Accelerated Solution of High Order Non-linear ODEs using Chebyshev Spectral Method Comparing with Adomian Decomposition Method

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1 Stdes Noler Sceces (): 9-, ISSN -9 IDOSI Pblctos, Accelerted Solto of Hgh Order No-ler ODEs sg Chebyshev Spectrl Method Coprg wth Ado Decoposto Method M.M. Khder Deprtet of Mthetcs, Fclty of Scece, Beh Uversty, Beh, Egypt Abstrct: I ths rtcle, ccrte Chebyshev spectrl ethod for solvg hgh order o-ler ODEs s preseted. Propertes of the Chebyshev polyols re tlzed to redce the coptto of the proble to set of lgebrc eqtos. Soe eples re gve to verfy d llstrte the effcecy d splcty of the ethod. We copred or ercl reslts gst the Ado decoposto ethod (ADM). Specl tteto s gve to stdy the covergece lyss of ADM. Nercl reslts were obted fro these two ethods show tht the proposed techqes re ecellet coforce ost cses. Also, fro the preseted eples, we fod tht the proposed ethod c be ppled to wde clss of hgh order o-ler ODEs. Key words: Chebyshev spectrl ethod hgh order o-ler ODEs Gss-Lobtto odes Ado decoposto ethod covergece lyss INTRODUCTION Chebyshev polyols re eples of egefctos of sglr Str-Lovlle probles. Chebyshev polyols hve bee sed wdely the ercl soltos of the bodry vle probles [5] d copttol fld dycs. The estece of fst Forer trsfor for Chebyshev polyols to effcetly copte tr-vector prodcts hs et tht they hve bee ore wdely sed th other sets of orthogol polyols. Chebyshev polyols re well ow fly of orthogol polyols o the tervl [-,] tht hve y pplctos [, 9]. They re wdely sed becse of ther good propertes the pproto of fctos. However, wth or best owledge, very lttle wor ws doe to dpt ths poly-ols to the solto of dfferetl eqtos. The well ow Chebyshev polyols [7] re defed o the tervl [-,] d c be detered wth the d of the followg recrrece forl: T () T () T (),,,... The frst three Chebyshev polyols re T (), T(), T() - The sqre tegrble fcto f() c be epressed Chebyshev seres: : f () c T () f(.5.5)t () c d p f(.5.5)t () c d,,,..., p Ths techqe hs bee eployed to solve lrge vrety of ler d o-ler dfferetl eqtos. Ths ethod s sed for solvg secod d forth-order ellptc eqtos []. Ths ethod s lso dopted for solvg frctol order tegro-dfferetl eqtos [9]. Also, ths procedre s sed to obt the ercl solto of ODEs wth o-lytc solto [6]. Ordry dfferetl eqtos hve bee the focs of y stdes de to ther freqet pperce vros pplctos fld echcs, vscoelstcty, bology, physcs d egeerg [6, 8, 9]. Coseqetly, cosderble tteto hs bee gve to the soltos of hgh-order ODEs of physcl terest. Most dfferetl eqtos do ot hve ect soltos, so pproto d ercl techqes [-], st be sed. Recetly, severl ercl ethods to solve the dfferetl eqtos hve bee gve sch s vrtol terto ethod [, 8,, ], hootopy pertrbto ethod [8, ], Ado decoposto Correspodg Athor: M.M. Khder, Deprtet of Mthetcs, Fclty of Scece, Beh Uversty, Beh, Egypt 9

