Solving Higher Dimensional Initial Boundary Value Problems by Variational Iteration Decomposition Method
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1 Availabl a hp://pvam.d/aam Appl. Appl. Mah. ISSN: 9-94 Vol. No. Dcmbr 8 pp. 54 Applicaios ad Applid Mahmaics: A Iraioal Joral AAM Solvig Highr Dimsioal Iiial Bodar Val Problms b Variaioal Iraio Dcomposiio Mhod Mhammad Aslam Noor Sd Tasf Mohd-Di Dparm of Mahmaics COMSATS Isi of Iformaio Tcholog Islamabad Pakisa oormaslam@homail.com sdasfs@homail.com Rcivd: Dcmbr 7 7; Accpd: J 8 Absrac I his papr w appl a rlaivl w chiq which is calld h variaioal iraio dcomposiio mhod VIDM b combiig h radiioal variaioal iraio ad h dcomposiio mhods for solvig highr dimsioal iiial bodar val problms. Th proposd mhod is a lga combiaio of variaioal iraio ad h dcomposiio mhods. Th aalical rsls of h problms hav b obaid i rms of covrg sris wih asil compabl compos. Th mhod is qi ffici ad is pracicall wll sid for s i hs problms. Svral ampls ar giv o vrif h accrac ad fficic of h proposd chiq. Kwords: Variaioal iraio mhod oliar problms iiial val problms bodar val problms Adomia s polomials AMS Sbjc Classificaio Nmbrs: 5 N 4 B 54
2 AAM: Ir. J. Vo. No. Dcmbr Irodcio Th mrical ad aalical solios of highr dimsioal iiial bodar val problms of variabl cofficis liar ad oliar ar of cosidrabl sigificac for applid scics. Eampls of liar modls ar Elr-Darbo qaio Lambropobs qaio ad Tricomi qaio s Millr 97 Solima Noor ad Mohd-Di 8f Wawa 999 Wilco 97 giv b: α β a b c rspcivl. Eampls of oliar modls ar irodcd i KdV qaio of variabl cofficis ad Claira s qaio s giv b m α β f rspcivl. I his papr w cosidr h highr dimsioal iiial bodar val problm of h form m m m F < < < < b < < c m > sbjc o h Dirichl bodar codiios b g f a f h g c h ad h iiial codiios φ η whr F is a liar or a oliar fcio. A cosidrabl si of rsarch work has b ivsd i hs sciific applicaios. Svral chiqs icldig h spcral characrisics dcomposiio ad variaioal homoop prrbaio hav b sd for solvig hs problms s Millr 97 Solima Noor ad Mohd-Di 8f Wawa 999 Wilco 97. H dvlopd h variaioal iraio mhod for solvig liar oliar iiial ad bodar val problms s H 999 H H H 7a.
3 5 Noor ad Mohd-Di I is worh mioig ha h origi of variaioal iraio mhod ca b racd back o Ioki Ski ad Mra s Ioki Ski ad Mra 978 b h r poial of h VIM was plord b H. Sic h bgiig of 98s h Adomia s dcomposiio mhod has b applid o a wid class of fcioal qaios s Wawa 999. I hs mhods h solio is giv i a ifii sris sall covrgig o a accra solio. Ispird ad moivad b h ogoig rsarch i his ara w appl h variaioal iraio dcomposiio mhod VIDM for solvig h highr dimsioal iiial bodar val problms. This mhod is a lga combiaio of variaioal iraio ad h Adomia s dcomposiio mhods for solvig highr dimsioal iiial bodar val problms. This ida has b sd implicil b Abbasbad s Abbasbad 7a Abbasbad 7b for solvig qadraic Riccai diffrial ad Kli-Gordo qaios. I a sbsq work Noor ad Mohd-Di dvlopd h lga coplig of Adomia s polomials wih h corrcioal fcioal of variaioal iraio mhod for solvig varios classs of siglar ad osiglar iiial ad bodar val problms s Mohd-Di Noor & Noor 8a Noor ad Mohd-Di 7c Noor ad Mohd-Di 8h. Svral ampls ar giv o illsra h rliabili ad prformac of h proposd mhod.. Variaioal Iraio Mhod To illsra h basic cocp of h chiq w cosidr h followig gral diffrial qaio L N g whr L is a liar opraor N a oliar opraor ad g is h ihomogos rm. Accordig o variaioal iraio mhod s H 999 H H H 7a H 7b Abbasbad 7a Abbasbad 7b Mohd-Di Noor& Noor 8a Noor ad Mohd-Di 7a Noor ad Mohd-Di 7b Noor ad Mohd-Di 8b Noor ad Mohd-Di 8c Noor ad Mohd-Di 8c Noor ad Mohd- Di 8c Noor ad Mohd-Di 8d Noor ad Mohd-Di 8 Noor ad Mohd-Di 8f Noor ad Mohd-Di 8g w ca cosrc a corrcio fcioal as follows λ L s N s g s ds Whr λ is a Lagrag mliplir s which ca b idifid opimall via variaioal iraio mhod. Th sbscrips do h h approimaio is cosidrd as a rsricd variaio i.. δ. Th rlaio is calld as a corrcio fcioal. Th solio of h liar problms ca b solvd i a sigl iraio sp d o h ac idificaio of h Lagrag mliplir. Th pricipls of variaioal iraio mhod ad is applicabili for varios kids of diffrial qaios ar giv s H 999 H H H
