ADOMIAN DECOMPOSITION METHOD AND TAYLOR SERIES METHOD IN ORDINARY DIFFERENTIAL EQUATIONS

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1 IJRRS 6 () gst wwwarpapresscom/volmes/vol6isse/ijrrs_6 pf DOMI DECOMPOSITIO METHOD D TYOR SERIES METHOD I ORDIRY DIFFERETI EQUTIOS José lbeiro Sáchez Cao Uiversia EFIT Departameto e Ciecias ásicas Meellí-Colombia Telé: (++7 ) 69 et josache@eafiteco STRCT I this paper it is reveale that omia ecompositio metho correspos to Talor series metho whe applie to the soltio of oliear iitial vale problems i the followig sese: the omia s polomials ca be obtai trogh Talor coefficiets Kewors: Talor series metho omia ecompositio metho oliear ifferetial eqatios ITRODUCTIO There are ma ew aaltical approimate methos to solve iitial vale problems i the literatre mog these omia s ecompositio metho [-] have bee receive mch attetio i recet ears i applie mathematics i geeral a easil hale a wie a class of liear or oliear problems The omia techiqe is base o a ecompositio of a soltio of a oliear fctioal eqatio i a series of fctios Each term of the series is obtaie from a polomial geerate b a power series epasio of a aaltic fctio The mai avatage of the DM is that it ca be applie irectl for all tpes of fctioal eqatios liear or oliear other importat avatage is capable of greatl recig the size of comptatio wor while still maitaiig high accrac of the merical soltio I [9] the athor compare the omia Decompositio Metho (DM) a the Talor series metho b sig some particlar eamples a showe that the omia s techiqe proce reliable reslts with a fewer iteratios whereas the Talor series metho sffere from comptatioal ifficlties t i this paper we will show that both omia ecompositio metho a Talor series metho are eqivalets a therefore their covergece is the same i both The followig reslt will be sefl i this paper: Faà i ro formla gives a eplicit eqatio for the th erivative of the compositio f(g(t)) Sch reslt will be se i all wor Theorem (Faà i ro formla set-partitio versio) et D f Where a f a g fctios with a sfficiet mber of erivatives the! g D f gt Dg! t D gt D gt!!!!! a the last sm is over all partitios of ie vales of that The proof of this formla is fo i [7] DOMI DECOMPOSITIO METHOD The omia s techiqe epes o ecomposig the oliear ifferetial eqatio F Ito the two compoets sch where a are the liear a oliear parts of F respectivel The operator is assme to be a ivertible operator Solvig for () leas to pplig the iverse operator o both sies of Eq () iels 68

2 IJRRS 6 () gst Cao omia Decompositio Metho 69 where is the costat of itegratio satisfies the coitio ow assmig that the soltio ca be represete as ifiite series of the form Frthermore sppose that oliear term ca be writte as ifiit series i terms of the omia polomials of the form where are omia polomials of (see [-6]) is give b followig formla: 6! i i i The sbstittig Eqs () a () i Eq() gives 7 ; s this gives the recrrece scheme of DM as 8 i orer to obtai the omia s polomials the followig algorithm will be se (see [])!! 9!!! Whe we trie to solve the eqatio i aaltical form the process is loger However i practice all the terms of series (7) caot be etermie a the soltio is approimate b the trcate series lthogh i ma sitatios o ca fo the soltio i close form sig este itegrals properties see for eample [8]

3 IJRRS 6 () gst Cao omia Decompositio Metho Propositio Cosier the ifferetial eqatio together the iitial coitio The the geeral soltio gives b Talor s series metho!! is precisel the omia Decompositio Metho: where! scheme: where : a s s! verifies! come etermie b the iterative so that Proof: Replacig the iitial coitio () ito Eq () oe gets f ow b erivatio Eq () with respect oe gets () b sig the iitial coitios: a The b mltiplig! i Eq () we obtai both sies of above eqatio a b sig the above otatio we have! ow the et step is itegrate to both sies the last eqatio That is sice s s s! s s s we have s!! erivatio Eq () agai we obtai () s s 7

4 IJRRS 6 () gst Cao omia Decompositio Metho Replacig have! i Eq () a ivie b! a b mltiplig to both sies of eqatio for!! Therefore b sig the otatio!!! itegratio to both sies of above eqatio we obtai The! s s s s s! cotiig of the same wa this process oe gets!! Itegrate both sies of above eqatio we have s s The above iterative scheme has bee costrcte: s s Where i i verifies! The the geeral soltio of iitial vale problem Eqs () -() becomes That is the omia ecompositio metho s PPICTIOS D RESUTS: IER EXMPE Sice the problems give i the article at referece [9] (Wazwaz 998) o the comptatios a aalsis of Talor series metho are relevat for or aim we rese some of the problems Cosier the liear iitial vale problem e sbject to the iitial coitios 7 The compariso will be mae b applig the two methos separatel 6 we 7

