate the perormace o HSAOR method let us cosider time-ractioal diusio equatio (TFDE s) be deied as xt xt xt a x b x c x xt () x x where a(x) b(x) ad c(x) are kow uctios or costats whereas is a parameter which reers to the ractioal order o time derivative. The outlie o this paper is as ollows: I Sectios ad 3 the prelimiaries ad the Hal-Sweep Caputo s implicit approximatio equatio are preseted. The Sectio 4 deals with derivatio o amily o AOR methods. I which ormulatio o the HSAOR iterative method is itroduced. I Sectio 5 shows umerical example ad its results ad coclusio is give i Sectio 6.. Prelimiaries Beore costructig the approximatio equatio o problem () the ollowig are Hal-Sweep Caputo s implicit iite dierece ad some basic deiitios give or ractioal derivative theory

3 Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio 347 Deiitio. [9] The Riema-Liouville ractioal itegral operator is deied as x J x x t t dt 0x 0 0 J o order- () Deiitio.[9] The Caputo s ractioal partial derivative operator D o order- is deied as x m t D x dt 0 (3) m m 0 x t with m m m N x 0 As metioed i the previous sectio o gettig umerical solutio o problem () irstly we costruct implicit iite dierece approximate equatio based o the Caputo s derivative deiitio with Dirichlet boudary coditios ad cosider the o-local ractioal derivative operator. This approximatio equatio ca be categoried as ucoditioally stable scheme. To solve problem () the solutio domai o the problem has bee restricted to the iite space domai 0 x γ with 0 whereas the parameter reers to the ractioal order o time derivative. Cosider problem () associated with the boudary ad iitial coditios as ollow (0t ) g0 t t g t ad the iitial coditio x0 x where g0 t g t ad x are give uctios. To get a discretize approximatio to the time ractioal derivative i Eq. () we cosider Caputo s ractioal partial derivative o order which is deied as [98] ( xt ) u x s t s ds t 0 0 (4) t 0 t 3. Hal-Sweep Caputo s Implicit Fiite Dierece Approximatio To derive the Caputo s implicit iite dierece approximatio sectio let us cosider Eq.(4) to show the ormulatio o Caputo s ractioal partial derivative o the irst order approximatio method which is give as t i k D σ ω (5) i i where σ k k ad ω. To acilitate us i discretizig problem () let the solutio domai o the problem be partitioed uiormly. To do this we cosider some positive itegers m ad i

4 347 A. Suarto et al which the grid sizes i space ad time directios or the iite dierece algorithm are γ 0 T deied as h x ad k t respectively. Accordig to these grid sizes m we develop the uiormly grid etwork o the solutio domai where the grid poits i the space iterval 0 γ are show as the umbers ih i 0... m ad the grid poits i the time iterval x i 0 T are labeled k t The the values o the uctio x t at the grid poits are deoted as i xi t. By usig the ormulatio i Eq. (5) ad the implicit iite dierece discretizatio scheme the Hal-Sweep Caputo s implicit iite dierece approximatio equatio o problem () to the reerece grid poit at xi t ih k ca be show as σ k ω i i (6) ai i i i bi i i ci i 4h 4h or i=4...m-. Actually this hal-sweep approximatio equatio is classiied as the ully implicit iite dierece approximatio equatio which is cosistet irst order accuracy i time ad secod order i space. To costruct the liear system the the approximatio equatio (6) ca be rewritte based o the speciied time level. For istace we have or : k ω i i p i i q i i r i i σ (7) where ai bi pi 4h 4h ai qi ci h ai bi ri 4h 4h Also we get or = pi i qi i ri i i 0 i 4...m (8) where ω qi σ k qi i0 σk i 0. For the coveiece i solvig liear systems i Eq. (7) ad (8) Let us cosider the approximatio equatio (8) beig used i costructig the tridiagoal matrix orm as A (9) where

