Australian Journal of Basic and Applied Sciences. Full-Sweep SOR Iterative Method to Solve Space-Fractional Diffusion Equations

Transcription

1 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: AENSI Jourals Australa Joural of Basc ad Aled Sceces ISSN: Joural home ae: Full-Swee SOR Iteratve Method to Solve Sace-Fractoal Dffuso Euatos 1 Ada Suarto, Jumat Sulama, 3 Azal Saud 1 verst Malaysa Sabah, Deartmet of Mathematcs wth Ecoomcs, Faculty of Scece ad Natural Resources, Kota Kabalu, Sabah, Malaysa. verst Malaysa Sabah, Deartmet of Mathematcs wth Ecoomcs, Faculty of Scece ad Natural Resources, Kota Kabalu, Sabah,, Malaysa. 3 verst Malaysa Sabah, Deartmet of Iformatcs, Faculty of Comut ad Iformatc, Kota Kabalu, Sabah, Malaysa. A R T I C L E I N F O Artcle hstory: Receved 3 Setember 14 Receved revsed form 17 November 14 Acceted 5 November 14 Avalable ole 13 December 14 Keywords: Cauto s Sace-Fractoal, Imlct fte dfferece, FSSOR method A B S T R A C T Backroud: Sace-Fractoal dffuso euatos are eeralzatos of classcal dffuso euatos whch are used model suerdffusve roblem. I ths aer, we eame the effectveess of the FSSOR teratve method to solve sace-fractoal dffuso euatos based o a ucodtoally mlct fte dfferece scheme. From the dscretzato of the oe dmesoal lear sace-fractoal dffuso euatos by us the Cauto s sace fractoal dervatve. We ca derve the Cauto s mlct fte dfferece aromato euato. The ths aromato euatos hece wll be used to eerated the corresod system of lear euatos. Two umercal eamles were used to make a comarso betwee FSGS ad FSSOR methods.based o comutatoal umercal results, t ca be cocluded that the roosed FSSOR teratve method s sueror to the FSGS teratve method 14 AENSI Publsher All rhts reserved. To Cte Ths Artcle: Ada Suarto, Jumat Sulama, Azal Saud, Full-Swee SOR Iteratve Method To Solve Sace-Fractoal Dffuso Euatos. Aust. J. Basc & Al. Sc., 8(4): , 14 INTRODCTION I recet years, may researchers are cocered to study of fractoal artal dfferetal euatos[(cho et al.,1), (Tadera et al.,6), (Mohammad et al.,14)]. The sace-fractoal dffuso euatos are tye of fractoal artal dfferetal euatos whch may researcher are solved umercally. For ett soluto of these roblems, may researchers were seek dfferet ways to solve these roblems. For stace (Azz et al.,13) used Chebyshev collocato method to dscretze sace-fractoal to obta a lear system of ordary dfferetal euatos ad used the fte dfferece method for solv the result system. The (Saadatmad et al.,11) used tau aroach, whle (She et al., 5) used sulated eds ad elct fte dfferece to solve sace-fractoal dffuso euatos. Also (Aslefallah et al.,14) aled theta scheme to solve sacefractoal dffuso euatos. I ths aer, we roose a dfferet aroach to et umercal solutos of the oe-dmesoal sacefractoal artal dfferetal euato (SFPDE s) teratvely whch defed as, t t a, t b, t c, t f, t (1) wth tal codto, f,, ad boudary codtos, t t, t T,,t 1 t, t T. Before develo the dscrete euato of Problem (1), we descrbe some ecessary basc deftos for fractoal dervatves whch are used the aer. Corresod Author: Ada Suarto, verst Malaysa Sabah, Deartmet of Mathematcs wth Ecoomcs, Faculty of Scece ad Natural Resources, Kota Kabalu, Sabah, Malaysa. Ph: , E-mal: ada99@mal.com.

2 154 Ada Suarto et al, 14 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: Defto 1.(Zha, 9) The Rema-Louvlle fractoal teral oerator, 1 J f () ( ) ( - t) 1 f t dt J of order- s defed as,, () Defto.(Azz et al.,13) The Cauto s fractoal artal dervatve oerator, as D of order - s defed (m) 1 f (t) D f dt, (3) m1 (m ) ( - t) wth m 1 m, mn, We have the follow roertes whe m 1 m, : D k, k s a costat, D, 1 1, for N for N ad ad fucto to deote the smallest teer reater tha or eual to, N =,1,,... ad. amma fucto. Accord to revous studes, may studes have bee coducted to vestate the effectveess of the FSSOR method [(Youssef, 1), (Su, 5), (Starke et al., 1991), (Haddmos, )]. However, there s o study that have bee coducted lterature for solv sace-fractoal dffuso euato Problem (1) by FSSOR method. Ths aer make a effort to eame the effectveess Full-Swee Successve Over- Relaato (FSSOR) teratve method whch s comared wth the Full-Swee Gauss-Sedel (FSGS) teratve method to solve Problem (1). For the umercal soluto of Problem (1), we derve umercal aromato euato based o the Cauto s sace-fractoal dervatve defto wth Drchlet boudary codtos ad also cosder the o-local fractoal dervatve oerator. Dervato of Cauto s mlct fte dfferece aromato euato ca be cateored as ucodtoally stable scheme. Dervato of Cauto s Imlct Fte Dfferece Aromato: Assume that h, k s ostve teer ad us secod order aromato, we et k t,t 1,s 1 t s s ( ) -1 1h 1-1, -, --1, h -s h h - = h 3 Let us defe h -,h 3 ad , -, --1, 1 - the the dscrete aromato of E. (4) -1,t,h - 1, -, s O --1, s the (4) Now we aromate Problem (1) by us Cauto s mlct fte dfferece aromato : 1 a, C, f,,-1,h -1, -, --1, b 1, h -1,

