International Journal of Modern Mathematical Sciences, 2013, 5(3): International Journal of Modern Mathematical Sciences

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1 Iraioal Joral of Mor Mahmaical Scic - Iraioal Joral of Mor Mahmaical Scic Joral hompagwwwmorsciificprcom/joral/ijmmap ISSN -X Floria USA Aricl Compario of Lagrag Mliplir for Noliar BVP Aif Mhmoo Farah Jab Awa a S Taf Moh-Di Dparm of Mahmaic Facl of Scic HITEC Uivri Taila Ca Pakia * Ahor o whom corrpoc hol b ar; af@homailcom Aricl hior Rciv Dcmbr Rciv i rvi form Fbrar Accp Fbrar Pblih 7 Fbrar Abrac I hi papr w compar ac a approima Lagrag mliplir o olv boar val problm BVP I i obrv ha of ac Lagrag mliplir agibl rc h compaioal work Nmrical ampl plicil ppor hi ia Kwor Noliar problm Approima Lagrag mliplir Eac Lagrag mliplir Variaioal iraio mho VIM Mahmaic Sbjc Claificaio Q79 Irocio I h rc pa hr ha b a prc vlopm i oliar cic [-] I h imilar co vral mrical a aalical mho iclig Homoop Aali HAM Prrbaio Moifi Aomia Dcompoiio MADM Variaioal Iraio VIM Variaio of Paramr Fii iffrc Fii volm hav b vlop o olv ch qaio [-] a h rfrc hri Mo of h chiq hav hir ibil ficici iclig valaio of h o-call Aomia polomial ivrg rl cciv applicaio of h igral opraor -raliic ampio o-compaibili wih h oliari of phical problm a vr lgh calclaio Ioki Ski a Mra [7] iroc gral of h Lagrag mliplir i oliar mahmaical phic a bql h am go covr io a iraiv chm call a Variaioal Iraio Mho VIM a ha b frql for fiig appropria olio of wi rag of oliar phical problm of Coprigh b Mor Sciific Pr Compa Floria USA

2 I J Mor Mah Sci - 7 compl ar [-7 -] Th baic problm i h implmaio of VIM i h ac iificaio of Lagrag mliplir which i a ral har ak Ipir a moiva b h ogoig rarch i hi ara w hav ma a ail compario of ac a approima Lagrag mliplir of corrcio fcioal of VIM o olv boar val problm W vlop h corrcio fcioal a calcla h olio of Boar Val Problm BVP ig VIM copl wih ac a approima Lagrag mliplir W vlop h corrcio fcioal a calcla h ac Lagrag mliplir opimall via variaioal hor which rc h cciv applicaio of h igral opraor a hc hac h fficic of h propo VIM I i obrv ha of ac Lagrag mliplir i h corrcio fcioal of VIM rc h compaioal work o a agibl lvl a hc icra i fficic wiho compromiig h phical ar of oliar problm of ivrifi compl ar Svral ampl ar giv which clarl rflc h igificac of ac Lagrag mliplir all h calclaio ar ma b makig of MAPLE Variaioal Iraio Mho VIM [-7 -] qaio To illra h baic cocp of h chiq w coir h followig gral iffrial L N g whr L i a liar opraor N a oliar opraor a g i h forcig rm Accorig o variaioal iraio mho [-7 -] w ca corc a corrc fcioal a follow ~ L N g whr i a Lagrag mliplir [-7 -] which ca b iifi opimall via variaioal iraio mho Th bcrip o h h approimaio ~ i coir a a rric variaio i ~ ; i call a corrcio fcioal Th olio of h liar problm ca b olv i a igl iraio p o h ac iificaio of h Lagrag mliplir Th pricipl of variaioal iraio mho a i applicabili for vario ki of iffrial qaio ar giv i [-7 -] I hi mho i i rqir fir o rmi h Lagrag mliplir opimall Th cciv approimaio of h olio will b rail obai po ig h rmi Lagrag mliplir a a lciv fcio coql h olio i giv b lim For rror ima a covrgc of Variaioal Iraio Mho VIM Ramo [] Coprigh b Mor Sciific Pr Compa Floria USA

3 I J Mor Mah Sci - Nmrical Applicaio I hi cio w appl Variaioal Iraio Mho VIM copl wih ac a approima Lagrag mliplir o olv boar val problm BVP W vlop h corrcio fcioal a calcla h ac Lagrag mliplir opimall via variaioal hor which rc h cciv applicaio of h igral opraor a hc hac h fficic of h propo VIM Nmrical rl ar vr coragig Eampl Coir Cach Racio Diffio Problm whr D p Si Co D p Co a Th corrcio fcioal for h abov problm i giv b ~ ~ p Makig corrcio fcioal aioar approima Lagrag mliplir ca b iifi a hc p Coql Co Co Co Co Co Co Co Co Co Co Coprigh b Mor Sciific Pr Compa Floria USA Co Co Co Co Co Th ri olio i giv b Co Co Co Co Co

4 I J Mor Mah Sci - 9 a h clo form olio i obai a Co Now ac Lagrag mliplir forcach Racio Diffio Problm ca b iifi b makig of aioar coiio a Hc Co Si Coql Co Co which i h ac olio Eampl Coir h followig homogo La-Em qaio // / wih iiial coiio Uig h raformaio qaio bcom / // Th corrcio fcioal i giv b / ~ Makig corrcio fcioal aioar approima Lagrag mliplir ca b iifi a hc Coql Coprigh b Mor Sciific Pr Compa Floria USA

