Variational Iteration Method for Solving Initial and Boundary Value Problems of Bratu-type

Transcription

1 Availabl a hp://pvamd/aam Appl Appl Mah ISSN: 9-9 Vol Iss J 8 pp Prviosl Vol No Applicaios ad Applid Mahmaics: A Iraioal Joral AAM Variaioal Iraio Mhod for Solvig Iiial ad Bodar Val Problms of Bra-p Mhammad Aslam Noor oormaslam@homailcom Sd Tasf Mohd-Di Dparm of Mahmaics COMSATS Isi of Iformaio Tcholog Islamabad Pakisa sdasfs@homailcom Rcivd Novmbr 7; accpd Fbrar 8 Absrac I his papr w prs a rliabl framwork o solv h iiial ad bodar val problms of Bra-p which ar widl applicabl i fl igiio of h combsio hor ad ha rasfr Th algorihm rss mail o a rlaivl w chiq h variaioal iraio mhod Svral ampls ar giv o cofirm h fficic ad h accrac of h proposd algorihm Kwords: Variaioal iraio mhod Bra s problm bodar val problms iiial val problms AMS Sbjc Classificaio Nmbrs: N B Irodcio This papr is cocrd wih bodar val problms ad a iiial val problm of h Brap I is wll kow ha Bra s bodar val problm i o-dimsioal plaar coordias is of h form s Aschr 99 Bod 98 Bckmir Jacobso Wawa Noor ad Mohd-Di 7a 89

2 9 Noor ad Mohd-Di wih bodar codiios Th sadard Bra p problm was sd o modl a combsio problm i a mrical slab Th Bra modls appar i a mbr of applicaios sch as h fl igiio of h hrmal combsio hor ad i h Chadraskhar modl of h pasio of h ivrs s Bckmir I simlas a hrmal racio procss i a rigid marial whr h procss dpds o h balac bw chmicall grad ha ad ha rasfr b codcio; s Argbsola Wawa Noor ad Mohd-Di 7 A sbsaial amo of rsarch work has b dircd for h sd of h Bra problm s Aschr 99 Bod 98 Bckmir Jacobso Wawa Noor ad Mohd-Di 7a ad h rfrcs hri Svral mrical chiqs sch as h fii diffrc mhod fii lm approimaio wighd rsidal mhod ad h shooig mhod hav b implmd idpdl o hadl h Bra modl mricall s Aschr 99 Argbsola 98 Bckmir Jacobso I addiio Bod 98 mplod Chbshv polomial pasios ad h Ggbar polomials as bas fcios Rcl Wawa sd dcomposiio mhod for solvig sch modls Mor rcl Noor ad Mohd-Di 7b mplod homoop prrbaio mhod ad h variaioal iraio dcomposiio mhod for fidig h solio of hs problms Ispird ad moivad b h ogoig rsarch i his ara w applid a rlaivl w chiq h variaioal iraio mhod for solvig iiial ad bodar val problms of Bra p modls Th ac solio o is giv as: s Aschr 99 Bod 98 Bckmir Jacobso Wawa Noor ad Mohd-Di 7a l cosh cosh whr saisfis cosh Th Bra problm has ro o or wo solios wh c c ad c rspcivl whr h criical val c saisfis h qaio

3 AAM: Ir J Vol Iss J 8 [Prviosl Vol No ] 9 c c sih I was valad i Aschr 99 Bod 98 Bckmir Jacobso ad Wawa ha h criical val c is giv b c 879 Th basic moivaio of h prs work is o irodc a rliabl ram of wo bodar val problms of Bra-p modl giv b wih bodar codiios ad wih bodar codiios i addiio a iiial val problm of h Bra-p wih iiial codiios I his papr or work sms mail from h variaioal iraio mhod s H 999 H ad W Ioki al 978 Mohd-Di 7 Noor ad Mohd-Di 7ab Th basic moivaio of his papr is o propos mahmaical chiq wiho imposig prrbaio rsriciv assmpios or liariaio Th variaioal iraio mhod which accral comps h sris solio is of gra irs o applid scics Th mhod provids h solio i a rapidl covrg sris wih asil compabl compos Th mai advaag of h mhod is ha i ca b applid dircl for all ps of oliar diffrial ad igral qaios homogos or ihomogos wih cosa or variabl cofficis Morovr h proposd mhod is capabl of gral rdcig h si of compaioal work whil sill maiaiig high accrac of h mrical solio Svral ampls ar giv o gag h ffcivss ad h sflss of h sggsd variaioal