3 155 Ada Suarto et al, 14 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: for =1,,,m-1. The we ca smlfy the scheme aromato euato as b -1, -, --1, 1, -1, C, 1, -1 a,h h, f, So we et : b -1, 1 c, b 1, a -1, -, - 1, f (5) a a, b,h b, c c, F f, h ad f,-1 F Accord to E. (5), the aromato euato s kows as the fully mlct fte dfferece aromato euato whch s cosstet secod order accuracy sace-fractoal. For smlcty, let E. (5) for 3be rewrtte as R s r f (6) R a 1 3-3, - -1, - 1 -,, --1, a, s a 1 a, b a a 1 a, a 1 a c, r a b., 1, The E. (6) ca be used to costruct a lear system matr form as A f (7) 1 A 1 r 1 3 r 3 r 3 m- m- m-1 r m- m-1 m1 m m,1 T m1,1, m,1 m1,1 m1 T m,1 f. Full-Swee Successve Over-Relaato Iteratve Method: Based o the trdaoal lear system E. (7), t s clear that the characterstcs of ts coeffcet matr has lare scale ad sarse. Essetally, the varous cocet of teratve methods has bee coducted by may researchers such, [(You, 1954,1971,197), (Hackbush, 1995), (Saad, 1996), (Evas, 1985), (Yousf ad Evas, 1995) ad (Othma ad Abdullah, )]. For solv the trdaoal lear system, (You, 1954,1971,197) tated Full-Swee Successve Over-Relaato method. Namely as stadard SOR s the most kow ad wdely used teratve techues to solve solv ay lear systems. To derve ths FSSOR method, let the coeffcet matr A (7) ca be eressed as summato of the three matrces A D-L-V

4 156 Ada Suarto et al, 14 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: D, L, ad V are daoal, lower traular ad uer traular matrces resectvely. The, Sor teratve method ca be defed eerally as k1 1 k D L [ V (1 )D] D L 1 f (8) (k) descrbed Alorthm 1. reresets a ukow vector at k th terato. The alcato of the FSSOR method ca be Alorthm 1: FSSOR Method:.Italze 1 ad 1.For =1,,,-1 ad =1,,,m-1 ass k1 1 k D L [ V (1 )D] 1 D L f.coverece test. If the coverece crtero.e k1 k 1 1 s satsfed, o ste (v) Otherwse o back to ste () v.dslay aromate solutos Numercal Eermet: I ths secto, we have selected two eamles of the sace-fractoal dffuso euatos to verfy the effectveess of the Full-Swee Gauss-Sedel (FSGS) ad Full-Swee Successve Over-Relaato (FSSOR) teratve methods. I comarso, three crtera wll be cosdered for both teratve methods such as umber teratos, the eecuto tme (secods) ad mamum error at three dfferet values of 1., 1.5 ad 1.8. Dur the mlemetato of the ot teratos, the coverece test cosdered the tolerace error, 1 1. Eamles 1 (Azz et al., 13): Let us cosder the follow sace-fractoal tal boudary value roblem 1.5, t, t d, t, (9) 1.5 t O fte doma 1, wth the dffuso coeffcet.5 d 1.5, the source fucto, t 1cos t 1 s t 1, wth the tal codto, 1s(1) ad the boudary codtos, t st 1, 1, t s t 1, The Eact soluto of ths roblem s, t 1s t 1 for t>. Eamles (Azz et al., 13) : Let us cosder the follow sace-fractoal tal boudary value roblem 1.8, t 1.8, t -t (1.) 3-1 e, (1) 1.8 t wth the tal codto 3, -, ad zero Drchlet codtos -t The eact soluto of ths roblem s, t (1 - )e All umercal results for roblems (9) ad (1), obtaed from alcato of FSGS ad FSSOR teratve methods are recorded Tables 1 ad by us the dfferet value of mesh sze, M=18, 56, 51, 14 ad 48. Table 1: Comarso of umber teratos, the eecuto tme ( secods) ad mamum errors for the teratve methods us eamle at 1.,1.5,1.8 = 1. = 1.5 = 1.8 M Method K Tme Ma K Tme Ma K Tme Ma 18 FSGS e e e- FSSOR 5.1.6e e e- 56 FSGS e e e-

