Revised Variational Iteration Method for Solving Systems of Ordinary Differential Equations

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1 Availabl at Appl. Appl. Math. ISSN: 9-9 Spcial Iu No. Augut 00 pp. 0 Applicatio ad Applid Mathatic: A Itratioal Joural AAM Rvid Variatioal Itratio Mthod for Solvig St of Ordiar Diffrtial Equatio Elha Salhpoor Dpartt of Mathatic Ilaic Azad Uivrit - Noohahr Brach Noohahr Ira lha_alhpor@ahoo.co Hoi Jafari Dpartt of Mathatic Uivrit of Mazadra Babolar Ira jafari@uz.ac.ir Mrih Abbaia Afrapoli Dpartt of Mathatic Ilaic Azad Uivrit - Chahahr Brach P.O Bo Chahahr Ira arih_abbaia@ahoo.co Rcivd: Ma 0 00; Accptd: Jul 5 00 Abtract A odificatio of th variatioal itratio thod applid to t of liar/o-liar ordiar diffrtial quatio which ild a ri olutio with acclratd covrgc ha b prtd. Illutrativ apl hav b giv. Kword: St of Ordiar Diffrtial Equatio Rvid Variatioal Itratio Mthod MSC000 No.: 5L05; 5D5 0

2 AAM: Itr. J. Spcial Iu No. Augut 00. Itroductio I 999 th variatioal itratio thod VIM wa propod b H i H 00. Thi thod i ow widl ud b a rarchr to tud liar ad o-liar partial diffrtial quatio. Nurou probl i Phic Chitr Biolog ad Egirig cic ar odld athaticall b t of ordiar diffrtial quatio.g. ri circuit chaical t with vral prig attachd i ri lad to a t of diffrtial quatio. I thi work w coidr H variatioal itratio thod a a wll kow thod for fidig both aaltical ad approiat olutio of t of diffrtial quatio. Thi tchiqu wa dvlopd b th Chi athaticia H 00. Th variatioal itratio thod i ud for olvig autooou ordiar diffrtial t i H 000. Applicatio of thi thod to th Hlholtz quatio i ivtigatd i Moai t al. 00. Thi thod i ud for olvig Burgr ad coupld Burgr quatio i Abdou t al I Abdou t al. 005 th applicatio of th prt thod to coupld Schrodigr KdV quatio ad hallow watr quatio ar providd. Mot ralitic diffrtial quatio do ot hav act aaltic olutio approiatio ad urical tchiqu thrfor ar ud tivl. Thi w itrativ thod ha prov rathr uccful i dalig with both liar a wll a o-liar probl a it ild aaltical olutio ad offr crtai advatag ovr tadard urical thod. Biazar t al 00 hav applid thi thod to a t of ordiar diffrtial quatio. I th prt papr w u th rvid variatioal itratio thod to obtai olutio of t of liar/ o-liar ordiar diffrtial quatio. W dotrat that th ri olutio thu obtaid covrg fatr rlativ to th ri obtaid b tadard VIM. Svral illutrativ apl hav b prtd. Th prt papr ha b orgaizd a follow. Th wll kow H variatioal itratio thod i rviwd i Sctio. Sctio dal with th aali of VIM applid to a t of ordiar diffrtial quatio. I ctio w itroduc rvid VIM for t of ordiar diffrtial quatio rpctivl. Sctio copar th rvid VIM ad tadard VIM with illutrativ apl. Thi i followd b th cocluio i Sctio 7.. H' Variatioal Itratio Mthod Accordig to th variatioal itratio thod w coidr th followig diffrtial quatio: Lu Nu gt whr L i a liar oprator N a o-liar oprator ad gt i th ourc ihoogou tr. Accordig to th variatioal itratio thod w ca cotruct a Corrctio fuctioal a follow u t ~ t u t λ τ{ Lu τ Nu τ g τ} dτ o

3 Salhpoor t al. whr λ i a gral Lagragia ultiplir which ca b idtifid optiall via th variatioal thor th cod tr o th right i calld th corrctio ad u ~ i coidrd a a rtrictd variatio i.. δ u ~ = 0.. VIM for St of Ordiar Diffrtial Equatio Coidr th followig t of ordiar diffrtial quatio: L u u... u N u u... u g L u u... u N u u... u g L u u... u N u u... u g. whr L L L ar liar oprator N N N ar o-liar oprator u u t ad g g t. W ca cotruct a corrctio fuctioal a follow: t u ~ ~... ~ u λ[ L u u... u N u u u g] dτ o t u u λ [ L u u... u N u ~ u~... u~ g ] d τ o t u [... ~ ~... ~ u L u u u N u u u g ] d o whr λ λ... λ ar a gral Lagragia ultipli which ca b idtifid optiall via th variatioal thor b uig th tatioar coditio th ubcript dot th th-ordr approiatio u ~ i coidrd a a rtrictd variatio i.. δ u ~ = 0.. Rvid VIM for a St of Ordiar Diffrtial Equatio I thi ctio w propo a odificatio of th Variatioal itratio w rwrit quatio i th for L u N u... u g i... i i i i whr L i ar liar oprator N i ar o-liar oprator i thi ca th fuctioal ar obtaid a t u ~ ~ i ui λi [ Li ui τ Ni u τ... u τ gi τ dτ o whr λ i i a gral Lagrag ultiplir δ u i 0 i... t ui ui i [ Li ui N i u... ui ui... u g i d o 5