5 Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio 3473 A q * p 4 p q r * 4 p q r 4 * 6 p r m 4 q * m 4 p m 4 m m 4 q m r m 4 * m p m m m x T T m m 4. Formulatio o Hal-Sweep Accelerated Over-Relaxatio Reer to the liear system (9) it ca be see that the characteristic o the coeiciet matrix o the liear system has large scale ad sparse. It meas that the iterative methods are suitable optio to solve the liear system [8]. Thereore we cosider the applicatio o HSAOR method as liear solver (9). Particularly the HSAOR method is essetially the extesio o the FSAOR iterative method. The mai purpose o the hal-sweep iteratio is to reduce the computatioal complexities durig iteratio process. Developmet o the HSAOR method is the combiatio betwee the halsweep iteratio cocept ad Accelerated Over-Relaxatio (AOR) method. Let the liear system (9) be expressed as summatio o the three matrices A D L V (0) where D L ad V are diagoal lower triagular ad upper triagular matrices respectively. Accordig to Eq. (0) the HSAOR iterative method ca be deied geerally as [8] : k k D ωl βv β ω L β D β D ωl () where k represets a ukow vector at k th iteratio. Also the implemetatio o the HSAOR iterative method may be described i Algorithm. Algorithm : HSAOR method i. Iitializig all the parameters. Set k 0. ii. For ad i 4 m 4m Calculate k k D ωl βv β ω L β D β D ωl iii. Covergece test. I the covergece criterio i.e k k 0 ε 0 is satisied go to Step (iv). Otherwise go back to Step (ii). iv. Display approximate solutios.

6 3474 A. Suarto et al 5. Numerical Experimet I order to ivestigate the perormace o the proposed iterative methods we cosider two examples o the time ractioal diusio equatios beig used to demostrate eectiveess o the HSAOR compared with FSAOR ad FSSOR iterative methods. To do this three criteria have bee cosidered such as umber o iteratios executio time (i secods) ad maximum absolute error at three dieret values o = ad For implemetatio o these three iterative schemes the covergece 0 test cosidered the tolerace error which is ixed as = 0. Examples :[30] Cosider the ollowig time ractioal iitial boudary value problem be give as xt xt 0 0 x γ t 0 () t x where the boudary coditios are give i ractioal terms kt kt (0t ) ( t ) (3a) ( ) ( ) ad the iitial coditio x0 x. (3b) From Problem () as takig it ca be see that problem () ca be reduced to the stadard diusio equatio xt xt 0 x γ t 0 (4) t x with the iitial ad boudary coditios x0 x (0t ) kt ( t ) kt. The the aalytical solutio o Problem (4) is obtaied as ollows ( xt ) x kt. Now by applyig the series m i i m ( x0 ) t m ( x0 ) t ( xt ) m i 0 t! i 0 t ( i ) to ( xt ) or 0 it ca be show that the aalytical solutio o problem (4) is give as t ( xt ) x k. (5) ( ) Examples : [30] Let us cosider the ollowig time ractioal iitial boudary value problem be deied as xt xt x 0 0 x γ t 0 (6) t x where the boudary coditios are give i ractioal terms t (0t ) 0 (t ) e (7a)

7 Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio 3475 ad the iitial coditio x0 x. (7b) From problem (6) as takig it ca be show that Eq. (6) ca also be reduced to the stadard diusio equatio xt xt x t x 0 x γ t 0. (8) The the aalytical solutio o problem (8) is obtaied as ollows ( xt ) x e t. Now by applyig the series ( xt ) m i i m ( x0 ) t m ( x0 ) t m i 0 t! i 0 t ( i ) to ( xt ) or 0 it ca be show that the aalytical solutio o problem (8) is stated as ( xt ) 3 t t t x ( ) ( ) ( 3 )... (9) All the simulatios were implemeted by C programmig laguage. Results o umerical simulatios which were obtaied rom implemetatio o FSSOR FSAOR ad HSAOR iterative methods have bee recorded i Tables ad at dieret values o mesh sizes m = ad 048. Table : Compariso o Number Iteratios (K) The Executio Time (secods) ad Maximum Errors or The Iterative Methods sig Examples at M Method = 0.5 = 0.50 = 0.75 K Time Max Error K Time Max Error K Time Max Error 8 FSSOR e e e-04 FSAOR e e e-04 HSAOR e e e FSSOR e e e-04 FSAOR e e e-04 HSAOR e e e-04 5 FSSOR e e e-04 FSAOR e e e-04 HSAOR e e e FSSOR e e e-04 FSAOR e e e-04 HSAOR e e e FSSOR e e e-04 FSAOR e e e-04 HSAOR e e e-04