5 157 Ada Suarto et al, 14 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: FSSOR e e e- 51 FSGS e e e- FSSOR e e e- 14 FSGS e e e- FSSOR e e e- 48 FSGS e e e- FSSOR e e e- Table : Comarso of umber teratos, the eecuto tme ( secods) ad mamum errors for the teratve methods us eamle at 1.,1.5,1.8 = 1. = 1.5 = 1.8 M Method K Tme Ma K Tme Ma K Tme Ma 18 FSGS e e e-3 FSSOR e e e-3 56 FSGS e e e- FSSOR e e e- 51 FSGS e e e- FSSOR e e e- 14 FSGS e e e- FSSOR e e e- 48 FSGS e e e- FSSOR e e e- Cocluso: I ths aer we roosed a Cauto s mlct fte dfferece ad scheme FSSOR method to solve the sace-fractoal dffuso euatos. We have aled the formulato of the Cauto s fte dfferece euatos to eerate a corresod lear system. The for solv the lear system, the formulato of FSGS ad FSSOR teratve methods have bee costructed based o the Cauto s dervatve oerator. From observato of all eermetal results by mos the FSGS ad FSSOR teratve methods, t ca be also observed tables 1 ad that the umber of teratos ad the eecuto tme for FSSOR teratve method have bee decled tremedously as comared wth FSGS teratve method. Ths s due to the mlemetatos of FSSOR teratve method have bee accelerated by us the otmal value of the wehted arameter, ω. Accord to the accuracy ad of both teratve methods, t ca be cocluded that ther umercal solutos are ood areemet. REFERENCES Aslefallah, M. ad D. Rostamy, 14. A Numercal Scheme for Solv Sace-Fractoal Euato by Fte Dfferece Theta-Method. Iteratoal Joural of Advaces Aled Mathematcs ad Mechacs, : 1-9. Azz, H. ad G.B. Lohma, 13. Numercal Aromato for Sace-Fractoal Dffuso Euatos va Chebyshev Fte Dfferece Method. Joural of Fractoal ad Alcatos, : Cho, H.W., S.K. Chu ad Y.J. Lee, 1. Numercal Solutos for Sace-Fractoal Dserso Euatos wth Nolear Source Terms. Bull.Korea Math.Soc., : Evas, D.J., Grou Elct Iteratve methods for solv lare lear systems. Iteratoal Joural Comuter Maths, 17: Hackbusch, W., Iteratve Soluto of Lare Sarse Systems of Euatos. New York: Srer- Verla. Haddmos, A.,. Successve Over-relaatos ad Relatve Methods. Joural Of Comutatoal ad Aled Mathematcs, 13: Othma, M. ad A.R. Abdullah,. A Effcet Four Pots Modfed Elct Grou Posso Solver. Iteratoal Joural Comuter Mathematcs, 76: Saad, Y., Iteratve method for sarse lear systems. Bosto: Iteratoal Thomas Publsh. Saadatmad, A. ad M. Dehha, 11. A Tau Aroach for Soluto of The Sace-Fractoal Dffuso Euato. Joural of Comuter ad Mathematcs wth Alcatos, 6: She, S. ad F. Lu, 5. Aalyss of a Elct Fte Dfferece Aromato for The Sace- Fractoal Dffuso Euato wth Isulated Eds. Joural ANZIAM, 46(E): C871-C878. Starke, G. ad W. Nethammer, SOR for AX XB = C. Joural of Lear Alebra ad Its Alcatos, 154: Su, L-y, 5. A Comarso Theorem for The SOR Iteratve Method. Joural of Comutatoal ad Aled Mathematcs, 181: Tadera, C. ad M.M. Meerschaert, 7. A Secod-order Accurate Numercal Method for The Two- Dmesoal Fractoal Dffuso Euato. Joural of Comutatoal Physcs, :

6 158 Ada Suarto et al, 14 Australa Joural of Basc ad Aled Sceces, 8(4) Secal 14, Paes: You, D.M., Iteratve Methods for Solv Partal Dfferece Euatos of Elltc Tye. Tras. Amer. Math. Soc., 76: You, D.M., Iteratve Soluto of lare Lear Systems. Lodo: Academc Press. You, D.M., 197. Secod-Deree Iteratve Methods for The Soluto of Lare Lear Systems. Joural of Aromato Theory, 5: Yousf, W.S. ad D.J. Evas, Elct De-Couled Grou Iteratve Methods ad Ther Imlemetatos. Parallel Alorthms ad Alcatos, 7: Youssef, I.K., 1. O The Successve Over-relaato Method. Joural of Mathematcs ad Statstcs, : Zha, Y., 9. A Fte Dfferece Method For Fractoal Partal Dfferetal Euato. Aled Mathematcs Ad Comutato, 15: