Modified Decomposition Method for Solution of Fractional Partial Differential Equations of Two-Sided
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1 Arile Ieraioal Joral of Moder Mahemaial Siee 4: 3-36 Ieraioal Joral of Moder Mahemaial Siee Joral homepage: ISSN: 66-86X Florida USA Modified Deompoiio Mehod for Solio of Fraioal Parial Differeial Eqaio of Two-Sided I. I. Gorial Deparme of Mahemai Ib Al Haiham College Edaio Baghdad Uiveriy Arile hiory: Reeived Ag Reeived i revied form 5 Sepember Aeped 7 Sepember Pblihed 8 Sepember. Abra: I hi paper a aalyial olio of he wo-ided fraioal parial differeial eqaio ha bee preeed. The algorihm for he aalyial olio for hi eqaio i baed o modified deompoiio mehod. The fraioal derivaive i deribed i Capo' ee. The aalyial mehod ha bee applied o olve a praial eample ad he rel have bee ompared wih he ea olio. Keyword: Modified deompoiio mehod Capo derivaive Fraioal parial differeial eqaio. Mahemai Sbje Claifiaio : 49M7. Irodio The fraioal all i ed i may field of iee ad egieerig [ 3]. The olio of differeial eqaio oaiig fraioal derivaive i mh ivolved ad i lai aalyi mehod are maily iegral raform h a aplae raform Forier raform Melli raform e.[4] I ree year Adomia deompoiio mehod i applied o olvig fraioal differeial eqaio. Thi mehod effiiely work for iiial vale or bodary vale problem for liear or oliear ordiary or parial differeial eqaio ad eve for ohai yem [5] a well. By ig hi mehod Saha Ray e al [6 7 8 ] olved liear differeial eqaio oaiig fraioal derivaive of order / or 3/ ad oliear differeial eqaio oaiig fraioal derivaive of order/. I hi paper we oider he wo-ided fraioal parial differeial eqaio of he form [3]: Copyrigh by Moder Sieifi Pre Compay Florida USA
2 I. J. Moder Mah. Si. 4: 3-36 Copyrigh by Moder Sieifi Pre Compay Florida USA 3 bje o he iiial odiio = f R ad he bodary odiio = k T R = k T where + - ad are kow fio of ad f i a kow fio of k ad k are kow fio of ad i a give fraioal mber. Whe = ad eq. beome he followig laial paraboli parial differeial eqaio: Similarly whe = ad eq. rede o he followig laial hyperboli parial differeial eqaio I he pree work we apply he modified deompoiio mehod for olvig eq. ad ompare he rel wih ea olio. The paper i orgaized a follow. Seio i he mahemaial ape; I Seio 3 i abo he bai idea of modified deompoiio mehod; I Seio 4he wo-ided fraioal parial differeial eqaio ad i olio by modified deompoiio mehod are preeed. I Seio 5 a merial eample i olved ig he modified deompoiio mehod. Fially he mmarized olio abo he olio of wo-ided fraioal parial differeial eqaio i i Seio 6.. Mahemaial Ape The mahemaial defiiio of fraioal all ha bee he bje of everal differe approahe [45]. The Capo fraioal derivaive operaor D of order i defied i he followig form:
3 I. J. Moder Mah. Si. 4: D f m m f m d where m m m. Similar o ieger-order differeiaio Capo fraioal derivaive operaor i a liear operaio: D f g D f D g where ad are oa. For he Capo' derivaive we have: D ad D R R R 3. Bai Idea of Modified Deompoiio Mehod Coider a geeral oliear eqaio [] R F g where i he operaor of he highe-ordered derivaive wih repe o ad R i he remaider of he liear operaor. The oliear erm i repreeed by F. Th we ge g R F 3 The ivere d i amed a iegral operaor give by The operaig wih he operaor o boh ide of Eqaio 3 we have f g R F 4 where f i he olio of homogeeo eqaio 5 Copyrigh by Moder Sieifi Pre Compay Florida USA