4 AAM: Ir. J. Vo. No. Dcmbr a H 7b. I his mhod i is rqird firs o drmi h Lagrag mliplir λ opimall. Th sccssiv approimaio of h solio will b radil obaid po sig h drmid Lagrag mliplir ad a slciv fcio cosql h solio is giv b lim.. Adomia s Dcomposiio Mhod Cosidr h diffrial qaio s Wawa 999 L R N g whr L is h highs-ordr drivaiv which is assmd o b ivribl R is a liar diffrial opraor of ordr lssr ordr ha L N rprss h oliar rms ad g is h sorc rm. Applig h ivrs opraor L o boh sids of ad sig h giv codiios w obai f L R L N whr h fcio f rprss h rms arisig from igraig h sorc rm g ad b sig h giv codiios. Adomia s dcomposiio mhod s Wawa 999 dfis h solio b h sris whr h compos ar sall drmid rcrrl b sig h rlaio f L R L N k. k k k Th oliar opraor N ca b dcomposd io a ifii sris of polomials giv b N A whr A ar h so-calld Adomia s polomials ha ca b grad for varios classs of oliariis accordig o h spcific algorihm dvlopd i s Wawa 999 which ilds
5 58 Noor ad Mohd-Di A d! N dλ i i λ. i λ 4. Variaioal Iraio Dcomposiio Mhod VIDM To illsra h variaioal iraio dcomposiio mhod w irodc Adomia s polomials i h corrcio fcioal. Hc w obai h followig iraiv schm for fidig h approima solios λ L A g d. This mhod is calld as h variaioal iraio dcomposiio mhod VIDM ad is obaid b h lga coplig of Adomia s polomials ad h corrcioal fcioal of variaioal iraio mhod s Mohd-Di Noor & Noor 8a Noor ad Mohd-Di 8f Noor ad Mohd-Di 8h Noor ad Mohd-Di 7c. 5. Nmrical Applicaios I his scio w appl h variaioal iraio dcomposiio mhod VIDM for solvig h highr dimsioal iiial bodar val problms. For h sak of compariso w cosidr h sam ampls as discssd i Wawa 999 ad Noor ad Mohd-Di 8f. Eampl 5.. Cosidr h wo dimsioal iiial bodar val problm < < > wih bodar codiios ad h iiial codiios
6 AAM: Ir. J. Vo. No. Dcmbr Th corrcio fcioal is giv as. λ d whr is cosidrd as a rsricd variaio. Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as λ ilds h followig iraio formla. d Cosql h followig approimas ar obaid!! 5! 4!!! !! 4! 5!! 7!. Th sris solio is giv b !! 4! 5!! 7! 8! ad i a closd form b:.
7 Noor ad Mohd-Di Eampl 5.. Cosidr h hr dimsioal iiial bodar val problm < < < sbjc o h Nma bodar codiios sih sih sih ad h iiial codiios. Th corrcio fcioal is giv as. 45 λ d whr is cosidrd as a rsricd variaio. Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as λ ilds h followig iraio formla. 45 d Cosql h followig approimas ar obaid! 5!! 5 7! 5!! 7 5 9! 7! 5!!
8 AAM: Ir. J. Vo. No. Dcmbr 8 Th sris solio is giv b 5 7 9! 5! 7! 9! ad i a closd form b sih. Eampl 5.. Cosidr h wo dimsioal oliar ihomogos iiial bodar val problm 5 < < > wih bodar codiios ad h iiial codiios. Th corrcio fcioal is giv as 5 λ d whr is cosidrd as a rsricd variaio. Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as λ ilds h followig iraio formla d. 5 Applig h variaioal iraio dcomposiio mhod
9 Noor ad Mohd-Di 5 Q d M whr Q ad M ar h so-calld Adomia s polomials ha rprs h oliar opraors ad ca b grad for all p of oliariis s Wawa 999 which ilds h followig Q Q Q Q M M M M. Cosql h followig approimas ar obaid. Th ac solio is obaid as. Eampl 5.4. Cosidr h hr dimsioal oliar iiial bodar val problm < < < sbjc o h Nma bodar codiios sih ad h iiial codiios.
10 AAM: Ir. J. Vo. No. Dcmbr 8 Th corrcio fcioal is giv as. λ λ d d Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as λ ilds h followig iraio formla. M Q H d d whr Q M ad H ar h so-calld Adomia s polomials ha rprs h oliar opraors ad ca b grad for all p of oliariis s Wawa 999. Cosql h followig approimas ar obaid Usig h ois rms phoma w obaid.. Coclsios I his papr w applid h variaioal iraio dcomposiio mhod VIDM for solvig liar ad oliar highr dimsioal iiial bodar val problms wih cosa cofficis variabl cofficis ad h fcio cofficis. Th proposd mhod is applid i a dirc wa wiho sig liariaio prrbaio or rsriciv assmpios. I all h mrical rsls ac solios wr obaid b sig h iiial codiios ol.
