Modified Variational Iteration Method for the Solution of nonlinear Partial Differential Equations

Transcription

1 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- ISSN Modifid Variaioal Iraio Mhod for h Solio of oliar Parial Diffrial Eqaios Olayiwola M O Akipl F O Gbolagad A W Absrac-Th Variaioal Iraio Mhod VIM has b show o solv ffcivly asily ad accraly a larg class of liar ad oliar problms wih approimaios covrgig rapidly o ac solios W prs a w Modifid Variaioal Iraio Mhod MVIM for h solio of som parial diffrial qaios of physical sigificac Id Trms : variaioal iraio mhod lagrag mliplir Taylor s sris parial diffrial qaio modifid variaioal iraio mhod corrcio fcioal INTRODUCTION Diffrial qaios play a crcial rol i applid mahmaics ad physics Th rsls of solvig hs qaios ca gid ahors o kow h dscribd procss dply B a ims i is difficl o obai h ac solios o hs problms I rc dcads hr has b gra dvlopm i h mrical aalysis ad ac mhod of solvig parial diffrial qaios For isac Adomia s dcomposiio mhod Homoopy prrbaio mhod paramr padig mhod c H dvlopd h variaioal iraio mhod for solvig liar oliar ad bodary val problms Th mhod was firs cosidrd by Ioki Ski ad Mra 978 ad flly plord by H J H I his mhod h solio is giv i a ifii sris sally covrgig o a accra solio Olayiwola al 9 sd modifid powr sris mhod for h solio of sysms of diffrial qaios I is obsrvd ha h mhod solv ffcivly asily ad accraly a class of liar oliar ordiary diffrial qaios wih approima solio which covrg vry rapidly o accra solio I his papr w prs a w modifid variaioal iraio mhod for h solio of oliar parial diffrial qaios VARIATIONAL ITERATION METHOD To illsra h basic cocp of h VIM w cosidr h followig gral oliar parial diffrial qaio L R N g whr L is a liar im drivaiv opraor R is a liar opraor which has parial drivaiv wih rspc o N is a oliar opraor ad g is a ihomogos rm Accordig o VIM w ca cosrc a corrc fracioal as follows: R ~ N ~ L g d whr is a Lagrag mliplir which ca b idifid opimally via variaioal iraio mhod Th sbscrip do h h approimaio ~ is cosidrd as a rsricd variaio i ~ Th IJSER hp://wwwijsrorg

2 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- ISSN IJSER hp://wwwijsrorg sccssiv approimaio of h solio will b radily obaid po sig h drmid Lagrag mliplir ad ay slciv fcio cosqly h solio is giv by: lim I Modifid VIM qaio bcoms: g g g 4 d g N R L ~ ~ 5 whr g ca b fod by sbsiig for i wh APPLICATION OF MVIM W cosidr h oliar homogos gas dyamics qaio Hossi al 8: Wih h iiial codiios Th iraio formla is d Makig corrcio fcioal saioary h Lagrag mliplir ca b idifid as: Usig 4 ad 5 i d 4 4

3 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- ISSN Similarly I is obvios ha a highr mbr of iraio maks covrgs o ac solio: Figr shows h graphs of ac solio agais for diffr im Figr shows h graphs of MVIM solio agais for diffr im whil Figr shows h graphs of ac solio ad MVIM solio agais for Figr IJSER hp://wwwijsrorg

4 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 4 ISSN Figr Figr W cosidr h oliar qaio: IJSER hp://wwwijsrorg

5 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 5 ISSN IJSER hp://wwwijsrorg Similarly This covrgs o: Figr 4 shows h graphs of ac solio agais for diffr im Figr 5 shows h graphs of MVIM solio agais for diffr im whil Figr 6 shows h graphs of ac solio ad MVIM solio agais for 8

6 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 6 ISSN Figr 4 Figr 5 IJSER hp://wwwijsrorg

7 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 7 ISSN IJSER hp://wwwijsrorg Figr 6 : Cosidr a oliar o-homogos parial diffrial qaio ; Doga l Applig MVIM o l 4 d 5 I h sam way w comp ohr compos for Figr 7 shows h graphs of ac solio ad MVIM solio agais for

8 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 8 ISSN Figr 7 4 CONCLUSIONS I his work h MVIM was applid o h solio of oliar parial diffrial qaios Th mrical rsls dmosrad ha h mhod is accra rliabl ad covrgs fasr wih lss compaio wh compard wih ohr mhods i h lirar REFRENCES [] Ablwafa EM Abdo MA Mahmod AA Noliar flid flows i pip-lik domai problms sig VIM class Solio Ad Fracals [] Ablwafa EM Abdo MA Mahmod AA Th solio of oliar coaglaio problm wih mass class Solio ad Fracals 69 - [] Awodola T O Variabl Vlociy Iflc o h Vibraio of Simply Sppord Brolli - Elr Bam Udr Epoially Varyig Magid Movig Load Joral of Mahmaics ad Saisics 4: 8-7 [4] Barari A A approima solio for bodary val problms i srcral Egirig ad Flid Mchaics Mahmaical problms i Egirig 8 [5] Baiha B Applicaio of Variaioal Iraioal mhod o a Gral Ricca Eqaio I Mahmaical Form 7 o [6] Biazar J Ghazii H H s Variaoal iraio mhod for solvig liar ad o-liar sysm of ordiary diffrial qaios Applid Mahs ad Compaio [7] Bildik N Koralp A Th s of variaio iraio mhod diffrial rasform mhod ad Adomai dcomposiio mhod for solvig diffr yps of oliar pol I J oliar Scic; Nmrical Simlaio [8] Clik E Nmrical mhod o solv chmical diffrial Allgrai qaio I J of Qam chmisry [9] Dihli K ad Ford NJ Aalysis of fracioal diffrial qaio J mah ad Appl [] Djidjli K Nmrical mhods for spcial oliar bodary val problms of ordr m J of Compaioal ad Appl Mahmaics Vol 47 6o 6-69 IJSER hp://wwwijsrorg