4 I. J. Moder Mah. Si. 4: ivolvig he oa of iegraio. The iegraio oa ivolved i he olio of homogeeo Eqaio 5 are o be deermied by he iiial or bodary odiio aordig a he problem i iiial-vale problem or bodary-vale problem. The ADM ame ha he kow fio a be epreed by a ifiie erie of he form ad he oliear operaor F a be deompoed by a ifiie erie of polyomial give by F A where will be deermied rerrely ad A are he o-alled polyomial of defied by A d F! d i i B he modified deompoiio mehod wa iroded by Wazwaz [9]. Thi mehod i baed o he ampio ha he fio f a be divided io wo par amely f ad f. Uder hi ampio we e f f f We apply hi deompoiio whe he fio f oi of everal par ad a be deompoed io wo differe par. I hi ae f i ally a mmaio of a polyomial ad rigoomeri or raedeal fio. A proper hoie for he par f i impora. For he mehod o be more effiie we ele f a oe erm of f or a lea a mber of erm if poible ad f oi of he remaiig erm of f. 4. Modified Deompoiio Mehod for Solvig he Two-Sided Fraioal Parial Differeial Eqaio We adop modified deompoiio mehod for olvig Eqaio. I he ligh of hi mehod we ame ha Copyrigh by Moder Sieifi Pre Compay Florida USA
5 I. J. Moder Mah. Si. 4: 3-36 Copyrigh by Moder Sieifi Pre Compay Florida USA 34 Now Eqaio a be rewrie a where whih i a eaily iverible liear operaor ad are he Capo derivaive of order. Therefore we a wrie The he modified deompoiio mehod MDM rerive heme i a follow f f 5. Nmerial Appliaio I hi eio we apply modified deompoiio mehod for fidig he aalyial olio of wo-ided fraioal parial differeial eqaio:
6 I. J. Moder Mah. Si. 4: Wih he oeffiie fio:.5 ad e. 8 e ad he ore fio: bje o he iiial odiio =.4 ad bodary odiio = ad.4 =.8e. Noe ha he ea olio o hi problem i: = e. Table how he aalyial olio for wo-ided fraioal parial differeial eqaio obaied for differe vale ad ompario bewee ea olio ad aalyial olio. Table. Compario bewee ea olio ad aalyial olio whe. 5 for wo-ided fraioal parial differeial eqaio Ea Solio Modified Adomia Deompoiio Mehod e - MADM Colio I hi paper aalyial olio for fraioal wo-ided fraioal parial differeial eqaio obaied by ig modified deompoiio mehod ha bee deribed ad demoraed. I i lear ha he modified deompoiio mehod i i high agreeme wih he ea olio. Copyrigh by Moder Sieifi Pre Compay Florida USA
7 I. J. Moder Mah. Si. 4: Referee [] Podlby I. Fraioal Differeial Eqaio Aademi Pre Sa Diego 999. [] Miller KS Ro B. A Irodio o he Fraioal Call ad Fraioal Differeial Eqaio Wiley New York 993. [3] Shimiz N Zhag W. Fraioal all approah o dyami problem of vioelai maerial JSME Serie C Mehaial Syem Mahie Eleme ad Mafarig 4999: [4] Da J. S. Time ad pae fraioal parial differeial eqaio J. Mah. Phy 465: [5] Adomia G. Noliear Sohai Operaor Eqaio Aademi Pre New York 986. [6] Saha R. S. Poddar B. P; Bera R.K. Aalyial olio of a dyami yem oaiig fraioal derivaive of order / by Adomia deompoiio mehod ASMEJ ApplMeh 75: [7] Saha R. S. Bera R.K. Aalyial olio of he Bagley Torvik eqaio by Adomia deompoiio mehod Appl. Mah. Comp. 685: [8] Saha R. S. Bera R.K. A approimae olio of a oliear fraioal differeial eqaio by Adomiade ompoiio mehod Appl. Mah. Comp. 675: [9] Wazwaz A. A reliable modifiaio of Adomia deompoiio mehod Appl. Mah. Comp. 999: [] Khader M. M. O he merial olio for he fraioal diffio eqaio Commiaio i Noliear Siee ad Nmerial Simlaio 6: [] Ray S. S. Aalyial olio for he pae fraioal diffio eqaio by wo-ep Adomia deompoiio mehod Comm. Noliear. Si. Nmer. Siml. 449: [] Wazwaz A. M. A reliable reame for mied Volerra-Fredholm iegral eqaio Applied Mahemai ad Compaio 7: [3] Meerhaer M. ad Tadjera C. Fiie Differee Approimaio for Two- Sided Spae- Fraioal Parial Differeial Eqaio Appl. Nmer. Mah. 566: 8-9. [4] Oldham K. B.ad SpaierJ. Fraioal Call: Theory ad Appliaio Differeiaio ad Iegraio o Arbirary Order Aademi Pre I. New York-odo 974. [5] Podlby I. A Irodio o Fraioal Derivaive Fraioal Differeial Eqaio Some Mehod of Their Solio ad Some of Their Appliaio Fraioal Differeial Eqaio Mahemai i Siee ad Egieerig Vol. 98 Aademi Pre Sa Diego 999. Copyrigh by Moder Sieifi Pre Compay Florida USA
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