The Variational Iteration Method for Analytic Treatment of Homogeneous and Inhomogeneous Partial Differential Equations
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1 Global Joral of Scic Frotir Rarch: F Mathmatic ad Dciio Scic Volm 5 I 5 Vrio Yar 5 Tp : Dobl Blid Pr Rviwd Itratioal Rarch Joral Pblihr: Global Joral Ic USA Oli ISSN: 9- & Prit ISSN: Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio B M O Olaiwola O Stat Uivrit Nigria Abtract- I thi papr th athor d th variatioal itratio mthod VIM to fid th aaltical oltio to homogo ad ihomogo partial diffrtial qatio Fw mrical ampl wr prtd to how th ffctiv ad fficic of th mthod It wa obrvd that b carfll cho a vr good choic of iitial g lad to a oltio i a clod form Th mthod i lgat ad rliabl Kword: variatioal itratio mthod iitial val problm partial diffrtial qatio GJSFR-F Claificatio : FOR Cod : MSC : A ThVariatioalItratioMthodforAalticTratmtofHomogoadIhomogoPartialDiffrtialEqatio Strictl a pr th compliac ad rglatio of : 5 M O Olaiwola Thi i a rarch/rviw papr ditribtd dr th trm of th Crativ Commo Attribtio- Nocommrcial Uportd Lic prmittig all o commrcial ditribtio ad rprodctio i a mdim providd th origial work i proprl citd
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2 Rf Hoi Jafari tal 8: Applicatio of Homotop Prtrbatio Mthod for Solvig Ga Damic Eqatio Applid Mathmatical Scic Vol8 No 89-9 Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio M O Olaiwola Abtract- I thi papr th athor d th variatioal itratio mthod VIM to fid th aaltical oltio to homogo ad ihomogo partial diffrtial qatio Fw mrical ampl wr prtd to how th ffctiv ad fficic of th mthod It wa obrvd that b carfll cho a vr good choic of iitial g lad to a oltio i a clod form Th mthod i lgat ad rliabl Kword: variatioal itratio mthod iitial val problm partial diffrtial qatio I Itrodctio Partial diffrtial qatio; liar or oliar homogo or ihomogo ha ma applicatio to ral lif problm that ari i cic girig ad tcholog Thr ar ma mrical mthod for th oltio of diffrt tp of diffrtial qatio ch a Adomia dcompoitio mthod [-] homotop prtrbatio mthod [] variatioal itratio mthod [5-9] modifid variatioal itratio mthod -5] Rlt b vario rarchr [-5] hav how rliabilit fficic ad applicabilit of th mthod I thi papr a variatioal itratio mthod for th oltio of homogo ad ihomogo partial diffrtial qatio i prtd It i to b otd that th Lagrag mltiplir rdc th itratio o itgral oprator ad alo miimi th comptatioal tim Th mthod rqir o traformatio ad or liariatio of a form Som mrical problm ad rlt ar prtd to how th rliabilit of th mthod II Th Variatioal Itratio Mthod Th baic ida of th H' Variatioal Itratio Mthod VIM [5-9] ca b plaid b coidrig th followig oliar partial diffrtial qatio L N g V V Yar 5 7Global Joral of Scic Frotir Rarch Volm XV I rio I F Athor: Dpartmt of Mathmatical ad Phical Scic Faclt of Baic ad Applid Scic Collg of Scic Egirig ad Tcholog Iformatio Maagmt ad Tcholog Ctr Offic of th Vic-Chacllor O Stat Uivrit Oogbo Nigria -mail: olaiwolaodi@iodg 5 Global Joral Ic US
3 Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio Whr L i th liar oprator N i th oliar oprator ad g i th ihomogo trm Accordig to th mthod w ca cotrct a corrctio fctioal a follow Th corrpodig variatioal itratio mthod for olvig i giv a whr λ L N g d λ i a Lagrag mltiplir which ca b idtifid optimall b variatioal Not 8Global Joral of Scic Frotir Rarch VolmYar 5 XV I V V rio I F itratio mthod Th bcript dot th th approimatio i coidrd a a rtrictd variatio i δ Th cciv approimatio of th oltio ca b ail obtaid b dtrmi th Lagrag mltiplir ad th iitial g coqtl th oltio i giv b III lim Applicatio ad Nmrical Rlt I thi ctio i problm will b prtd to illtrat th fficic of th mthod Eampl Coidr th followig ihomogo qatio Th corrctio fctioal i giv b λ d Makig th corrctio fctioal tatioar to obtai λ hc th itrativ formla bcom d 5 Coqtl followig approimat ar obtaid 7 8 th clod form ad act oltio i giv a 9 5 Global Joral Ic US