8 3476 A. Suarto et al From the umerical result recorded i Table by imposig the FSSOR FSAOR ad HSAOR iterative methods it is obvious at 0. 5 that umber o iteratios have declied approximately by % correspods to the HSAOR iterative method as compared with FSSOR methods. Particularly i terms o executio time implemetatios o HSAOR method are much aster about.0-99.% tha FSSOR methods. It meas that the HSAOR method requires the less amout or umber o iteratios ad computatioal time as compared with FSAOR ad FSSOR iterative methods. For other value o it ca be observed that their coclusios are i lie with Table : Compariso o Number Iteratios (K) The Executio Time (secods) ad Maximum Errors or The Iterative Methods sig Examples at M Method = 0.5 = 0.50 = 0.75 K Time Max Error K Time Max Error K Time Max Error 8 FSSOR e e e-0 FSAOR e e e-0 HSAOR e e e-0 56 FSSOR e e e-0 FSAOR e e e-0 HSAOR e e e-0 5 FSSOR e e e-0 FSAOR e e e-0 HSAOR e e e-0 04 FSSOR e e e-0 FSAOR e e e-0 HSAOR e e e FSSOR e e e-0 FSAOR e e e-0 HSAOR e e e-0 From the umerical result recorded i Table 0. 5 it ca be observed 0. 5 that umber o iteratios have declied approximately by % correspods to the HSAOR iterative method as compared with FSSOR methods. Similar to executio time implemetatios o HSAOR method are much aster about % tha FSSOR methods. It meas that the HSAOR method requires the less amout or umber o iteratios ad executio time at as compared with FSAOR ad FSSOR iterative methods. For other value o it ca be cocluded that their coclusios are i lie with 0. 5.

9 Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio Coclusio For the umerical solutio o the time-ractioal diusio problems this paper applied the derivatio o the Caputo s implicit iite dierece approximatio equatios i which this approximatio equatio leads a tridiagoal liear system. Based o the umerical result recorded i Tables ad it ca be poited out that the HSAOR method requires the less amout or umber o iteratios ad executio time at as compared with FSAOR ad FSSOR iterative methods. The observatio o the accuracy o proposed iterative methods show that their umerical solutios are i good agreemet. Reereces [] Maiardi F. 997 Fractals ad Fractioal Calculus Cotiuum Mechaics Spriger-Verlag Heidelberg. pp [] Diethelm K. ad Freed A.D. 999 Scietiic Computig i Chemical Egieerig II Computatioal Fluid Dyamic Reactio Egieerig ad Molecular Properties Spriger-Verlag Heidelberg. pp [3] Meerschaert M.M. ad Tadera C. 004 Fiite Dierece Approximatio or Fractioal Advectio-Dispersio Flow Equatios Joural Computatioal ad Applied Mathematics. 7 pp [4] Chaves A. 998 Fractioal Diusio Equatio to Describe Levy light Physics Letter. 39 pp.3-6. [5] Agrawal O.P. 00 Solutio or a Fractioal Diusio-Wave Equatio Deied i a Bouded Domai Noliear Dyamic. 9 pp [6] Yuste S.B. ad Acedo L. 005 A Explicit Fiite Dierece Method ad a New Vo Neuma-type Stability Aalysis or Fractioal Diusio Equatios SIAM Joural Numerical Aalysis. 4 pp [7] Yuste S.B. 006 Weighted Average Fiite Dierece Method or Fractioal Diusio Equatios Joural o Computatioal Physics. 6 pp [8] Youg D.M. 97 Iterative Solutio o Large Liear Systems Academic Press Lodo. [9] Hackbusch W. 995 Iterative Solutio o Large Sparse Systems o Equatios Spriger-Verlag New York. [0] Saad Y.996 Iterative Method or Sparse Liear Systems Iteratioal Thomas Publishig Bosto. [] Evas D.J. 985 Group Explicit Iterative Methods or Solvig Large Liear Systems Iteratioal Joural Computer Mathematics. 7 pp [] Ibrahim A. ad Abdullah A.R. 995 Solvig the two dimesioal diusio equatio by the Four Poit Explicit Decoupled Group (EDG) iterative method Iteratioal Joural Computer Mathematics. 58 pp

