International Journal of Pure and Applied Sciences and Technology

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1 I. J. Pr Appl. Sci. Tchol. 4 pp. -4 Iraioal Joral of Pr a Appli Scics a Tcholog ISSN 9-67 Aailabl oli a Rsarch Papr Variaioal Iraio Mho for Solig So Mols of Noliar Parial Diffrial Eqaios A.S.J.AL-Saif * a T.A.K. Hai Dpar of Mahaics Collg of Ecaio Uirsi of Basrah Basrah Iraq. * Corrspoig ahor -ail: saaralsaif@ahoo.co Rci: 6--; Accp: 8-4- Absrac: I his papr h ariaioal iraio ho is ipl for solig oliar iiial al probls. Aalic solios of h oliar parial iffrial qaios wih iiial aa ar obai. I has b show ha h ho is qi ffici a is pracicall wll si for s i hs probls. Sral apls ar gi o rif h accrac a fficic of h propos chiq. Kwors: Variaioal iraio ho Noliar iiial al probls.. Irocio I 978 Ioki al. [4 propos a gral Lagrag liplir ho o sol o-liar probls which was h firs propos o sol probls i qa chaics. I 998 h Lagrag liplir ho is oifi b H [8- io a iraio ho ha is call ariaioal iraio ho VIM. I is s o sol ffcil asil a accral a larg class of o-liar probls wih approiaios corgig rapil o accra solios whr h approia solio of h VIM i h ai is rail obai po sig h obai Lagrag liplir a o h slci iiial approia. Th ariaioal iraio ho chags h iffrial qaio o a rcrrc sqc of fcios whr h lii of ha sqc is cosir as h solio of h parial iffrial qaios. Th ai aaag of h ho is ha i ca b appli ircl o all ps of oliar iffrial a igral qaios hoogos or ihoogos wih cosa or ariabl cofficis [ 5-7. Moror h propos ho is capabl

2 I. J. Pr Appl. Sci. Tchol. 4 pp. -4 of gral rcig h siz of copaioal work whil sill aiaiig high accrac of h rical solio. I his work h o-liar probls ha h followig qaios; A h o-liar probls ha h followig ss of qaios; 4 will b si b iplig VIM whr a is h solio of qaio a >. Eqaio is o of a class of o-liar ha rasfr qaios ariss i sral ipora phsical chical racios a ahaical biolog applicaios [ 8 9. This class of qaio is sol b sig Li grop ho of ifiisial rasforaios [. Eqaio is h hprbolic oliar which scribs h shock phoo ihr i is solio. This qaio is hal b sig sral chiqs sch as iffrial qarar ho DQM[5 Aoia coposiio hoadm[ fii iffrc ho[8. Eqaio is o of a class of qasiliar parabolic qaios. This qaio is sol ricall b fii iffrc sch [7 a horicall b iffr hos [. Ss of qaios 4 copl Brgr s qaios. I has b fo o scrib arios kis of phoa sch as a ahaical ol of rblc h approia hor of flow hrogh a shock wa ralig i a iscos fli [6. This ss is hal b ADM [6 a iffrial rasfors ho [. Ss of qaios 5 arisig i arios siplifi ols of fli flow a rblc a i ca b sol b arios rical hos sch as ADM [ a DQM[5. Th ai of his papr is o appl h ariaioal iraios ho o sol oliar iiial al probls ha o o appl h ho for solig i b sig VIM ac solio a approia solios of h probls ha b obai i rs of corg sris wih asil copabl copos. Th orgaizaio of his papr is as follows; scio gis brif ias of VIM. I scio h sffici coiios ar prs o gara h corgc of h ho. I scio 4; for apls ar gi o illsra h ffciss a h sfl of h ariaioal iraio ho. I scio 5 w prs iscssio of or work. Coclsios ar prs i h las scio.. Basic Cocps of Variaioal Iraio Mho: Ia of ariaioal iraio ho ps o h gral Lagrag's liplir ho [4. This ho has a ai far which is h solio of a ahaical probl wih liarizaio asspio s as iiial approiaio or rial fcio. This approiaio corgs rapil o a accra solio [. 5

