omia Decomposiio Meho wih iffere polyomials for oliear Klei Goro eqaio a a sysem of oliear parial iffereial eqaios Mohaa Riyah Saa Deparme of Mahemaics College of Sciece Uiversiy of Basra Iraq bsrac I his paper he omia Decomposiio Meho DM wih moifie polyomial [] is applie for oliear moels firs we apply i for solvig oliear parial iffereial eqaio Klei Goro eqaio wih a qaraic o -liear erm he we iscsse he solio of sysem of oliear parial iffereial eqaios wih his moificaio. The merical resls obaie by his polyomial have bee compare wih merical resls i [] a [] o show he efficiecy i applicaios. Keywors: Noliear Klei Goro eqaio Sysem of oliear parial iffereial eqaios DM.
معا ختلفة لحل Klei Goro DM في م DM [] ي ية Klei Goro eqaio wih a qaraic o-liear erm [] ل [] نا تها مع.
. Irocio Sice he begiig of he 98s he DM has bee applie for a wie class of fcioal eqaios []. omia gives he solio as ifiie series sally covergig o a accrae solio. The procere of DM ee a polyomial i applicaios may researchers se omia polyomial i he implemeaio of DM. Kalla [] iscsse a ew polyomial for DM as we see i secio. I his paper we se DM wih Kalla polyomial for fiig he solio of oliear Klei Goro eqaio wih a qaraic oliear erm a he solio of a sysem of oliear parial iffereial eqaio. Tes problems are iscsse [ ] we se Maple sofware for his prpose he obaie resls sgges ha Kalla polyomials iroces a promisig ool a powerfl improveme for solvig oliear parial iffereial eqaios a sysems of oliear parial iffereial eqaios.. The omia ecomposiio meho DM Le s cosier he followig eqaio L N R f where L is a iverible liear operaor N represes he oliear operaor a R is he remaiig liear par from eqaio we have L f N R ow applyig he iverse operaor L o boh sies of eqaio he se he iiial coiios we fi g L N L R where L s a g represes he erms havig from iegraig he remaiig erm f a from sig he give iiial or boary coiios. The DM assmes ha he oliear operaor N ca N be ecompose by a ifiie series of polyomials give by... where are he omia s polyomials [] efie as: N! i hi... So he cae be give as: N
N N N N N N N N N N N lso cae be epresse by a ifiie series of he form ieifyig he remaiig compoes for... ca be eermie by sig recrrece relaios g L L [ R ]... The oher polyomials ca be geerae i a similar way. The solio will be he approimaios k k k wih lim. k k Ibrahim L. El-Kalla [] iroce a ew formla for omia polyomials he claime ha he omia solio sig his ew formla coverges faser ha sig omia polyomials. Kalla polyomial give i he followig form: Where S.... For eample if formlas a are compe o be: Usig formla : i N S i N he firs hree polyomials sig
5 Usig formla : so o. These formlas are easy o compe by sig Maple sofware.. Solio of oliear Klei Goro eqaio by DM Le s cosier he oliear Klei Goro eqaio f N a where a is real cosa N is a give oliear fcio a z y a z y. The iiial coiios are b a b by DM we sppose ha a N where efie as so f L b b a N L al L. The erm r r coverge o he solio as.. Tes problems I his secio we se DM wih polyomial for eamples.١ [] a. [].
Eample. Cosier he eqaio Wih he iiial coiios a si 5. The eac solio of 5 is si. The Nmerical a Eac solios are show i Figres.a.b a.c. also able shows he compariso bewee he resls by DM D wih polyomial DM M wih he polyomial a he eac solio E. Eample. Cosier he cople sysem of oliear physical eqaios Wih he iiial coiios The eac solios are v v v v v k e k e e e k c z k c e v a k k e k c e v z. k c e where k is cosas. Wih a fie vales of k a c a for iffere vales of ime he merical solios of a v by DM wih polyomial are show i Figres..a a.b while he merical solios of a v by DM wih polyomial are show i Figres.a a.b. The behavior of he eac solios of a v are show i Figres.a a.b. Table shows he compariso bewee he resls by DM D a Dv wih polyomial DM M a Mv wih he polyomial a he eac solio E a Ev.