4 9 Noor ad Mohd-Di iraio mhod Th fac ha variaioal iraio chiq solvs oliar problms wiho sig Adomia polomials ca b cosidrd as a clar advaag of his mhod ovr h dcomposiio mhod H 999 dvlopd h variaioal iraio mhod for solvig liar oliar iiial ad bodar val problms I is worh mioig ha h mhod was firs cosidrd b Ioki Ski ad Mra 978 b h r poial of h variaioal iraio mhod was plord b H I his mhod h solio is giv i a ifii sris sall covrgig o a accra solio s H 999 H ad W Ioki al 978 Noor ad Mohd-Di 7a ad h rfrcs hri W appl h variaioal iraio mhod o solv h iiial ad bodar val problms of h Bra p I is show ha h proposd chiq provids h solio i a rapid covrg sris wih asil compabl compos I is obsrvd ha h sggsd mhod solv ffcivl asil ad accral a larg class of liar oliar parial drmiisic or sochasic diffrial qaios wih approima solios which covrg vr rapidl o accra solios 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 forcig rm Accordig o variaioal iraio mhod H 999 H ad W Ioki al 978 w ca cosrc a corrc fcioal as follows: ~ L s N s g s ds whr is a Lagrag mliplir s H 999 H ad W Ioki al 978 which ca b idifid opimall via variaioal iraio mhod Th sbscrips do h h approimaio ~ is cosidrd as a rsricd variaio i ~ ; is calld as a corrc 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 i H 999 H ad W Ioki al 978 Noor ad Mohd-Di 7a I his mhod i is rqird firs o drmi h Lagrag mliplir 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

5 AAM: Ir J Vol Iss J 8 [Prviosl Vol No ] 9 Nmrical Applicaios I his scio w appl h variaioal iraio mhod rviwd i scio for solvig iiial ad bodar val problms of h Bra-p Nmrical rsls idica ha h proposd variaioal iraio mhod is vr rliabl ad ffici Th fac ha h sggsd chiq solvs oliar problms wiho sig Adomia polomials ca b cosidrd as a clar advaag of his mhod ovr h dcomposiio mhod Morovr w hav also cosidrd a ampl of highr dimsioal iiial bodar val problm Noor ad Mohd-Di 7a For h sak of compariso w ak h sam ampls as cosidrd b Wawa Noor ad Mohd-Di 7b Eampl Cosidr h followig Bra-p modl wih iiial codiios Th corrc fcioal is giv as d d d whr is o giv b will b drmid b sig h ohr bodar codiios; ~ is cosidrd as a rsricd variaio Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as ilds h followig iraio formla d ~ d d Cosql h followig approimas ar obaid a A

6 9 Noor ad Mohd-Di a Th sris solio is giv b or qivall!!!!!! 7 Imposig h bodar codiio = lads o obai ad cosql h closd form solio is giv as cos l Eampl Cosidr h followig Bra-p modl wih iiial codiios Th corrc fcioal is giv as ~ d d d

7 AAM: Ir J Vol Iss J 8 [Prviosl Vol No ] 9 whr is o giv b will b drmid b sig h ohr bodar codiios; ~ is cosidrd as a rsricd variaio Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as ilds h followig iraio formla d ~ d d Cosql h followig approimas ar obaid A A A A Th sris solio is giv as A or qivall!!!!!! 7 Imposig h bodar codiios a lads o obai ad cosql h closd form solio is giv as l si