9 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- 9 ISSN [] Doga Kaya Th s of Adomia dcomposiio mhod for solvig a spcific oliar parial diffrial qaios Bll Bg Mahs Soc [] Dragasc G E capalasa V Noliar rlaaio phomo i polycrysalli solid i J oliar Nmrical Simlaio [] Gaji DD A Compariso of Variaioal iraio mhod wih Adomai dcomposiio mhod i som highly oliar qaios Iraioal Joral of Scic ad Tchology Vol No [4] Gbadya J A& ST Oi Dyamics bhavior of bam ad rcaglar plas dr movig loads Joral of Sod ad Vibraio [5] H JH Approima aalyical solio for spag flow wih fracioal drivaivs i poros mdia Comp Mah App Mch Eg [6] H JH Approima solio of oliar diffrial qaios wih covolio prodc oliariis Comp Mah Apply Mch Eg 67: [7] H JH Variaioal iraio mhod for aoomos ordiary diffrial sysm App Mah ad Compaio ; 4:5- [8] Hossi Jafari al Applicaio of Homoopy Prrbaio Mhod for Solvig Gas Dyamic Eqaio Applid Mahmaical Scics Vol8 No [9] Jmal Pradz A Nmrical Algorihm for a Kirchhoff-Typ Noliar Saic Bam Joral of Applid MahmaicsVolm 9 [] Ioki M al 978: Gral s of h Lagrag mliplir i oliar mahmaical physics i: S Nma Nassr Ed Vaaioal mhod i h mchaics of solid Prgamo Prss 56-6 [] Ji-Ha H 7: Variaioal iraio mhods som rc rsls ad w irpraio J of compiio ad Applid marhras 7-7 [] Ji-Ha H 999: Variaioal Iraio mhod: a kid of o-liar aalyical chiq: Som ampls I Joral of No-liar mchaics [] Lva Yilmaz Som Cosidraios o h sris solio of Diffrial qaios ad is girig Applicaios RMZ Marials ad Go-virom Vol 56 No [4] L JF Variaioal Iraio mhod for solvig wo-poi bodary val problms Joral of compaioal ad Applid Mahmaics [5] Ma TF ad Silva J Da Iraiv solio for a bam qaio wih oliar bodary codiios of hird ordr Applid mahmaics ad Compaio Vol 59 4o -8 [6] Mhammd AN Syd TM Variaioal Iraio Dcomposiio mhod for solvig Eigh-Ordr Bodary Val Problm Diff Eqaio ad Noliar mchaics 7 Vol Id 959 [7] Nra Gzk Bayram M Powr Sris solio of oliar firs ordr diffrial qaios sysms Trakya Uiv JS [8] Odiba ZM Momai S Applicaio of Variaio iraio mhod o oliar diffrial qaio of fracioal ordr I J o Noliar Scic Nmrical Simlaio [9] Olayiwola MO al Applicaio of modifid powr sris mhod for h solio of sysm of diffrial qaios Joral of Nigria Associaio of Mahmaical Physics Vol 5 9pp 9-94 [] Olayiwola MO Gbolagad AW O h Chaoic bhavoior of Hamoically driv Oscillaor Far Eas J of Dyamical Sysms Vol 8 iss 47-5 [] Podhiby I Fracioal Diffrial qaios Acadmic Prss 999 Sa Digo [] Saharay S Bra RK Solio of a raordiary diffrial qaio by Adomai dcomposiio mhod J of Applid Mahs 44-8 [] Siddiqi SS Twizll EH Spli solio of liar igh-ordr bodary val problms Compr mhods i Apply Mchaics ad girig Vol 996o [4] Taari M Dligha M O h Covrgc of H s Variaioal iraio mhod Joral of compaioal ad applid Mahmaics 7-8 [5] Wag L ad Ni Q Vibraio of Sldr Srcrs Sbjcd o Aial Flow or Aially Towd i Qisc Flid Advacs i Acosics ad Vibraio Volm 9 Aricl ID 44 [6] Wazwaz A M A rliabl algorihm for obaiig posiiv solio for oliar bodary val problms Comp Mah App [7] Wazwaz A M Aalyic ram for variabl cffici forh-ordr parabolic parial diffrial qaios Appl mah Comp : pp 9-7 [8] Wazwaz A M Parial diffrial qaios mhods ad applicaios A A Balkma Pblishrs [9] Wi-Xia Qia H s Iraio Formlaio for solvig Noliar Algbraic qaios Joral of Physics: Cofrc sris IJSER hp://wwwijsrorg

10 Iraioal Joral of Sciific & Egirig Rsarch Volm Iss Oc- ISSN [4] Yj Ci Ymi Zo Eisc ad Uiqss of solios for Forh-Ordr Bodary-Val Problms i Baach Spacs Elcroic Joral of Diffrial Eqaios Vol 99 No pp 8 IJSER hp://wwwijsrorg