On the Effective Region of Convergence of the Decomposition Series Solution

Transcription

1 Joral of Algorihms & Compaioal Techology Vol. 7 No. 7 O he Effecive Regio of Covergece of he Decomposiio Series Solio J-Sheg Da a,b *, Radolph Rach c,, Zhog Wag a a School of Mahemaics ad Iformaio Scieces, Zhaoqig Uiversiy, Zhaoqig, Gag Dog 566, P.R. Chia b College of Sciece, Shaghai Isie of Techology, Shaghai 48, P.R. Chia c 6 Soh Maple Sree, Harford, Michiga , U.S.A. Received 9/6/; Acceped 8// ABSTRACT I his paper we ivesigae he domai of covergece of he Adomia series solio based o he compaioal resls for several examples. We demosrae how he domai of covergece ca be exeded by irodcig a parameer c i he defiiio of he zeroh-order ad firsorder solio compoes ad. Frhermore we geeralize he cocep of he covergece parameer c from a wo-erm pariio of he iiial codiio o a mliple-erm pariio wih he desig of expadig he domai of covergece of he Adomia series solios for oliear differeial eqaios. Keywords: Adomia decomposiio mehod; Adomia polyomials; Noliear differeial eqaio; Solio coiaio. INTRODUCTION We develop a ew framework for adjsig or expadig he effecive regio of covergece of he Adomia decomposiio solio for oliear ordiary differeial eqaios. This ew approach is based o he oio of covergece parameer [] ad adapig he Wazwaz modified recrsio scheme [, ], b isead of he prpose for sppressig he occasioal pheomeo of a rapidly damped oscillaig covergece for he Adomia decomposiio series solio as origially proposed by Wazwaz [], he paramerized modified recrsio scheme is irodced for he prpose of expadig he effecive regio of covergece for he Adomia decomposiio series as a impora goal i is ow righ. *Correspodig ahor. dajssd@sia.com, apsrike@rio.e

2 8 O he Effecive Regio of Covergece of he Decomposiio Series Solio Firs we describe he procedre of he Adomia decomposiio mehod (ADM) for solvig oliear differeial eqaios [4 ], ad he compare he Adomia recrsio scheme wih he Wazwaz modified recrsio scheme. We shall se he firs-order oliear ordiary differeial eqaio as he vehicle o illsrae or ew approach o how we ca adjs or expad he regio of covergece for he Adomia decomposiio series solio, d d () + ()() + f (, ()) = g (); ( ) = C, α () where he fcios α, g ad f are aalyical. Firs cosider he ADM. We rewrie Eq. () i he sal operaor-heoreic oaio of Adomia: L + R + N = g, d where L = (), he L = () d, R = α ()(), N = f(, ()). d we rewrie Eq. () as L = g R N, () Nex ad apply he iegral operaor L, L L = L g L R L N, where L L = Φ ad LΦ =. I his pariclar case of a firs-order ordiary differeial eqaio, we have Φ =( ) = C. Therefore = + L g L Φ R L N. () For coveiece, we defie he γ fcio []: γ =Φ+L g, he = L γ R L N. (4) I he ADM, he solio () is represeed by a decomposiio series () = () = (5)

3 Joral of Algorihms & Compaioal Techology Vol. 7 No. 9 ad he olieariy comprises he Adomia polyomials N = A (). = (6) We remark ha he covergece of he Adomia series has already bee prove by several ivesigaors [ 6]. For example, Abdelrazec ad Peliovsky [6] have pblished a rigoros proof of covergece for he ADM der he aegis of he Cachy-Kovalevskaya heorem. I poi of fac he Adomia decomposiio series is fod o be a compaioally advaageos rearrageme of he Baach-space aalog of he Taylor expasio series abo he iiial solio compoe fcio. Frhermore covergece of he ADM is o limied o cases whe oly he fixed-poi heorem applies, which is far oo resricive for mos physical applicaios. The defiiioal formla for he Adomia polyomials was firs pblished i 98 [7]: A () =! λ f (, (; λ)), λ= (7) where (; λ) = λ (), = f (, (; λ)) = λ A = (),! () (; λ)). λ = λ= (8) The firs six Adomia polyomials for N = f(, ()) are A() = f(, ()), A f () = () (, ()), A f () f () = () (, ()) + (,! ( )), A() = () f (, ()) + () () (, ()) f () + (, (! f )),