2 Stdes Noler Sc., (): 9-, ethod [-, 8,, -6] d collocto ethod [, ]. I ths pper, dfferetl eqtos wth ore cople olerty re cosdered. We cosder the ercl solto of hgh-order ODEs whch cotg the polyol fctos of ow d ts dervtves d c be wrtte the for: r () P,r () () () r (r) () Q,r () () () f() r sbject to the followg tl codtos: () s s d L, s,,,... s d (the hghest dervtve wth respect to ) d R re ler opertors,.e., t s possble to fd bers >, > sch tht L S, R. The oler ter N() s Lpschtz cotos wth: N() N( θ) r θ, J [,] for y costt r >. Apply the verse opertor L -S whch defed by: () () c,,,,..., () () () (), () ( )( ) ( ) s s -s L (.)... (.) (dt )...(dt s- )(dt s) (5) d () s ow fcto fro C [,]. Kow fctos P,r (), Q,r () d f() re defed o the closed tervl [,]. Ths s o-ler dfferetl eqto of order. The frst d the secod order probles re ofte ecotered the ltertre. Iportt o-ler eqtos of ths type re Rcct, Abel, Ede-Fowler, Dffg, V der Pol, Rylegh d Yerovs eqtos. These eqtos rse dfferet res of physcs d egeerg sceces. The orgzto of ths pper s s follows: I the et secto, bref revew of the Ado decoposto ethod s trodced. Secto srzes the covergece lyss of ADM. I secto, soe llstrtve eples re gve d solved sg the two ethods, Chebyshev spectrl ethod d ADM to clrfy the ethods. Also coclso s gve secto 5. Note tht we hve copted the ercl reslts sg Mthetc progrg. A BRIEF REVIEW OF THE ADOMIAN DECOMPOSITION METHOD Cosder the o-ler tl vle proble of ODEs the followg geerl for: wth the tl codtos: S L() R() N(()) g() () () () f,,,,,s- () for soe costts f, 9 to both sdes of () gves: s f () L s [R() N(()) g()]! the frst prt fro the rght hd sde of Eq. (6) s obted fro the solto of the hoogeos dfferetl eqto L S () sg the tl codtos. Ado decoposto ethod [, ] defes the solto () s fte seres the for: (6) () () (7) the copoets () c be obted recrsve for. The o-ler ter N() c be decoposed by fte seres of polyols gve by: N() A (8) the copoets A c be obted sg the followg forl: d A N( ),,,,... λ (9)! dλ λ Sbsttte by (7) d (8) to Eq.(6) gves:

3 Stdes Noler Sc., (): 9-, s f () L s [g()]! L s [R () A ] () Fro ths eqto, we c obt the copoets () of the solto () by the followg recrrece forl: s f () L s [g()]! () L s [R () A ], () The covergece of ADM s trodced y ppers, for eple [, 7]. CONVERGENCE ANALYSIS OF ADM I ths secto, the sffcet codtos re preseted to grtee the covergece of ADM, whe ppled to solve o-ler ODEs, the pot s tht we prove the covergece of the seres, whch s geerted by sg ADM. Theore : The o-ler proble () hs qe solto, wheever <α<,, s ( ) s! the costts d re defed bove. Proof: Deotg (C[J],. ) the Bch spce of ll cotos fctos o J wth the or defed by: f() f() J Defe ppg F:E E * s * * F F L [R( ) N() N( )] J * * [ R( ) N() N( ) ]...s fold... d J * * ( )...s fold... d J s ( ) * * J s! Uder the codto <α<, the ppg F s cotrcto, therefore, by the Bch fed-pot theore for cotrcto, there est qe solto to proble () d ths copletes the proof. Hosse d Nsbzdeh trodced sple ethod to detere the rte of covergece of Ado decoposto ethod []. I ths secto, we dpt t to see here. Theore : Let N be opertor fro Hlbert spce to tself d () be the ect solto of Eq.(), the, the pprote solto whch s obted by () coverges to () f: <γ< d stsfy the followg codto: γ,,,,... Proof: The proof of ths theore c be fod []. ILLUSTRATIVE EXAMPLES () I ths secto, we trodce three eples of o-ler ODEs. We fd the ercl solto of these eples sg the Chebyshev spectrl ethod d ADM d plot the crves of these soltos. Eple : Let s cosder V der Pol eqto [9]: () ().5( ()) '() (), < < () -S F() s() L[R() N(()) g()] s f σ ()! s the solto of the hoogeeos dfferetl eqto L S () sg the tl codtos. Now, let s sse tht, d * be two dfferet soltos to (), the, we c obt: wth the codtos (), ().5..I: Procedre solto sg Chebyshev spectrl ethod: We solve the o-ler ODEs of the for () wth the gve tl codtos by sg Chebyshev spectrl ethod. For ths prpose sce the Gss- Lobtto odes le the co-pttol tervl [-,], the frst step of ths ethod, the trsforto ( η ) s sed to chge Eq.() to the followg for: 9