5 IJRRS 6 () gst Cao omia Decompositio Metho The Talor series metho The Talor series metho itroces the soltio b a ifiite series give b a 8 I [9] the athor replaces Eq (7) ito Eq (6) to obtai! a a a The coefficiets are etermie b eqatig coefficiets of lie powers of throgh etermiig a formal recrrece relatio The athor shows that a eplicit recrrece relatio is ifficlt alterativel term a term he mltiplies the series ivolve Istea of sig the soltio propose b [9] ie mltiplig the ep series b soltios series we cosier the followig steps: Sbstittig the iitial coitio o Eqs (7) i Eq (6) we obtai erivatio Eq (6) iels e e 9 Replacig a ito Eq (9) we have Cotiig this process we obtai the first erivatives at : iv v 6 The the approimate soltio is give b!!!! 6! 6! The omia ecompositio metho ccorig [79] Eq (6) ca be writte i a operator form as e of () we fi that! Where The iverse of is therefore s s () s pplig w we wws The ecompositio metho cosists of ecomposig Sbstittig Eq () ito both sies of Eq() iels s to both sies ito a sm of compoets give b the ifiite series w we ww s et we eqate selecte compoets o both sies sig the followig recrsive relatioship: 7

6 IJRRS 6 () gst Cao omia Decompositio Metho e ccorig we fi 6 ( ) e!! Which agai gives the soltio obtaie i Eq( ) PPICTIO: OIER EXMPE Cosier the oliear iitial vale problem!! 6 6 I the same maer as was oe i eample we will compare this eample with the two methos We iitiall assme the fctio () spports erivatives of all orers The Talor series metho I [] the athor replaces Eq () ito Eq () to obtai a a a a 6 The athor shows that Eq (6) presets comptatioal ifficlties istea of sig the soltio propose b [9] ie mltiplig the epoetial series b soltios series we cosier the followig steps: Eq () is eqivalet to eqatio 7 Sbstittig the iitial coitio i Eq (7) we obtai erivatio Eq (7) iels 8 Replacig i Eq (8) we have erivatio agai Eq (8) o gets 9 Replacig the vales i Eq (9) oe gets 6 erivatio oce more iv Replacig the vales a 6 The the approimate soltio is give b i Eq () oe gets iv 7

7 IJRRS 6 () gst Cao omia Decompositio Metho 7! 8!!!! gai we ca calclate recrretl so ma coefficiets i the series as ecessar to proce a soltio with a esire accrac for all real sfficietl close = However we caot epress the geeral coefficiet a of the series b meas of a eplicit fctio of a therefore we caot irectl calclate the rais of covergece of this soltio Sch sitatios occr whe o are looig for soltios i series of powers of oliear eqatios omia ecompositio metho gai we ca pt Eq () i a operator form as I this case is the oriar ifferetial operator a s is a itegral operator pplig as before to both sies of Eq () a sig the iitial coitios we fi The ecompositio metho epas each of the oliear terms a formall i a power series give b were a are the so-calle omia polomials correspoig to the oliear terms a respectivel The omia polomials are give b a Sbstittig Eqs () () a () ito the fctioal eqatio () gives Here we ca assme the covergece of the series both sies of Eq () will match b settig the recrsive relatioship This leas to 8 The soltio of Eq() i a series form is therefore! 8 The et eample shows that the omia ecompositio is more complicate tha Talor series metho becase its polomial caot fo easil bt oe ca se the coefficiets fo b Talor a the b sig omia metho ca fi the soltio Cosier the iitial vale problem t t

8 IJRRS 6 () gst Cao omia Decompositio Metho This problem has eact soltio: l t t Proceeig as before we write the ifferetial eqatio i the form: t repeatig the process carrie ot with the previos eamples Doig the sccessive erivatives we fo that t t t t t t e 7 t t t t t So the soltio i Talor series is therefore: t 7 P t t! We fi that the rais of covergece of the series is give b a lim a t lim! t lim t e This same problem with a best reslt has bee treate i [] lthogh the ecompositio metho provies the same aswer obtaie b the Talor metho it ivolves more comptatioal wor I aitio a recrrece relatio was ot eas to obtai b sig omia metho We cocle that althogh the omia ecompositio metho provies the same aswer obtaie b Talor series with the same comptatioal wor We shol to be patiet COCUSIO The omia ecompositio metho after all cotie to be a goo metho for solvig oliear iitial vale problems; it shol be ote that certai oliear problems ca be solve aalticall b sig this metho a certai properties of itegratio sch as este itegral [8] whereas Talor series soltio wol approimate soltio 6 CKOWEDGEMETS The athor wish to tha the Direcció e Ivestigació Docecia e la Uiversia EFIT for their spport i evelopig this wor 7 REFERECES [] K bbaoi Y Cherralt Covergece of omia s metho applie to oliear eqatios Math Compt Moellig (9) (99) 6-7 [] omia G (986) oliear Stochastic Operator Eqatios caemic Press oo [] omia G (99) Solvig Frotier Problems of Phsics: The Decompositio Metho Klwer caemic Pblishers Dorrecht [] omia G (98) Stochastic Sstems caemic Press ew Yor Y [] J-Sheg Da () ew ieas for ecomposig oliearities i ifferetial eqatios ppl Math Compt [6] M zreg-ïo Develope ew lgorithm for Evalatig omia Polomials CMES vol o pp-8 9 [7] Roma S (98) The formla e Faà i ro mer Math Mothl [8] J Sáchez Cao () omia ecompositio metho for a class oliear problem ISR pplie Mathematics Vol rticle ID 797 pages oi: //797 [9] M Wazwaz (998) compariso betwee omia ecompositio metho a Talor series metho i the series soltio ppl Math Compt [] M Wazwaz ew algorithm for calclatig omia polomials for oliear operators pplie Mathematics a Comptatio () -69 7