5 I J Mor Mah Sci - Coprigh b Mor Sciific Pr Compa Floria USA Th ri olio i giv b Ivr raformaio il 7 a clo form olio i obai a Now ac Lagrag mliplir for ca b iifi b makig of aioar coiio a Hc Coql Eampl Coir h co orr iffrial qaio

6 I J Mor Mah Sci - Coprigh b Mor Sciific Pr Compa Floria USA / Si Th corrcio fcioal i giv b ~ ~ Makig corrcio fcioal aioar approima Lagrag mliplir ca b iifi a hc Coql followig approima ar obai Si Si Si 7 Si Th ri olio i giv b 7 Si a h clo form olio i obai a Si Coh Now ac Lagrag mliplir for ca b iifi b makig of aioar coiio a Sih Hc Sih Coql followig approima ar obai Si Si Coh which i h ac olio Eampl Coir h hir orr iffrial qaio

7 I J Mor Mah Sci - Coprigh b Mor Sciific Pr Compa Floria USA wih iiial coiio / // Th ac olio of h problm i Th corrcio fcioal i giv b ~ Makig corrcio fcioal aioar approima Lagrag mliplir ca b iifi a hc Coql followig approima ar obai Now ac Lagrag mliplir for ca b iifi b makig of aioar coiio a Hc Coql followig approima ar obai

8 I J Mor Mah Sci - which i h ac olio Eampl Coir h followig hir orr iffrial qaio 7 wih iiial coiio / // Th ac olio of h problm i Th corrcio fcioal i giv b 7 ~ ~ ~ Makig corrcio fcioal aioar approima Lagrag mliplir ca b iifi a hc Coql followig approima ar obai Now ac Lagrag mliplir for7 ca b iifi b makig of aioar coiio 9 Coprigh b Mor Sciific Pr Compa Floria USA

9 I J Mor Mah Sci - 7 A Hc 7 Coql followig approima ar obai 7 Coclio Eac a approima Lagrag mliplir ar compar o olv oliar boar val problm BVP I i obrv ha of ac Lagrag mliplir agibl rc h compaioal work Nmrical ampl plicil r-cofirm h claim Rfrc [] S Abbaba Nmrical olio of oliar Kli-Goro qaio b variaioal iraio mho Ira J Nmr Mch Egg [] S Abbaba A w applicaio of H variaioal iraio mho for qaraic Riccai iffrial qaio b ig Aomia polomial J Comp Appl Mah [] M A Abo a A A Solima Nw applicaio of variaioal iraio mho Ph D - - [] J H H Som ampoic mho for rogl oliar qaio I J Mo Ph -99 [] J H H Variaioal iraio mho- Som rc rl a w irpraio J Comp Appl Mah [] J H H a X W Variaioal iraio mho Nw vlopm a applicaio Comp Mah Appl 7-9 [7] M Ioki H Ski a T Mra Gral of h Lagrag mliplir i oliar mahmaical phic i S Nma-Nar E Variaioal mho i h Mchaic of oli Prgamo pr Nw York 97 p- [] W X Ma Complio olio o h Korwg- Vri qaio Ph L A - [9] W X Ma a B Fchir Eplici a ac olio of Kolmogorov-ProvkII-Pikov qaio I J Noli Mch Coprigh b Mor Sciific Pr Compa Floria USA

10 I J Mor Mah Sci - [] S T Moh-Di M A Noor a K I Noor Som rlaivl w chiq for oliar problm Mahmaical Problm i Egirig 99 ; Aricl ID 9 oi/9/9 [] S Momai No-prrbaiv aalical olio of h pac- a im-fracioal Brgr qaio Chao Solio& Fracal 9 97 [] S T Moh-Di O h covrio of parial iffrial qaio ZichriffürNarforchg A- A Joral of Phical Scic a 9-9 [] S T Moh-Di Variaioal Iraio Mho for Hiroa-Sama Mol Uig H Polomial ZichriffürNarforchg A A Joral of Phical Scic -7 - [] S T Moh-Di Moifi variaioal iraio mho for igro-iffrial qaio a copl m ZichriffürNarforchg A A Joral of Phical Scic a 77- [] S T Moh-Di A Yilirim G Dmirli Aalical olio of wav m i R wih coplig corollr Iraioal Joral of Nmrical Mho for Ha a Fli Flow 9- [] S T Moh-Di Variaioal iraio chiq for boar val problm VDM Vrlag ISBN [7] S T Moh-Di M A Noor K I Noora M M Hoii Solio of iglar qaio b H variaioal iraio mho Iraioal Joral of Noliar Scic a Nmrical Simlaio - [] J I Ramo O h variaioal iraio mho a ohr iraiv chiq for oliar iffrial qaio Appl Mah Comp [9] A M Wazwaz Th compoiio mho for h approima olio o h Gora problm Appl Mah Comp [] A M Wazwaz a A Gorgi A aalical of Fihr qaio b ig Aomia compoiio mho Appl Mah Comp 7 9 Coprigh b Mor Sciific Pr Compa Floria USA