8 9 Noor ad Mohd-Di Eampl Cosidr h followig iiial val problm of h Bra-p wih iiial codiios Th corrc fcioal is giv b d ~ d d whr ~ is cosidrd as a rsricd variaio Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as ilds h followig iraio formla d ~ d d Cosql h followig approimas ar obaid Th sris solio is giv as or qivall

9 AAM: Ir J Vol Iss J 8 [Prviosl Vol No ] Th ac solio is giv b l cos Eampl Cosidr h hr dimsioal iiial bodar val problm sbjc o h Nma bodar codiios sih sih sih ad h iiial codiios Th corrc fcioal is giv as d whr ~ is cosidrd as a rsricd variaio Makig h abov fcioal saioar h Lagrag mliplir ca b drmid as ilds h followig iraio formla d Cosql h followig approimas ar obaid!

10 98 Noor ad Mohd-Di!!!! 7 7!!! 7 7! 9 9! Th sris solio is giv b!! 7 7! 9 9! ad i a closd form b sih Coclsio I his papr w applid h variaioal iraio mhod for solvig h iiial ad bodar val problms of Bra-p Th rsls clarl idica h rliabili ad accrac of h proposd chiq W also applid h proposd chiq o a hr dimsioal iiial bodar val problm Th sggsd mhod is sd dircl wiho sig prrbaio liariaio or rsriciv assmpios Morovr h variaioal iraio mhod is mor rliabl ad ffici ha h dcomposiio mhod Th fac ha h mhod solvs oliar problms wiho sig Adomia polomials ca b cosidrd as a clar advaag of his mhod ovr h dcomposiio mhod Ackowldgm: Th ahors ar highl grafl o boh h rfrs ad Prof Dr Aliakbar Moar Haghighi for hir cosrciv comms W wold lik o hak Dr S M Jaid Zaidi Rcor CIIT for providig cll rsarch faciliis ad virom REFERENCES Aschr U M R Mahij ad RD Rssll 99 Nmrical solio of bodar val problms for ordiar diffrial qaios SIAM Philadlphia PA Bod J P Chbshv polomial pasios for simlaos approimaio of wo brachs of a fcio wih applicaio o h o-dimsioal Bra qaio Appl Mah Comp pp89-

11 AAM: Ir J Vol Iss J 8 [Prviosl Vol No ] 99 Bod J P 98 A aalical ad mrical sd of h wo-dimsioal Bra qaios Joral of Sciific Compig pp 8- Bckmir R Ivsigaios of osadard Micks-p fii-diffrc schms for siglar bodar val problms i clidrical or sphrical coordias Nm Mh P Diff Eqs9 pp 8-98 Jacobso J ad K Shmi Th Liovill-Bra-Glfad problm for radial opraors J Diff Eqs 8 pp 8-98 Wawa A M Adomia s dcomposiio mhod for a rliabl ram of h Brap qaios Appl MahComp pp - H J H 999 Variaioal iraio mhod- a kid of o-liar aalical chiq: som ampls I J Noli Mch pp H J H Variaioal iraio mhod for aoomos ordiar diffrial ssms Appl Mah Comp pp - H J H Som asmpoic mhods for srogl oliar qaio I J Nod Ph pp -99 H J H ad X H W Cosrcio of soliar solios ad compaco-lik solio b variaioal iraio mhod Chos Sol Frcs 9 pp 8- Ioki M H Ski ad T Mra 978 Gral s of h Lagrag mliplir i o-liar mahmaical phsics i: S Nma-Nassr Ed Variaioal Mhod i h Mchaics of Solids Oford: Prgamo Prss pp - Mohd-Di S T 7 A rliabl approach for Blasis qaio Procdigs of ICMS pp - Noor M A ad ST Mohd-Di 7a A ffici mhod for forh ordr bodar val problms Comp Mah Appl pp - Noor M A ad ST Mohd-Di 7b Variaioal iraio chiq for solvig highr ordr bodar val problms Appl Mah Comp 89 pp 99-9