Variational iteration method: A tools for solving partial differential equations

Transcription

1 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o Th Joral of Mahmaics a Compr Scic Availabl oli a hp:// Th Joral of Mahmaics a Compr Scic Vol. o Variaioal iraio mho: A ools for solvig parial iffrial qaios Elham Salhpoor a Hossi Jafari b a Dparm of Mahmaics Islamic Aza Uivrsiy ooshahr Brach ooshahr Ira lham_salhpor6@yahoo.com b Dparm of Mahmaics Uivrsiy of Mazara Babolsar Ira Jafari@mz.ac.ir Rciv: Ags Rvis: ovmbr Oli Pblicaio: Jaary Absrac I his papr H's variaioal iraio mho VIM has b s o obai solio oliar gas yamics qaio a Sfa qaio. This mho is bas o Lagrag mliplirs for iificaio of opimal vals of paramrs i a fcioal. Usig his mho cras a sqc which s o h ac solio of h problm. Kywors Variaioal iraio mho; oliar gas yamics qaio; Sfa qaio; Parial iffrial qaio. Irocio Aalyical mhos commoly s o solv oliar qaios ar vry rsric a mrical chiqs ivolvig iscrizaio of h variabls o h ohr ha givs ris o roig off rrors. Rcly iroc variaioal iraio mho by H [5678] which givs rapily covrg sccssiv approimaios of h ac solio if sch a

2 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o solio iss has prov sccssfl i rivig aalyical solios of liar a oliar iffrial qaios. This mho is prfrabl ovr mrical mhos as i is fr from roig off rrors a ihr rqirs larg compr powr/mmory. H [67] has appli his mho for obaiig aalyical solios of aoomos oriary iffrial qaio oliar parial iffrial qaios wih variabl cofficis a igro-iffrial qaios. Th variaioal iraio mho was sccssflly appli o Svh orr Sawaa-Kora qaios [] o Schroigr-KV graliz KV a shallow war qaios [] o liar Hlmholz parial iffrial qaio []. Liar a oliar wav qaios KV K Brgrs a cbic Bossisq qaios hav b solv by Wazwaz [45] sig h variaioal iraio mho. I h prs papr w mploy VIM mho for solvig followig qaios. Cosir h homogos gas yamics qaio wih iiial coiio g. Also w mploy VIM mho for solvig followig qaios. Cosir h Sfa qaio s wih iiial coiio s s. Frhr w compar h rsl wih giv solios sig ADM [3 4]. Th papr has b orgaiz as follows. Scio givs a brif rviw of VIM. Scio 35 cosiss of mai rsls of h papr i which variaioal iraio mho of oliar gas yamics qaio a oliar Sfa qaio hav b vlop. I Scio 46 illsraiv ampls ar giv.i Scio 6illsraiv ampls ar giv. Coclsios ar prs i Scio 7. H's variaioal iraio mho Accorig o h variaioal iraio mho w cosir h followig iffrial qaio: L g 4 Whr L is a liar opraor a o-liar opraor a g is h sorc ihomogos rm. Accorig o h variaioal iraio mho w ca cosrc a Corrcio fcioal as follow { L g } 5 o 3

3 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o Whr λ is a gral Lagragia mliplir which ca b iifi opimally via h variaioal hory h sco rm o h righ is call h corrcio a is cosir as a rsric variaio i.. δ =. So w firs rmi h Lagrag mliplir ha will b iifi opimally via igraio by pars. Th sccssiv approimaios of h solio will b raily obai po sig h obai Lagrag mliplir a by sig ay slciv fcio.cosqly h solio 6. lim 3 Applyig VIM for oliar gas yamics qaio I his scio w shall iroc a rliabl algorihm bas o VIM o hal siglar iiial val problm i a ralisic a ffici way cosirig gas yamics qaio as a mol problm. Is corrcio variaioal fcioal i -ircio o obai h solio of gas yamics qaio by variaioal iraio mho ca b prss as follows: 7 whr. akig variaio wih rspc o h ip variabl oicig ha. 8 This yils h saioary coiios 9. This i r givs. - - Sbsiig his val of h Lagrag mliplir io h fcioal 3givs h iraio formla hs w ca obai approimaio solios for cosirig boary coiios a isr. 4 Illsraiv Eampl To mosra h ffcivss of h mho w cosir hr Eqs. [3] wih giv iiial coiio. Eampl 4. Cosir h oliar gas yamics qaios wih h iiial coiio