4 O he Effecive Regio of Covergece of he Decomposiio Series Solio = + + A f () (, ()) () 4 4 () () f (, ())! + () + f () (, ()) 4 4 f (, ()),! 4! 4 A = + ( + ) f 5() 5 (, ()) () () () 4() f (, ()) + () () + () ()!! f (, ()) () f () () (, ()) 5 5!! f (, ()) Differe algorihms for he Adomia polyomials have bee developed by Rach [, 8], Adomia ad Rach [9], Wazwaz [], Abdelwahid [] ad several ohers [ 5]. Recely ew algorihms ad sbroies i MATHEMATICA for fas geeraio of he Adomia polyomials o high orders have bee developed by Da [, 6, 7]. From Eqs. (4) (6) he solio compoes are deermied by he Adomia recrsio scheme: () = γ (), = L R () L A (),. + (9) () Nex we cosider he Wazwaz modified recrsio scheme i he ADM. I he Wazwaz modified decomposiio, he solio () is also represeed by he Adomia decomposiio series () = () =, b where he solio compoes are isead deermied by he modified recrsio scheme of Wazwaz []: () = γ () L R () L A (), () = γ (), () () () = L R () L A (),, ()

5 Joral of Algorihms & Compaioal Techology Vol. 7 No. where Wazwaz has offered a sefl pariio γ () = γ () + γ () for cases sppressig he occasioal pheomeo of damped oscillaig or oisy covergece for ihomogeeos eqaios [8, 9]. Ths we have deermied he solio i he form of a differe decomposiio series by sig a differe recrsio scheme. Of corse he sm of he wo disic decomposiio series, which differ oly i he formlas of heir respecive solio compoes, boh eqal he exac solio * () as desiged. Usally Wazwaz s pariio [] γ () = γ () + γ () is aimed a L g for he o-homogeeos eqaio, whe a oisy covergece is aicipaed. I he Wazwaz-El-Sayed modified recrsio scheme [], he Taylor expasio of γ () abo is sed o faciliae he calclaio of iegrals. I he ex secio, we irodce he paramerized recrsio scheme. I Secio we ivesigae he choice of he parameer c i he paramerized recrsio scheme ad is impac o he domai of covergece for hree oliear examples. Secio 4 smmarizes or fidigs.. PARAMETRIZED RECURSION SCHEME We geeralize he cocep of covergece parameer proposed i []. We irodce a parameer c ad pariio he iiial vale C as wo pars: C = ( C c) + c. (4) The we disribe he compoes C c ad c of C i ad, respecively, hs obaiig he paramerized modified recrsio scheme: () = γ () c, (5) () = c L R () L A (), (6) () = L R () L A (), (7) The covergece parameer c is desiged o be varied so as o expad he effecive regio of covergece of he Adomia decomposiio series. We seek o deermie a appropriae choice ad preferably a opimal choice of he covergece parameer c.