3 I. J. Pr Appl. Sci. Tchol. 4 pp. -4 To illsra h basic cocps of h VIM w cosir h followig oliar iffrial qaio: L N g 6 whr L is a liar opraor a N is a oliar opraor a g is a ihoogos r. Accorig o h VIM [4 w ca cosrc a corrcio fcioal as follows: λ L N ~ [ g 7 whr λ is a gral Lagragia liplir[56 which ca b iifi opiall ia h ariaioal hor a igraio b pars. Th sbscrip os h h-orr approiaio ~ is cosir as a rsric ariaio i.. δ ~ [8. So w firs ri h Lagrag liplir λ ha will b iifi opiall ia igraio b pars. Th sccssi approiaios of h solio will b rail obai po sig h obai Lagrag liplir a b sig a slci fcio. Cosql h Solio li 8. Corgc Aalsis of h VIM: Hr w will s h corgc aalsis as h sa ar i [ of h ariaioal iraio ho o h oliar qaios. L s cosir h Baach spac X wih h s of applicaios : Ω R wih a h associa or: < Ω Ω Th VIM is corg if h coiios of h followig hor ar saisfi. Thor [: Baach s fi-poi hor Ass ha X b a Baach spac a A : X X is a oliar appig a sppos ha A A γ for so cosaγ <. Th A has a iq fi poi. [ [ Accorig o h hor for h oliar appig [ λ F A A sffici coiio for corgc of h ariaioal iraio ho is h sricl coracio of A sch ha for X w ha M a M for M >.

4 I. J. Pr Appl. Sci. Tchol. 4 pp Applicaios of VIM: I his scio w appl h ariaioal iraio ho for solig fi ols of oliar iiial al probls. Thr coais o-liar parial iffrial qaios a h ohrs coai o-liar ss of parial iffrial qaios. 4. Th oliar qaios: To phol or work w iroc hr apls Eapl : Cosir h followig o-liar parial iffrial qaio; 9 Accorig o h VIM w ca cosrc h corrcio fcioal 7 of qaio 9 as follows: λ ~ ~ [ whr λ is a gral Lagrag liplir. Th al of λ ca b fo b cosirig ~ a ~ as a rsric ariaios i.. δ ~ δ~ i qaio h igraig h rsl b par o obai λ. Th h corrcio fcioal bcos i h followig iraio forla: [ Cosql h followig approias ar obai b sig h abo iraio forla sar wih h iiial approiaio sall rs M sall rs a so o. Th solio i a clos for is rail fo o b [9.

5 I. J. Pr Appl. Sci. Tchol. 4 pp Tabl : Copariso bw h DQM ADM a h VIM solios a.. ac 4 VIM 4 ADM DQM Eapl : Cosir h followig o-liar parial iffrial qaio [5 Th corrcio fcioal 7 of qaio 4 is gi as: 4 λ ~ ~ [ 5 Procig as bfor w fi λ. Sbsiig his al io h corrcio fcioal 5 gis h followig iraio forla: [ 6 W sar wih iiial approiaio a sig h iraio forla 6 w ca obai: sall rs 7 sall rs M a so o. Th solio i a clos for is rail fo as; 8

6 I. J. Pr Appl. Sci. Tchol. 4 pp Tabl : Copariso bw h DQMADM a h VIM solios a.. ac VIM 4 ADM 6 DQM Eapl : Cosir h followig oliar parial iffrial qaio [79; > 9 Accorig o h VIM w ca cosrc h corrcio fcioal 7 of qaio 9 as follows: λ ~ ~ ~ ~ [ Procig as bfor w fi λ. Sbsiig his al io h corrcio fcioal gis h followig iraio forla: [ Usig h abo iraio forla a h iiial approiaios w obai h followig sccssi approiaios: M!!! a so o. Th solio i a clos for is gi b:

7 I. J. Pr Appl. Sci. Tchol. 4 pp Tabl : Copariso bw h DQM a h VIM solios 5. a.. ac VIM 5 ADM 5 DQM Th o-liar sss qaios W ow iroc wo apls for o-liar ss qaios Eapl 4: Cosir h followig o-liar ss qaios [ 6 4 Followig h aalsis prs abo w cosrc a corrcio fcioal for 4 as: λ λ ~ ~ ~ ~ ~ [ ~ ~ ~ ~ ~ [ 5 Procig as bfor w fi λ. Th corrcio fcioal 5 bcos i h followig iraio forla: [ [ 6 Cosql h followig approiaios ar obai b sig h abo iraio forlas 6 wih h iiial approiaios si. M!! si! si si si 7 So o. Accorig o h qaio 8 h solios ar gi as: si 8

8 I. J. Pr Appl. Sci. Tchol. 4 pp Tabl 4: Copariso bw h DQM ADM a h VIM solios a.. ac VIM 5 ADM 5 DQM Eapl 5: Cosir h followig wo-isio o-liar ss qaios [ 9 To sol h ss 9 b as of ariaioal iraio ho w cosrc a corrcio fcioal 7 as: λ λ ~ ~ ~ ~ [ ~ ~ ~ ~ [ Procig as bfor w fi λ. Th corrcio fcioal bcos i h followig iraio forla: [ [ Cosql h followig approias ar obai b sig h abo iraio forlas wih h iiial approiaios M rs sall rs sall So o. Th solio i a clos for is rail fo as; /

9 I. J. Pr Appl. Sci. Tchol. 4 pp Tabl 5: Copariso bw h DQM ADM a h VIM solios a.. ac 8 VIM ADM DQM Discssio I his s apls ar sol b h VIM o h bo rgio or. Th obai rsls fro h ariaioal iraio ho ar lis i h Tabls -5 a.. Tabls show h copariso bw or solio sig VIM a h solio of h sa probl b sig ADM a DQM. Th rsls wr obai show ha VIM solios wih ls iraios corg rapil o h ac solio or a.. I apls h forh iraios ar wh cosirig for arriig a h accra solios a i apls 45 w fifh a ighh iraios rspcil. Fro h copariso bw h rsls of VIM a ac solio w oic ha h approia solios obai b sig VIM ar r wll accra wh h br of iraios is icrasig wih salls al of i. Cosql w cocl ha h rrors shol b sallr wih sallr i. Moror h probls i his s ar s b h fi poi hor. Th rsls cofir ha A has a iq fi poi a saisfig h corgc coiio i h hor wih h al of γ for; Eapl : γ δ δ 6M Eapl : γ δ δ M δ M δ Eapl : γ Eapl 4: γ δ δ M δ δ 4 Eapl 5: γ δ M δ δ whr δ ' s ar h absol als of iffrial opraors ha appar i parial iffrial qaios. 6. Coclsios I his papr h ariaio iraio ho has b sccssfll plo o obai h approia aalical solios of o-liar boar a iiial al probls. Th ho has b appli ircl wiho sig liarizaio or a rsrici asspios. Th copariso of h rical rsls of VIM wih ohr solios b sig ohr hos show ha h ariaioal iraio ho is a powrfl ahaical ool o solig oliar parial iffrial qaios a fasr i corgc o ac solio.

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11 I. J. Pr Appl. Sci. Tchol. 4 pp [7 M.O. Miasari M.E. Miasari A. Barari a D.D. Gagi Applicaio of H s ariaioal iraio ho o oliar Hlholz a fifh-orr K Eqaio J. Appl. Mah. Sa. If [8 B. J. No Nrical Solio of Parial Diffrial Eqaios Norh Holla Soh Asralia 98. [9 A.D. Polai a V.F. Zais Habook of Noliar Parial Diffrial Eqaios Chapa a Hall/CRC U.S.A 4. [ A.S.V. Rai Kah a K. Ara Diffrial rasfor ho for solig liar a oliar sss of parial iffrial qaios Phs. L. A [ A.A. Saarskii V.A. Galakioo S.P. Kro a A.P. Mikhailo Blow- p i Probls for Qasiliar Parabolic Eqaios Walr Grr Brli 995. [ H. Sphai Diffrial Eqaios hir Solio sig Sris E. M. MacCall C.U.P. Cabrig 989. [ M. Taari M. Dhgha O h corgc of H s ariaioal iraio ho J. Cop. a Appl. Mah