10 3478 A. Suarto et al [3] Yousi W.S. ad Evas D.J. 986 Explicit Group Iterative Methods or solvig elliptic partial dieretial equatios i 3-space dimesios Iteratioal Joural Computer Mathematics. 8 pp [4] Yousi W.S. ad Eva D.J. 995 Explicit De-coupled Group Iterative Methods ad Their Implemetatios Parallel Algorithm ad Applicatios. 7 pp.53-7 (995). [5] Abdulah A.R. 99 The Four Poit Explicit Decoupled group (EDG) Method: A Fast Poisso Solver Iteratioal Joural Computer Mathematics. 38 pp [6] Othma M. ad Abdullah A.R. 998 The Hal-Sweep Multigrid Method as a ast Multigrid Poisso Solver Iteratioal Joural Computer Mathematics. 69 pp [7] Sulaima J. Hasa M.K. ad Othma M. 005 The Hal-Sweep Iterative Alteratig Decompositio Explicit (HSIADE) Method or Diusio Equatios Computatioal ad Iormatio Sciece Lecture Note i Computer Sciece. 334 pp [8] Aruchua E. ad Sulaima J. 0 Hal-Sweep Cougate Gradiet method or Solvig First Order Liear Fredholm Itegro-dieretial Equatios Australia Joural o Basic ad Applied Scieces. 5 pp [9] Aruchua E. ad Sulaima J. Applicatios o the Cetral-Dierece with Hal-Sweep Gauss-Seidel Method or Solvig First Order Liear Fredholm Itegro-Dieretial Equatios World Academy o Sciece Egieerig ad Techology. 6 pp [0] Muthuvalu M.S. ad Sulaima J. 0 Computatioal Methods Based o Complexity Reductio Approach or First Kid Liear Fredholm Itegral Equatios with Semi-Smooth Kerel Joural o Moder i Numeric Mathematics. 3 pp [] Saudi A. ad Sulaima J. 03 Robot Path Plaig sig Laplacia Behaviour-Based Cotrol (LBBC) via Hal-Sweep SOR Techological Advaces i Electrical Electroic ad Computer Egieerig (TAEECE) 03 Iteratioal Coerece o (IEEE Koya) pp [] Saudi A. ad Sulaima J. 0 Path Plaig or Idoor mobile Robot sig Hal-Sweep SOR via Nie-Poit Laplacia (HSSOR9L) IOSR Joural o Mathematics. 3 pp [3] Saudi A. ad Sulaima J. 009 Path Plaig or mobile Robot with Hal- Sweep Successive Over-Relaxatio (HSSOR) Iterative Method I Proceedig o Symposium io progress i iormatio & Commuicatio Techology (SPICT 009) Kuala Lumpur pp [4] Fauzi N.I.M. ad Sulaima J. 0 Hal-Sweep Modiied Successive Over Relaxatio Method or Solvig Secod Order Two-Poit Boudary Value Problems usig Cubic Splie Iteratioal Joural Cotemporary Mathematics Scieces. 7 pp [5] Hasa M.K. ad Yit Hoe N. 03 Impact o Alteratig Let Right Strategy o Hal-Sweep Successive-Over-Relaxatio or The Solutio o Two

11 Caputo s Implicit Solutio o Time-Fractioal Diusio Equatio 3479 Dimesioal Boudary Value Problems Australia Joural o Basic ad Applied Scieces 7 pp [6] Sulaima J. Hasa M.K. Othma M. ad Karim S.A.A. 03 Numerical Solutios o Noliear Secod-Order Two-Poit Boudary Value Problems sig Hal-Sweep SOR With Newto Method Joural o Cocrete ad Applicable Mathematics. 557 pp.-0. [7] Akhir M.K.M. Othma M. Sulaima J. ad Maid Z.A. 0 Hal- Sweep Successive Over Relaxatios or Solvig Two-Dimesioal Helmholtz Equatio Australia Joural o Basic ad Applied Sciece. 5 pp [8] Hadidimos A. 978 Accelerated Over-Relaxatio Method Mathematics o Computatio. 3 pp [9] Zhag Y A Fiite Dierece Method For Fractioal Partial Dieretial Equatio Applied Mathematics ad Computatio. 5 pp [30] Ali S.E. Ozgur B.E. ad Korkmaz E. 03 Computatioal Methods Based o Complexity Reductio Approach or First Kid Liear Fredholm Itegral Equatios with Semi-Smooth Kerel Boudary Value Problem a Spriger Ope Joural. 68 pp

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