VARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS. Hunan , China,

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1 Mahemacal ad Compuaoal Applcaos Vol. 5 No. 5 pp Assocao for Scefc Research VARIATIONAL ITERATION METHOD FOR DELAY DIFFERENTIAL-ALGEBRAIC EQUATIONS Hoglag Lu Aguo Xao Yogxag Zhao School of Mahemacs ad Compuaoal Scece Xaga Uversy Hua 45 Cha lhl@xu.edu.c xag@xu.edu.c School of Mahemacs ad Compuer Scece Three Gorges Uversy Chogqg 44 Cha zyxlly8@.com Absrac- Varaoal erao mehod s appled o solve a class of delay dfferealalgebrac equaos. The obaed sequece of erao s based o he use of Lagrage mulplers. The correspodg covergece resuls are obaed ad successfully cofrmed by some umercal examples. Keywords- Delay Dffereal-Algebrac Equaos Varaoal Ierao Mehod Covergece. INTRODUCTION The varaoal erao mehod (VIM) was frs proposed by He [ ] ad has bee exesvely dscussed by may auhors [3-]. Applcaos of hs mehod have bee elarged due o s flexbly coveece ad effcecy. Some auhors have appled VIM o delay dffereal equaos [7] ad dffereal-algebrac equaos [3] bu VIM for delay dffereal-algebrac equaos (DDAEs) has o bee cosdered. I fac DDAEs are a very mpora class of mahemacal models ad ofe arse from he felds of compuer aded desg crcu aalyss mechacal sysems ec. Some resuls heorecal aalyss ad umercal soluos of DDAEs have bee gve whch clude sably of Ruge-Kua mehods for eural delay egro-dffereal-algebrac equaos [] he classcal covergece resuls of BDF mehods ad Ruge-Kua mehods for dex- DDAEs [] ad collocao mehods for rearded dfferealalgebrac equaos [3]. I hs paper we apply VIM o a class of DDAEs o oba approxmae aalycal soluos. The covergece resuls of he VIM for DDAEs are obaed. Some llusrave examples cofrm he heorecal resuls.. MAIN RESULTS Cosder he al value problem of a DDAE x '() = f( x() x( α()) y() y( β())) T = gx ( x( α) y ) T () x () = ϕ() τ y () = ψ() τ where he delay fucos α () ad β () sasfy τ α() τ β() : R f

2 H. Lu A. Xao ad Y. Zhao 835 R R R R g: R R R R are smooh vecor fucos o he real Eucldea spaces ad have bouded dervaves he al value fucos ϕ :[ τ] R ad ψ :[ τ ] R are couous gy ( x x( α ) y) s verble ad bouded a eghbourhood of he rue soluo. We assume ha he problem () has a smooh soluo x() y(). Throughou hs paper deoes he sadard Eucldea orm ad he marx orm s subordae o. Accordg o he VIM we ca cosruc he correco fucoal as follows x = x λ( s )( x ( ( ) ( ))) s f x s x α s y s y β s ds (a) = gx ( () x ( α()) y ()) (b) where λ () s s a geeral Lagrage mulpler whch ca be defed opmally by varaoal heory ad f deoes he resrcve varao.e. δ f =. Thus we have δx () = δx () λ()( s δx ()) s ds ad he saoary codos are obaed as λ() s λ ( s ) = =. s= s Moreover he geeral Lagrage mulpler ca be readly defed by λ () s =. Therefore he varaoal erao formula ca be wre as α x () = x () f( x () s x ( ()) s y () s y ( β())) s ds (3a) = gx ( () x ( α()) y ()). (3b) Theorem Le x () x() ( C[ τ T]) y () y() ( C[ τ T]) =. The he sequeces { x} = { y} = defed by (3) wh x() = ϕ() τ y() = ψ () τ coverge o he soluo of (). Proof. From he sysem () we obvously have x() = x() f(() x s x( α()) s y() s y( β())) s ds (4a) = g( x() x( α ()) y()). (4b) Iroduce Ex = x x Ey = y y = where Ex = Ey = < =. From (3)-(4) we oba E x = ( f( x ( s) x ( α( s)) y ( s) y ( β( s))) f( x( s) x( α( s)) y( s) y( β( s)))) ds = gx ( () x ( α()) y ()) gx ( () x( α()) y ()).