4 Stdes Noler Sc., (): 9-, () ( η ).5 ( ( η) )'( η) ( η ), -<η< The trsfored tl codtos re gve by: (-), '(-).5 () ( η ) ( η), ( η ) () (5) ϕ.5 d d l ϕ j j l ϕ j j j j (8) ( d l ϕ ) j j j Ths schee s o-ler syste of lgebrc eqtos ows ϕ, whch the solved sg Newto's terto ethod. After solvg ths syste d sbsttte ϕ Eq.(7), we c obt the ercl solto of Eq.(). The covergece of ths ethod s gve []. d (η) s the ow fcto fro C [-,]. Where the dfferetto Eqs.()-(5) wll be wth respect to the ew vrble η. Or techqe s ccoplshed by strtg wth Chebyshev pproto for the hghest order dervtve, () (η) d geertg pprotos to the lower order dervtves (),,, s follows: Settg, () (η) ϕ(η), the by tegrto we obt: η () ( η ) ϕη ( )dη c η η ( η ) ϕη ( ) dηηη d ( ) c c (6).II: Procedre solto sg ADM: I order to obt the ercl soltos of Eq.() sg ADM, we follow the followg steps: : Eq.() c be rewrtte the opertor for: L() R() N(()) (9) d L, R (), N().5( ()) '() d : Apply the verse opertor L - whch defed by (5) to both sdes of Eq.(9) gves: Fro the tl codtos (5), we c obt the costts of tegrto c,,, c, c Therefore, we c gve pprotos to Eq.() s follows: () () ()-L - [()] L - [N()]! Sbstttg by Eqs.(7) d (8) Eq.() gves: () () d, d l ϕ l ϕ (7) j j j j j j for ll,,,,, l b, l b, d ( η )c c, d c j j j j b ( η η )b j j j () () ()-L - ()! L - A () sbstttg by tl codtos, the, the copoets () of the solto () c be wrtte s: d b j re the eleets of the tr B s gve Ref. []. By sg Eq. (7), oe c trsfor Eq. () to the followg syste of o-ler eqtos the hghest dervtve: ().5, () -L-[ ()] L - [A ], A c be obted by the forl (9). 9 ()

5 Stdes Noler Sc., (): 9-, Tble : The Chebyshev solto, Ch d the solto sg ADM, ADM X ADM Ch Tble : The covergece behvor of the trcted soltos sg ADM Method ADM : The copoets () of the solto () sg the terto forl () re gve s follows: ().5 () () So, the solto () c be pproted s: () ψ () () () The trcted solto ψ () s gve by ψ () () () () sbject to the followg tl vles (), '() -, ''() The ect solto of ths proble s gve by () e -..I: Procedre solto sg Chebyshev spectrl ethod: We solve the o-ler ODEs of the for () wth tl codtos by sg Chebyshev spectrl ethod. For ths prpose sce the Gss-Lobtto odes le the copttol tervl [-,], the frst step of ths ethod, the trsforto ( η ) s sed to chge Eq.() to the followg for: () η ( η ) 6 ( η ) e ( η) -η 5 e, - <η< The trsfored tl codtos re gve by: (5) () () (-), (-) -, (-) (6) () (η) (η), (η) d (η) s the ow fcto fro C [-,]. Where the dfferetto Eqs.(5)-(6) wll be wth respect to the ew vrble η. Or techqe s ccoplshed by strtg wth Chebyshev pproto for the hghest order dervtve, () d geertg pprotos to the lower order dervtves (),,,, s follows: Settg () (η) ϕ(η), the by tegrto we obt () (η), () (η) d (η) s follows: The behvor of the ercl soltos sg Chebyshev spectrl ethod, Ch, wth, copred wth the pprote solto sg ADM, ADM, wth three copoets ( ) re preseted Tble. The covergece lyss of the pprote solto sg ADM s gve Tble, ters of Theore sch tht the L -or whch s defed s () () d Eple : Cosder the o-ler tl vle proble: () () 6 () e () 5 e -, < < () 95 () η ( η ) ϕη ( )dη c () η η ( η ) ϕη ( ) dηηη d ( ) c c η η η ( η ) ( η ) ( η ) ϕη ( ) dηdηdη c c c!! (7) Fro the tl codtos (6), we c obt the costts of tegrto,,, c, c -,, c