Fig..a The solio of wih polyomial Fig..b The solio of wih polyomial Fig..c The Eac solio of Fig..a he solio of by DM Fig..b he solio of v by DM wih polyomial wih polyomial 7
Fig..a he solio of by DM Fig..b he solio of v by DM wih polyomial wih polyomial Fig..a he Eac solio of Fig..b he Eac solio of v Table =. =. =. =. =.5 =..5E.5D.87E.87D.9E.9D.57E.57D.787E.787D.5M.87M.9M.57M.787M =..9E.9D.8E.8D.975E.975D.7E.7D.558E.558D.9M.8M.975M.7M.559M 8
=. =. =. =. =.5 =..599E.599D.9798E.9798D.97E.97D.859E.859D.995E.995D.599M.9798M.97M.859M.99M =..58779E.58779D.7557E.7557D.7E.7D.5E.5D.989E.989D.58778M.7557M.7M.5M.989M =.5.77E.77D.E.D.E.D.88E.88D.555E.555D.77M.M.M.88M.555M =..89E.89D.8E.8D.75E.77D.7E.8D.58E.58D.89M.8M.75M.7M.59M =.7.89E.89D.78E.78D.7E.7D.5E.58D.55E.55D.89M.78M.7M.5M.55M =.8.95E.955D.9E.957D.857E.8555D.8E.8D.7558E.755D.955M.9M.857M.8M.7558M =.9.9879E.9895D.9758E.9789D.97E.9D.9575E.95D.98E.9858D.9878M.9758M.97M.957M.98M =.9E.8D.E.D.E.8D.E.D.5E.5D.M.M.M.M.5M 9
Table = =. =. =. =. =.5 =.5 E.5 D.5M.75 Ev.75 Dv.75Mv.5979 E.5979 D.5979M.78 Ev.78 Dv.78Mv.598 E.598 D.598M.8888 Ev.8888 Dv.8888Mv.57 E.57 D.57M. Ev. Dv.Mv.59887 E.59887 D.59887M.8 Ev.8 Dv.8Mv.59 E.59 D.59M.879 Ev.879 Dv.879Mv =..5 E.59 D.59M.75 Ev. Dv.5Mv.5979 E.5995 D.5995M.78 Ev.877 Dv.88Mv.598 E.5 D.5M.8888 Ev.99 Dv.958Mv.57 E.587 D.587M. Ev.77 Dv.75Mv.59887 E. D.M.8 Ev.9 Dv.97Mv.59 E.9 D.9M.879 Ev.7 Dv.78Mv =..5 E.585 D.585M.75 Ev.88 Dv.589Mv.5979 E.557 D.557M.78 Ev.8 Dv.8Mv.598 E.57 D.57M.8888 Ev.9888 Dv.8Mv.57 E.5999 D.5999M. Ev.7598 Dv.778Mv.59887 E.7 D.7M.8 Ev.58 Dv.58Mv.59 E.8 D.8M.879 Ev. Dv.78Mv =..5 E.57 D.57M.75 Ev.99 Dv.75Mv.5979 E.57 D.57M.78 Ev.88 Dv.58Mv.598 E.5879 D.5879M.8888 Ev.9875 Dv.9Mv.57 E.988 D.988M. Ev.778 Dv.885Mv.59887 E.899 D.899M.8 Ev.59 Dv.Mv.59 E.558 D.558M.879 Ev.57 Dv.9Mv =..5 E.555 D.555M.75 Ev.975 Dv.755Mv.5979 E.578 D.578M.78 Ev.77 Dv.55Mv.598 E.5957 D.5957M.8888 Ev.997 Dv.9Mv.57 E.85 D.85M. Ev.7557 Dv.88Mv.59887 E.8 D.8M.8 Ev.55 Dv.89Mv.59 E.77 D.77M.879 Ev.58 Dv.Mv =.5.5 E.55 D.55M.75 Ev.5 Dv.57Mv.5979 E.58 D.58M.78 Ev.887 Dv.9Mv.598 E.8 D.8M.8888 Ev.9 Dv.7Mv.57 E.597 D.597M. Ev.789 Dv.8Mv.59887 E.89 D.89M.8 Ev.578 Dv.7Mv.59 E.79 D.79M.879 Ev. Dv.888Mv
=. = =. =. =. =. =.5.5979 E.598 E.57 E.59887 E.5998 D.57 D.9 D.5 D.5998M.57M.9M.5M.78 Ev.8888 Ev. Ev.8 Ev. Dv.899 Dv.7 Dv.5 Dv.9Mv.99Mv.7985Mv.597Mv.5 E.59 D.59M.75 Ev. Dv.Mv.59 E.777D.777M.879 Ev. Dv.7 Mv =.7.5 E.575879 D.575879M.75 Ev.8DV.7MV.5979 E.598 D.598M.78 Ev.9758DV.98MV.598 E.8 D.8M.8888 Ev.78DV.999MV.57 E.87 D.87M. Ev.57DV.7MV.59887 E.559 D.559M.8 Ev.57DV.555MV.59 E.79D.79M.879 Ev.7 DV.7 MV =.8.5 E.585 D.585M.75 Ev.985DV.957MV.5979 E.9 D.9M.78 Ev.779DV.79MV.598 E. D.M.8888 Ev.5777DV.88MV.57 E.95 D.95M. Ev.9DV.57MV.59887 E.587 D.587M.8 Ev.7DV.859MV.59 E.79D.79M.879 Ev.55 DV.8 MV 5. Coclsios I his paper he DM was applie for fiig he solios of he oliear parial iffereial eqaios a he sysem of oliear parial iffereial eqaios. I s clear ha DM wih iffere polyomials give iffere solios. The DM wih polyomial is beer ha DM wih polyomial. lso polyomial are more simpler ha polyomial i calclaios. I may be cocle ha DM wih polyomial is very powerfl a efficie i applicaio for wie classes of problems. Refereces [] Ibrahim L. El-Kalla Error aalysis of omia series solio o a class of Noliear iffereial eqaios Mah. ppl. 7 7 -. [] K. C. Basak P. C. Ray a R. K. Bera Solio of oliear Klei-Goro eqaio wih a qaraic oliear erm by omia ecomposiio meho Commicaios i Noliear Sciece a Nmerical Simlaio 9 78-7 [] S.. El-Wakil & M.. bo New applicaios of omia ecomposiio meho Chaos Solios a Fracals 7 5-5 [] G. omia Solvig froier problems of physics: The Decomposiio meho Klwer caemic Pblishers Dorrech 99.