3 83 Varaoal Ierao Mehod for Delay Dffereal-algebrac Equaos Based o he fac ha he fucos f g are smooh ad he marx deduce = α 3 4 g y s verble we E x () ( fexs () f Ex( ()) s f Eys () f Ey( β())) s ds (5a) E y () = ( g ) ge x () ( g ) ge x( α ()) (5b) 3 3 where f ( = 34) deoes he paral dervave of he fuco f o h varable g ( = 3) deoes he paral dervave of he fuco g o h varable. We ca derve () Exs 3 () ds E ( ()) x l l l l Ex 4 α s ds E () kl kl3 () kl kl y Eys 4 ( ()) ds Ey β s ds where l = max f ( = 34) k = max( ( g ) g ( g ) g ). Therefore 3 3 () Exs () ds Ex l l ( ()) 3 l l Exα s ds 4 Ey () kl kl 3 Eys 4 () ds kl kl Ey ( β ()) s ds Moreover we have where l l l max 3 l E xs 4 τ s T. () kl ( l) kl ( l) max E ys 3 4 τ s T Ex () max Exs ( τ T ) ρ τ s T E ()! y max Eys τ s T τ T max E x( s) max E y( s) k l ( = 34) are cosas ρ s he τ s T τ s T specral radus of he las marx he above equaly (). By usg he Srlg's formula we have hus ( Ex Ey ) T as.. Ex () ( T) e max τ ρ Exs ( ) τ s T Ey () π ( O(/ )) max Eys τ s T (7) (8)

4 H. Lu A. Xao ad Y. Zhao ILLUSTRATIVE EXAMPLES I hs seco some llusrave examples are gve o show he effcecy of he VIM for DDAEs. Example Cosder he followg al value problem y' = y y ( ) x ( ) = y ( ) x x() = y() =. We apply he VIM o (9) ad cosruc he correco fucoal (9) s s = y () y () ( y () s y x ) ds (a) = y x ( ). (b) Moreover he erao sequece sars wh he al approxmaos y () = x () = x () = ad s obaed from () as follows y 3 () = x () = y () = o( ) x () = o( ) y () = o( ) x () = o( ) From he above erao sequece we ca show ha lm y = s lm x = cos. () Example Cosder he followg al value problem x' = y z y' = x z ( ) z () = y xz x() = y() = z() =. The exac soluo of he sysem () s x() = ( ) e = o( ) y () = ( e ) = o z () = e = o

5 838 Varaoal Ierao Mehod for Delay Dffereal-algebrac Equaos We apply he VIM o () ad cosruc he correco fucoal x () = x () ( y z ()) s ds (3a) s s = y () y () ( x () s z z ()) s ds (3b) = y x z ( (3c) ). Moreover he erao sequece sars wh he al approxmaos x () = x () = y = y() = z = z() = ad s obaed from (3) as follows x () = y() = z () = x() = 3 y() = z() = 3 4 o( ) 3 4 x3() = y3() = 3 o( ) z3() = 3 o( ) x 4() = o( ) y4() = o( ) z4 = o () The above erae sequece shows ha he VIM yelds a very good approxmao o he exac soluo. 4. CONCLUSIONS I hs paper we successfully apply VIM o a class of DDAEs ad oba hghly accurae soluos wh few eraos. VIM hadles DDAEs whou ay especal assumpo o he delay em hus s a promsg mehod for DDAEs. 5. ACKNOWLEDGEMENT Ths work s suppored by projecs from NSF of Cha (No.9775) Specalzed Research Fud for he Docoral Program of Hgher Educao of Cha (No.943) NSF of Hua Provce (No.9JJ3) ad Research Projecs of Xaga Uversy (No.8xz5).

6 H. Lu A. Xao ad Y. Zhao 839. REFERENCES. J. H. He Varaoal erao mehod - a kd of o-lear aalycal echque: Some examples Ieraoal Joural of Nolear Mechacs J. H. He Approxmae soluo of olear dffereal equaos wh covoluo produc oleares Compuer Mehods Appled Mechacs ad Egeerg F. Solaa S. M. Karbass M. M. Hosse Applcao of He's varaoal erao mehod for soluo of dffereal-algebrac equaos Chaos Solos ad Fracals D. K. Salkuyeh Covergece of he varaoal erao mehod for solvg lear sysems of ODEs wh cosa coeffces Compuers ad Mahemacs wh Applcaos A. M. Wazwaz The varaoal erao mehod for aalyc reame for lear ad olear ODEs Appled Mahemacs ad Compuao A. Saadamad M. Dehgha Varaoal erao mehod for solvg a geeralzed paograph equao Compuers ad Mahemacs wh Applcaos Z. H. Yu Varaoal erao mehod for solvg he mul-paograph delay equao Physcs Leers A R. Mokhar M. Mohammad Some Remarks o he Varaoal Ierao Mehod Ieraoal Joural of Nolear Sceces ad Numercal Smulao S. M. Goh M. S. N. Noora I. Hashm e al. Varaoal Ierao Mehod as a Relable Treame for he Hyperchaoc Rossler Sysem Ieraoal Joural of Nolear Sceces ad Numercal Smulao I. Aes A. Yldrm Applcao of varaoal erao mehod o fracoal alvalue problems Ieraoal Joural of Nolear Sceces ad Numercal Smulao Y. Xu J. J. Zhao Sably of Ruge-Kua mehods for eural delay egro dffereal-algebrac sysem Mahemacs ad Compuers Smulao U. M. Ascher L. R. Pezold The umercal soluo of delay-dffereal-algebrac equaos of rearded ad eural ype SIAM Joural o Numercal Aalyss R. Hauber Numercal reame of rearded dffereal-algebrac equaos by collocao mehods Advaces Compuaoal Mahemacs