6 Stdes Noler Sc., (): 9-, Therefore, we c gve pprotos to Eq. () s follows: for ll,,,,, () () d, d, d l ϕ l ϕ l ϕ (8) j j j j j j j j j ( η ) l b, b, b,d c, d ( )c c, d j j l j j l j j η c! - ( η η ) j b ( η η )b, b b j j j j j! d b j re the eleets of the tr B s gve Ref. []. By sg Eq.(8), oe c trsfor Eq.(5) to the followg syste of o-ler eqtos the hghest dervtve: η η ϕ 6 d e d 5 e l j j ϕ j l j j ϕ j (9) Ths schee s o-ler syste of lgebrc eqtos ows ϕ, whch the solved sg Newto's terto ethod. After solvg ths syste d sbsttte ϕ Eq.(8), we c obt the ercl solto of Eq.()..II: Procedre solto sg ADM: I order to obt the ercl soltos for Eq.() sg ADM, we follow the followg steps: : Eq.() c be rewrtte the opertor for: L() R() N(()) g() () d L,R 6(), N() e (), g() 5 e - d Frst, to overcoe the coplcted ectto fro the fctos e - d e, whch c cse dffclt tegrtos d prolferto of ters, we se the Tylor epso of the fctos t, the followg for: e -.5, e : Apply the verse opertor L - whch defed by (5) to both sdes of Eq.() gves: () () () L - [5(-.5 )] L - [R()] L - [N()]! () Sbstttg by Eqs.(7) d (8) Eq.() gves: 96

7 Stdes Noler Sc., (): 9-, () () () L - [5(-.5 )]-6L - () L - A! () sbstttg by tl codtos, the, the copoets () of the solto () c be wrtte s: ().5 - L - [5(-.5 )], () -6L - () A, () A c be obted by the forl (9). : The copoets () of the solto () sg the terto forl () re gve s follows: () () () Tble : The ect solto, the Chebyshev solto, Ch d the solto sg ADM, ADM ADM Ch e Tble : The covergece behvor of the trcted soltos sg ADM Method ADM So, the solto () c be pproted s: () ψ () () () The trcted solto ψ () s gve by ψ () () () () The behvor of the ercl soltos sg Chebyshev spectrl ethod, Ch, wth, copred wth the pprote solto sg ADM, ADM, wth three copoets ( ) re preseted Tble. The covergece lyss of the pprote solto sg ADM s gve Tble, ters of Theore sch tht the L -or whch s defed s: 97 () () d Eple : Cosder the o-ler tl vle proble: (5) () e - (), < < (5) sbject to the followg tl vles () (),,,,, d the ect solto of ths proble s gve by () e X..I: Procedre solto sg Chebyshev spectrl ethod: We solve the o-ler ODEs of the for (5) wth tl codtos by sg Chebyshev spectrl ethod. For ths prpose sce the Gss-Lobtto odes le the copttol tervl [-,], the frst step of ths ethod, the trsforto ( η ) s sed to chge Eq.(5) to the followg for: 5 (5) ( η ) e -η ( η ), -<η< The trsfored tl codtos re gve by: () (-),,,,, () ( η ) ( η), ( η ) (6) (7)