11 4 Noor ad Mohd-Di I ma b cocldd ha mhod is vr powrfl ad ffici i fidig h aalical solios for a wid class of iiial bodar val problms. Th mhod givs mor ralisic sris solios ha covrg vr rapidl i phsical problms. Ths w cocld ha h VIDM ca b cosidrd as a ffici alraiv for solvig iiial ad bodar val problms. Ackowldgm: Th ahors ar highl grafl o boh h rfrs ad Prof Dr A. M. Haghighi for hir cosrciv comms. W wold lik o hak Dr S. M. Jaid Zaidi Rcor CIIT for h provisio of cll rsarch virom ad faciliis. REFERENCES Abbasbad S. 7a. A w applicaio of H s variaioal iraio mhod for qadraic Riccai diffrial qaio b sig Adomia s polomials J. Comp. Appl. Mah. 7 pp Abbasbad S. 7b. Nmrical solios of oliar Kli-Gordo qaio b variaioal iraio mhod Ira. J. Nmr. Mh. Egrg. 7 pp H J. H Variaioal iraio mhod A kid of o-liar aalical chiq som ampls Ira. J. Noliar Mch. 4 4 pp H J. H.. Variaioal iraio mhod for aoomos ordiar diffrial ssms Appl. Mah. Comp. 4 - pp.5-. H J. H. 7a. Variaioal iraio mhod: Nw dvlopm ad applicaios Comp. Mah. Appl. 54 pp H J. H. 7b. Variaioal iraio mhod--- som rc rsls ad w irpraios J. Comp. Appl. Mah. 7 pp. -7. H J. H.. Som asmpoic mhods for srogl oliar qaio I. J. Mod. Phs. pp Ioki M. Ski H. ad Mra T Gral s of h Lagrag mliplir i oliar mahmaical phsics i: S. Nma-Nasr Ed. Variaioal mhod i h Mchaics of solids Prgamo prss Nw York pp. 5-.
12 AAM: Ir. J. Vo. No. Dcmbr 8 5 Millr J. W. 97. Smmris of diffrial qaios: Th hpr gomric ad Elr- Darbo qaios SIAM. J. Mah. Aal. 4 pp Mohd-Di S. T. Noor M. A. ad Noor K. I. 8a. Travllig wav solios of svhordr gralid KdV qaios b variaioal iraio mhod sig Adomia s polomials I. J. Mod. Phs. B. Noor M. A. ad Mohd-Di T. S. 7a Variaioal iraio chiq for solvig highr ordr bodar val problms Appl. Mah. Comp. 89 pp Noor M. A. ad Mohd-Di T. S. 7b A ffici mhod for forh ordr bodar val problms Comp. Mah. Appl. 54 pp. -. Noor M. A. ad Mohd-Di T. S. 8b. Variaioal iraio mhod for solvig highrordr oliar bodar val problms sig H s polomials I. J. Noli. Sci. Nm. Siml. 9 pp Noor M. A. ad Mohd-Di T. S. 8c. Variaioal iraio mhod for solvig fifh-ordr bodar val problms sig H s polomials Mah. Prob. Egg. 8. Noor M. A. ad Mohd-Di T. S. 8d. Modifid variaioal iraio mhod for ha ad wav-lik qaios Aca Applda. Mahmc. Noor M. A. ad Mohd-Di T. S. 8. Modifid variaioal iraio mhod for solvig forh-ordr bodar val problms J. Appl. Mah. Compg. Noor M. A. ad Mohd-Di T. S. 8f. Variaioal homoop prrbaio mhod for solvig highr dimsioal iiial bodar val problms Mah. Prob. Egg. 8. Noor M. A. ad Mohd-Di T. S. 8g. Variaioal iraio mhod for sad flow of gas hrogh a poros mdim sig H s polomials ad Pad approimas Comp. Mah. Appl. Noor M. A. ad Mohd-Di T. S. 8h Solio of siglar ad osiglar iiial ad bodar val problms b modifid variaioal iraio mhod Mah. Prob. Egg. 8. Noor M. A. ad Mohd-Di T. S. 7c. Variaioal iraio dcomposiio mhod for solvig ighh-ordr bodar val problms Diff. Eqs. Noli. Mch. 7. Solima A. A.. A mrical simlaio ad plici solios of KdV-Brgr s ad La s svh-ordr KdV qaios Chaos Solios & Fracals 9 pp Wawa A. M Th dcomposiio mhod for solvig highr dimsioal iiial bodar val problms of variabl cofficis I. J. Comp. Mah. 7 pp
13 Noor ad Mohd-Di Wilco R. 97. Closd form solio of h diffrial qaio a b c b ormal ordrig poial opraors J. Mah. Phs. 4 pp. 5-7.
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