VARIATIONAL ITERATION METHOD: A COMPUTATIONAL TOOL FOR SOLVING COUPLED SYSTEM OF NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS

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1 Joral of Sciece a Ars Year 6 No. 336 pp ORIGINAL PAPER ARIATIONAL ITERATION METHOD: A COMPTATIONAL TOOL FOR SOLING COPLED SYSTEM OF NONLINEAR PARTIAL DIFFERENTIAL EQATIONS MORF OYEDNSI OLAYIOLA Mascrip receive:.8.6; Accepe paper: 6.8.6; Pblishe olie: Absrac. I his research work a compaioal meho which is base o he ariaioal Ieraio Meho IM is se for he merical solio of he cople sysem of oliear parial iffereial eqaios. Nmerical eamples are presee o show he covergece a accracy of he meho. Keywors: variaioal ieraio meho sysem of oliear parial iffereial eqaios Lagrage mliplier.. INTRODCTION The wie applicabiliy of sysem of parial iffereial eqaios has mae is sy arace mch aeio i a variey of applie scieces. Researchers [-] have se iffere merical mehos o solve forms of parial iffereial eqaios. These sysems were formally moele o ivesigae wave propagaio a o corol he shallow waer waves [6-]. Ioki e al [3] propose a Lagrage mliplier meho o solve oliear problems i qam mechaics i 978. He [4] laer moifie i io a variaioal ieraio meho o solve a large class of liear a oliear iffereial eqaios. I his work he variaioal ieraio meho iroce by He [4] will be se o solve cople oliear parial iffereial eqaios merically.. ARIATIONAL ITERATION METHOD Accorig o he variaioal ieraio meho [4] we cosier he iffereial eqaio: L N g where L is a liear operaor N is a oliear operaor a g is a ihomogeeos erm. A correcio fcioal ca be cosrce as follows: Os Sae iversiy College of Sciece Egieerig a Techology Facly of Basic a Applie Scieces Deparme of Mahemaical a Physical Scieces Osogbo Nigeria. olayiwola.oyesi@ios.e.g. ISSN: Mahemaics Secio

2 ariaioal ieraio meho Morf Oyesi Olayiwola Mahemaics Secio 44 g N L where is a geeral Lagragia mliplier which ca be ieifie opimally via variaioal heory. The seco erm o he righ is calle he correcio a is cosiere as a resrice variaio i.e.. 3. APPLICATIONS I his secio he IM will be se o solve sysems of oliear parial iffereial eqaios. Eample 3.. Cosier he cople sysem of oliear homogeeos parial iffereial eqaios [] of he form: 3 Therefore accorig IM he correcio fcioal ca be erive as follows: 4 here a are geeral Lagrage mlipliers a eoe resrice variaios i.e. Makig he above correcio fcioal saioary he 5

3 ariaioal ieraio meho Morf Oyesi Olayiwola 45 a herefore 6 7 The Lagrage mlipliers ca be ieifie as: 8 Sbsiig eqaio 8 io eqaio 4 resls i he followig ieraio formla. 9 wih he iiial coiios si. The followig erms ca be obaie coscos coscos cos cos cos cos cos cos cos cos. ISSN: Mahemaics Secio

4 46 ariaioal ieraio meho Morf Oyesi Olayiwola 3 3 cos...! 3! 3 3 cos...! 3! Hece i a close form he eac solios of he cople sysems are give by e si e si 3 Eample 3.. Cosier he sysem of homogeos a ihomogeeos oliear parial iffereial eqaios of he form: 4 wih iiial coiios 5 Accorig o IM he ieraio formla for eqaios 4 is 6 where 7 The followig erms ca be obaie from eqaios Therefore he eac solios are : Mahemaics Secio

5 ariaioal ieraio meho Morf Oyesi Olayiwola 47 Eample 3.3. Cosier he ihomogeeos sysem of parial iffereial eqaios. 3 3 wih iiial coiios e e The correcio fcioal of eqaio is: 3 3 Makig he above correcio fcioal saioary he he saioary coiios become: 3 which gives: 4 This gives he ieraio formla as follows: Therefore he followig sccessive erms were obaie. e e 6 ISSN: Mahemaics Secio

6 48 ariaioal ieraio meho Morf Oyesi Olayiwola e e e 7 e e e e e e e! e e! oise erms oise erms 8 By cacellig he oise erms bewee... a bewee... he eac 3 3 solio e e 9 ca be obaie. CONCLSION I his paper he variaioal ieraio meho has bee se o solve cople sysem of oliear parial iffereial eqaios homogeeos ihomogeeos a boh irecly wiho ay resricive assmpios or liearizaio. This meho is very effecive a accelerae he coverge of solio o eac for he eamples presee. The meho also ca be applie o sysem ha comprises of more ha wo liear a oliear parial iffereial eqaios of higher orer. REFERENCES [] Mahmo S.R. Sheh M. Ieraioal Joral of Pre a Applie Mahemaics [] Jasei F. Applie Mahemaics Scieces [3] IokiM. Sekie H. MraT.Geeral se of he Lagrage Mliplier i oliear Mahemaical Physics i Nema-Nasse S. E. ariaioal Meho i he Mechaics of Solis Pergamo Press N.Y SA 978. [4] He J.H. Comper Mehos Applicaio Mech Eg [5] He J.H. Ieraioal Joral of Noliear Mech [6] Debah L. Noliear Parial iffereial eqaios for scieiss a Egieers Birkhaser Boso 997. [7] Loga J.D. A Irocio o oliear parial iffereial eqaios iley- Iersciece New-York 994. [8] hiham G.B. Liear a oliear aves iley New York 974. [9] Lbich C. Oserma A. BIT Nmerical Mahemaics [] aewalle R.P. Applicaio Ner Mahs Mahemaics Secio