8 Stdes Noler Sc., (): 9-, d (η) s ow fcto fro C [-,]. Where the dfferetto Eqs.(6)-(7) wll be wth respect to the ew vrble η. Or techqe s ccoplshed by strtg wth Chebyshev pproto for the hghest order dervtve, (5) d geertg pprotos to the lower order dervtves (),,,,, s follows: Settg (5) (η) ϕ(η), the by tegrto we obt () (η),,,,, s follows: () η () η ϕ ()d η η c () η η () η ϕη ( ) dηη d ( η ) c c () η η η ( η ) () η ϕη ( ) dηdηdη c! - () η η ηη ( η ) () η ϕη () dηdηdηdη c! - η η η η η ( η ) ( η) ϕη ( ) dηηηηη d d d d c-! (8) Fro the tl codos (7), we c obt the costts of tegrto c,,,,, Therefore, we c gve pprotos to Eq.(6) s follows: - c () () d, d, d l ϕ l ϕ l ϕ, j j j j j j j j j () () l ϕ d, l ϕ d j j j j j j for ll,,,,, ( η ) 5 l b, l b, l b, l b, l b,d c, j j j j j j j j j j! - ( η ) ( η ) d c, d c d ( η )c c, d c! -! - ( η η ) ( η η ) ( η η ) j j 5 j b ( η η )b, b b,b b,b b j j j j j j j j j!!! (9) d b j re the eleets of the tr B s gve Ref. []. By sg Eq.(9), oe c trsfor Eq.(6) to the followg syste of o-ler eqtos the hghest dervtve: 5 η e d ϕ l j ϕ j () j 98

9 Stdes Noler Sc., (): 9-, Tble 5: The ect solto, the Chebyshev solto, Ch d the solto sg ADM, ADM ADM Ch e Tble 6: The covergece behvor of the trcted soltos sg ADM Method ADM Ths schee s o-ler syste of lgebrc eqtos ows ϕ, whch the solved sg Newto's terto ethod. After solvg ths syste d sbsttte ϕ Eq.(9), we c obt the ercl solto of Eq.(5)..II: Procedre solto sg ADM: I order to obt the ercl soltos for Eq. (5) sg ADM, we follow the followg steps: : Eq.(5) c be rewrtte the opertor for: 5 L() N(()) () 5 5 d L, N() e - () 5 d () () () L -5 A! () sbstttg by tl codtos, the, the copoets () of the solto () c be wrtte s: ()!!! () L -5 A, A c be obted by the forl (9). () : The copoets () of the solto () sg the terto forl () re gve s follows: ()!!! () () So, the solto () c be pproted s: () ψ () () (5) Frst, to overcoe the coplcted ectto fro the fctos e -, whch c cse dffclt tegrtos d prolferto of ters, we se the Tylor epso of the fctos t, the followg for e -X -.5. : Apply the verse opertor L -5 whch defed by (5) to both sdes of Eq. () gves: () () () L -5 [N()]! Sbstttg by Eqs.(7) d (8) Eq.() gves: () 99 The trcted solto ψ () s gve by ψ () () () () The behvor of the ercl soltos sg Chebyshev spectrl ethod, Ch, wth, copred wth the pprote solto sg ADM, ADM, wth three copoets ( ) re preseted Tble 5. The covergece lyss of the pprote solto sg ADM s gve Tble 6, ters of Theore sch tht the L -or whch s defed s () () d