4 AAM: Itr. J. Spcial Iu No. Augut 00 for i =. I fact th updatd valu u u... u ar ud for fidig i u i +. Thi tchiqu acclrat th covrgc of th t of quc. 5. Illutrativ Eapl To giv a clar ovrviw of th rvid thod w prt th followig apl. W appl th rvid VIM ad copar th rult with th tadard VIM. i Coidr th followig t of liar quatio: co Th corrctio fuctioal for abov t rad 0 λ d 0 { co } λ d 0 { p } { } d. 0 Thi ild th tatioar coditio λ 0 λ λ 0 λ λ 0 λ A a rult w fid λ λ λ. Subtitutig th valu of th Lagrag ultiplir ito th fuctioal abov giv th itratio forula { co } 0 d { p } 0 d { } 0. 0 d

5 Salhpoor t al. W gt th act olutio i ad co.. Accordigl w obtai th followig ucciv approiatio: i i co i co Th u of th odifid thod rult i th quc 0 { co} d 0 0 { p } d 0 { } d W obtai th followig ucciv approiatio: i co i co 0

6 AAM: Itr. J. Spcial Iu No. Augut 00 5 i co ii Coidr th followig t of o-liar diffrtial quatio: Th corrctio fuctioal for abov t rad λ { } 0 d λ d 0 { p } λ { }. 0 d Thi ild th tatioar coditio λ 0 λ λ 0 λ λ 0 λ A a rult w fid λ λ λ. Subtitutig th valu of th Lagrag ultiplir ito th fuctioal abov giv th itratio forula { } 0 d { p } 0 d { }. 0 d W gt act olutio ad ucciv approiatio: Accordigl w obtai th followig

7 Salhpoor t al. ih ih 8 8 Th u of th odifid thod rult i th quc 0 { } d 0 { p } d 0 0 { } d W obtai th followig ucciv approiatio: Stat if for o-liar tr rplac Adoia poloial 0 { A } d 0 0 { p } d 0

8 AAM: Itr. J. Spcial Iu No. Augut 00 7 whr 0 { } d A A A A 0 0 W obtai th followig ucciv approiatio: W draw blow graph of ad ad copar th act olutio olutio giv b th tadard VIM thod ad olutio giv b th rvid VIM thod. I Fig ad w put which ar th act olutio dot olutio * * * obtaid b rvid VIM ad dot olutio obtaid b th tadard VIM. 8

9 8 Salhpoor t al. iii Coidr th followig t of liar diffrtial quatio: Th corrctio fuctioal for abov t rad λ d 0 { } λ d 0 { }. Thi ild th tatioar coditio λ 0 λ λ λ 0 λ λ 0 0. A a rult w fid λ λ. Subtitutig th valu of th Lagrag ultiplir ito th fuctioal abov giv th itratio forula { d 0 }

10 9 AAM: Itr. J. Spcial Iu No. Augut 00 d 0 } { W gt th act olutio ad. co accordigl w obtai th followig ucciv approiatio: Th u of th odifid thod rult i th quc d 0 } { d 0. } { W obtai th followig ucciv approiatio: 0 0 0

11 0 Salhpoor t al I Fig. w plot ad co which ar th act olutio i dot olutio obtaid b rvid VIM ad * i i dot olutio obtaid b th tadard VIM. i. Cocluio Variatioal itratio i a powrful thod which ild a covrgt ri olutio for liar/o-liar probl. Th olutio obtaid b th variatioal itratio thod i a ifiit powr ri for appropriat iitial coditio which ca i tur b prd i a clod for th act olutio. Th rult how that th variatioal itratio thod i a powrful athatical tool to olvig t of ordiar diffrtial quatio. I our work w u th Mathatica to calculat th ri obtaid fro th variatioal itratio thod. Thi thod do ot rquir larg coputr powr. I th prt papr w plo th rvid VIM for olvig a t of ordiar diffrtial quatio. Th rvid thod ild a ri olutio which covrg fatr tha th ri obtaid b tadard VIM. Th illutrativ apl clarl dotrat thi. Mathatica ha b ud for graph prtd i thi papr.

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