4 Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio Eampl Coidr th followig ihomogo qatio Th corrctio fctioal ad itrativ formla bcom: Not λ d th followig approimat ar obtaid Th act oltio i giv b Eampl Coidr th followig qatio Th corrctio fctioal i giv b 5 lim 7 d 8 from 8 followig itratio ar obtaid 9 V V Yar 5 9Global Joral of Scic Frotir Rarch Volm XV I rio I F 5 Global Joral Ic US
5 Th ri oltio i giv b! 5!!!!! 5!!!! 5 5 ad th clod form oltio i which i th act oltio Eampl Coidr th followig qatio Th corrctio fctioal ad formla i giv b d 5 Coqtl followig approimat ar obtaid 7 8 Thi i th act oltio to Eampl 5 Coidr th followig qatio 9 Th corrctio fctioal i giv b d thrfor Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio 5 Global Joral Ic US Global Joral of Scic Frotir Rarch Volm Yar 5 F XV I rio I V V! Not
6 Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio Not Th ri oltio i giv b!! ad th clod form oltio i Eampl Coidr th followig qatio Th corrctio fctioal i giv b 5!!! 5!! 5 i i i d 7 Coqtl followig approimat ar obtaid i co 8 i co i 9! i co i co!! i co i co i!!! 5 5 i co i co i co!!! 5! Th ri oltio i giv b 5 i co!!! 5! ad th clod form oltio i which i th act oltio i co i co i Global Joral of Scic Frotir Rarch F Volm XV I V V rio I Yar 5 5 Global Joral Ic US
7 Th Variatioal Itratio Mthod for Aaltic Tratmt of Homogo ad Ihomogo Partial Diffrtial Eqatio IV Coclio Th papr ha ccfll dcribd ad applid th variatioal itratio mthod to om partial diffrtial qatio of phical igificac Th mthod provid th oltio i trm of rapidl covrgt ri It i alo clar ad rmarkabl that approimat oltio ar i good agrmt with aaltical oltio Th VIM wa d i a dirct wa withot ig liariatio prtrbatio or rtrictiv amptio Th mthod i lgat ad rliabl Rfrc Référc Rfrcia Not Global Joral of Scic Frotir Rarch F Volm XV I V V rio I Yar 5 G Adomia 978: Noliar diipativ wav qatio Applid Mathmatic Lttr Vol pp 5- Haa N A Imail Kamal Ralam & Aia A Abd Rabboh Adomia Dcompoitio Mthod for Gralid Brgr -Hl ad Brgr -Fihr Eqatio Applid Mathmatic ad Comptatio I H Abdl-Halim Haa 8: Compario of diffrtial traformatio tchiq with Adomia dcompoitio mthod for liar ad oliar iitial val problm Chao Solito ad Fractal Vol pp 5-5 Hoi Jafari tal 8: Applicatio of Homotop Prtrbatio Mthod for Solvig Ga Damic Eqatio Applid Mathmatical Scic Vol8 No Iokti M 978 Gral of th Lagrag mltiplir i oliar mathmatical phic i: S Nmat Nar Ed Variatioal mthod i th mchaic of olid Prgamo Pr 5- H J H 998 Approimat aaltical oltio for pag flow with fractioal drivativ i poro mdia Compt Math App Mch Eg H J H 998 Approimat oltio of oliar diffrtial qatio with covoltio prodct oliariti Comp Math Appl Mch Eg 7: Ji-Ha H999 Variatioal Itratio mthod: a kid of o-liar aaltical tchiq: Som ampl It Joral of No-liar mchaic H J H Variatioal itratio mthod for atoomo ordiar diffrtial tm App Math ad Comptatio 5- Amit Goal AlkaRama Gpta ad CNagaraja Kmar Solitar Wav Soltio for Brgr-Fihr tp Eqatio with Variabl Cofficit WASET 7-7 D Kaa SMElSad A mrical imlatio ad plicit oltio of Th gralid Brgr-Fihr qatio ApplMathCompt 5 - H N AImail A A A Rabboh A rtrictiv Pad approimatio for Th oltio of th gralid Fihr ad Brgr-Fihr qatio Appl MathCompt 5 - Javidi M Modifid Pdopctral Mthod for gralid Brgr -Fihr Eqatio Itratioal Mathmatical Form No Olaiwola M O : A Improvd Algorithm for th Soltio of Gralid Brgr-Fihr Eqatio Applid Mathmatic Vol 5 pp 9-5 Olaiwola M O - Aaltical Approimat to th Soltio of Som Noliar Partial Diffrtial Eqatio Joral of th Nigria Aociatio of Mathmatical Phic Vol 8 Pp Global Joral Ic US
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