4 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o Sbsiig io Eq. w obai h followig sccssiv approimaios - - Fially - h w ca wri -. w ca s ha calcla solio will b h followig ac solio: -. Rmark : oliar gas yamics qaio has b solv by ADM by Evas a Bl [3]. hy obai ac solio afr som iraio b w obai ac solio afr iraio. 5 Applyig VIM for Sfa problm I his scio w apply VIM o sfa prablmeq.3. Is corrcio variaioal fcioal i -ircio o obai h solio of sfa prablm 3 by variaioal iraio mho ca b prss as follows: 3 whr. akig variaio wih rspc o h ip variabl oicig ha 5-4 This yils h saioary coiios 6. This i r givs. Sbsiig his val of h Lagrag mliplir io h fcioal[3] givs h iraio formla 7 hs w ca obai approimaio or ac solio for. 6 Illsraiv Eampl

5 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o To mosra h ffcivss of h mho w cosir hr Eqs. [3] wih giv iiial coiio. Eampl 6. Cosir Sfa problm 3 wih h I.C.. 8 Sbsiig [8] io [7] w obai h followig sccssiv approimaios cosh w hav cosh fially solio. which is ac 7 Coclsio Variaioal iraio mho is a powrfl ool which is capabl of halig liar/oliar parial iffrial qaios. Th mho hav b sccssflly appli o oliar gas yamics qaio a Sfa qaio. Also comparisos wr ma bw H's variaioal iraio mho a Aomia composiio mho ADM for oliar gas yamics qaio a Sfa qaio. Th VIM rcs h volm of calclaios wiho rqirig o comp h Aomia polyomials. H's variaioal iraio mho facilias h compaioal work a givs h solio rapily if compar wih Aomia mho. Rfrcs [] M.A. Abo A.A. Solima w applicaios of variaioal iraio mho Physica D 5 8. [] G. Aomia Solvig Froir Problms of Physics: Th Dcomposiio Mho Klwr 994. [3] D.J. Evas a H.BlA w approach o h gas yamics qaio:a applicaio of h composiio mho Appl.Comp.Mah [4] R.Grzymkowski a DSloa Sfa problm solv by Aomia composiio mho Chaos Solios a Fracals [5] J.H. H Variaioal iraio mho -Som rc rsls a w irpraios J. Comp. Ir. J. Mah

6 Elham Salhpoor Hossi Jafari/ TJMCS Vol. o [6] J.H. H Variaioal pricipls for som oliar parial iffrial qaios wih variabl cofficis Chaos Solios a Fracals [7] J. H. H Variaioal iraio mho for aoomos oriary iffrial sysms Appl. Mah. Comp [8] J. H. H Som asympoic mhos for srogly oliar qaios I. J. Mo. Phys. B [9] W. Hrma P.P. Barj A. Korpl G. Assao A. va Immrzl a A. Mrpol Eac soliary wav solios of oliar volio a wav qaios sig a irc algbraic mho J. Phys. A: Mah. G [] M. Ioki al. Gral s of h Lagrag mliplir i o-liar mahmaical physics i: S. ma-assr E. Variaioal Mho i h Mchaics of Solis Prgamo Prss Ofor 978 pp [] H. Jafari al. Applicaio of H s Variaioal Iraio Mho for Solvig Svh Orr Sawaa-Kora Eqaios Appli Mahmaical Scics [] S. Momai a S. Abasa Applicaio of H's variaioal iraio mho o Hlmholz qaio Chaocs Solios a Fracals [3] E. J. Parks a B.R. Dffy A aoma ah-fcio mho for fiig soliary wav solios o o-liar volio qaios Comp. Phys. Comm [4] A. M. Wazwaz Th variaioal iraio mho: A rliabl aalyic ool for solvig liar a oliar wav qaios Comprs a Mahmaics wih Applicaios [5] A. M. Wazwaz Th variaioal iraio mho for raioal solios for KV K Brgrs a cbic Bossisq qaios J. Comp. Appl. Mah