10 CONCLUSION AND REMARKS I ths pper, the Chebyshev spectrl ethod d the Ado decoposto ethod re pleeted to obt the ercl soltos of the hgh-order o-ler ODEs d the ercl soltos wth respect to these ethods re copred. It s geerlly very dffclt to fd the lytcl soltos of hgherorder o-ler ODEs wth vrble coeffcets. So, we terest ths rtcle wth sg the two proposed ethods to solve erclly sch these eqtos. Sce, we ow tht the Chebyshev polyol pproto ethod s vld the tervl [-,], so, we sed the trsforto ( η ) to chge the tervl [,]. The Chebyshev spectrl ethod redces the cosdered o-ler dfferetl eqto to o-ler syste of lgebrc eqtos, whch solved sg the well ow ethod, ely, Newto terto ethod. Also, by sg ADM the soltos y te the closed for of the ect solto. I geerl sce the ADM solves the probles o few steps lter of terto stsfyg the desred precso, t does ot eed ore clclto order to solve the dfferetl eqto. Specl tteto s gve to stdy the covergece of ADM d stsfy ths theoretcl stdy vew the trodced ercl eples. I the ed, fro or ercl reslts sg two proposed ethods we c coclde tht, the soltos re ecellet greeet wth the ect solto ost cses. Also, the obted reslts deostrte relblty d effcecy of the proposed ethods. REFERENCES. Abssy, T.A.,. Iproved Ado decoposto ethod. Copters d Mthetcs wth Applctos, 59 (): -5.. Abbsbdy, S. d M.T. Drvsh, 5. A ercl solto of Brger's eqto by odfed Ado ethod. Appled Mthetcs d Coptto, 6: Ado, G., 989. Noler Stochstc Systes d Applctos to Physcs. Klwer Acdec Pblshers, Dordrecht.. Ado, G. d R. Rce, 99. Nose ters decoposto solto seres. Copt. Mth. Appl., : Ayüz, A. d H. Ysl,. The solto of hgh-order oler ordry dfferetl eqtos by Chebyshev seres. Appled Mthetcs d Coptto, 7: Stdes Noler Sc., (): 9-, 6. Bbol, E. d M.M. Hosse,. A odfed spectrl ethod for ercl solto of ODEs wth o-lytc solto. Appl. Mth. Copt., : Bell, W.W., 967. Specl Fctos for Scetsts d Egeers. New Yor Toroto Melbore. 8. Bhttchryy, R.K. d R.K. Ber,. Applcto of Ado ethod o the solto of the elstc wve propgto elstc brs of fte legth wth rdoly d lerly vryg Yog's odls. Appled Mthetcs Letters, 7: Dog, C.S., O. Ar d A. Kj, 99. Pretrc ectto of copttol ode of the lepfrog schee ppled to the V der Pol eqto. J. Copt. Phys., 7: El-Ged, S.E., 969. Chebyshev solto of dfferetl, tegrl d tegro-dfferetl eqtos. Copter J. : He, J.H., 999. Vrtol terto ethod- d of o-ler lytcl techqe: Soe eples. Itertol Jorl of No-Ler Mechcs, : Hosse, M.M. d H. Nsbzdeh, 6. O the covergece of Ado decoposto ethod. Appl. Mth. Copt., 8: Khlf, A.K., E.M.E. Elbrbry d M.A. Abd- Elrze,. Chebyshev epso ethod for solvg secod d forth-order ellptc eqtos. Appled Mthetcs d Coptto, 5: Khder, M.M.,. O the ercl soltos for the frctol dffso eqto. Coctos Noler Scece d Nercl Slto, 6: Khder, M.M. d R.F. Al-Br,. Approte ethod for stdyg the wves propgtg log the terfce betwee r-wter. Mthetcl Probles Egeerg, pp: Khder, M.M.,. Nercl solto of oler lt-order frctol dfferetl eqtos by pleetto of the opertol tr of frctol dervtve. Stdes Noler Sceces (): Lesc, D.,. Covergece of Ado's decoposto ethod: Perodc Tepertres. Copters d Mthetcs wth Applctos, : Swel, N.H., M.M. Khder d R.F. Al-Br, 7. Nercl stdes for lt-order frctol dfferetl eqto. Physcs Letters A, 7: 6-.

11 Stdes Noler Sc., (): 9-, 9. Swel, N.H. d M.M. Khder,. A Chebyshev psedo-spectrl ethod for solvg frctol order tegro-dfferetl eqtos. ANZIM, 5: Swel, N.H., M.M. Khder d R.F. Al-Br, 8. Noler focsg Mov systes by vrtol terto ethod d Ado decoposto ethod. Jorl of Physcs: Coferece Seres, 96: -7.. Swel, N.H. d M.M. Khder, 9. Ect soltos of soe copled oler prtl dfferetl eqtos sg the hootopy pertrbto ethod. Copters d Mthetcs wth Applctos, 58: -.. Swel, N.H. d M.M. Khder,. Approte soltos to the oler vbrtos of ltwlled crbo otbes sg Ado decoposto ethod. Appled Mthetcs Coptto, 7: Swel, N.H., M.M. Khder d F.T. Mohed,. O the ercl soltos of two desol Mwell s eqtos. Stdes Noler Sceces, (): Wzwz, A.M.,. A ew techqe for clcltg Ado's polyols for oler polyols. Appl. Mth. Copt., : Wzwz, A.M.,. The ercl solto of ffth-order bodry vle probles by the decoposto ethod. J. Copt. Appl. Mth., 6: Whth, G.B., 97. Ler d Noler Wves, New Yor, Wley.