6 O he Effecive Regio of Covergece of he Decomposiio Series Solio For complicaed olieariies, he choice i () () γ() = γ () + γ (), γ () = C, will grealy simplify operaios, i.e. leadig o simple-o-iegrae series. I his case, correspodig o he wo-erm pariio (4) of he iiial vale C, we obai he paramerized modified recrsio scheme () = c+ γ () L R () L A (), () = C c, () = L R () L A (), (8) (9) () We also cosider differe mliple-erm pariios of he iiial vale C, coaiig a embedded parameer c, i order o expad he effecive regio of covergece eve frher, whe i is eeded. For example, correspodig o a paramerized hree-erm pariio C = c + c+ c, () where c = C c, ad where, for example, c = c/4 ad c = c /4, he paramerized modified recrsio scheme he becomes () = γ () c, () () = c L R () L A (), () () = c L R () L A(), (4) () = L R () L A (),, (5) while correspodig o a paramerized ifiie mliple-erm pariio C c = =, (6)

7 Joral of Algorihms & Compaioal Techology Vol. 7 No. where c = C c ad where, for example, we sppose a geomeric seqece sch as c = c/ for, he paramerized modified recrsio scheme he becomes () = γ () c, (7) () = c L R () L A (),. + + (8) I effec we have disribed he correcio over he seqece of solio compoes.. THE EFFECTIVE REGION OF CONVERGENCE I he followig examples he paramerized recrsio scheme wih a embedded parameer c is sed. I he case of c = he sal recrsio scheme is obaied. We firs solve a specific Riccai eqaio, he a oliear differeial eqaio wih a egaive power o-lieariy ad he Lae-Emde eqaio. Example. Cosider he Riccai eqaio d d + = ; ( ) =. (9) The exac solio is * () =, >. + () Applyig he iegral operaor L = ( )d o boh sides of Eq. (9) yields Nex we decompose he solio = = ad he olieariy f ( ) = = A, where he Adomia polyomials A are = = L. A =, A =, A = +,, A = k k. k= ()

8 4 O he Effecive Regio of Covergece of he Decomposiio Series Solio Applyig he paramerized recrsio scheme (5) (7) = c, = c L A, = L A,, () we calclae he paramerized solio compoes = c ( c), = c( c) + ( c), 4 = c + c( c) ( c), = c ( c) 4c( c) + ( c),. The -erm paramerized approximaio is example, φ = k = (; c) (; c). k For φ 5 (; c) ( c)( c c ) ( c)( c) 5 4 = ( c). I he case of c =, he solio compoes are comped o be = ( ) ad he decomposiio solio is (; ) = ( ), = () which has a domai of covergece of < <. I order o see he effec of varyig he vale of c o he domai of covergece, i Fig. we plo he crves of he exac solio * () ad he -erm modified Adomia approximaios φ (; c) for c =,.,.,.4, ad i Fig. he crves of he exac solio * () ad he 6-erm modified Adomia approximaios φ 6 (; c) for c =,.,.,.4. We see ha for he hree vales of c,.,. ad.4, he modified decomposiio solios have a larger domai of covergece ha for c =. We fid ha as c < he domai of covergece arrows. For or paramerized approximaio φ 6 (; c), he opimal vale of c is ear.. Frher we have checked he maxima M of (;.) o he ierval.9 ad have fod ha M <.9 for all =,,..., 5. Nex, for he hree-erm pariio of he iiial vale c c = c + c+ c, c = c, c = ad c =, 4 4 (4)

9 Joral of Algorihms & Compaioal Techology Vol. 7 No Figre. The exac solio * () (solid lie) ad he -erm modified Adomia approximaios φ (; ) (do lie), φ (;.) (dashed lie), φ (;.) (do-side lie) ad φ (;.4) (do-do-side lie) Figre. The exac solio * () (solid lie) ad he 6-erm modified Adomia approximaios φ 6 (; ) (do lie), φ 6 (;.) (dashed lie), φ 6 (;.) (do-side lie) ad φ 6 (;.4) (do-do-side lie). we cosider he ew paramerized recrsio scheme c c = c, = L A, = L A, = L A. 4 4 (5) We fid ha he resls ca ow be improved by icreasig he parameer c frher. I Fig. we plo he exac solio * () ad he -erm ew modified

10 6 O he Effecive Regio of Covergece of he Decomposiio Series Solio Figre. The exac solio * () (solid lie) ad he -erm ew modified Adomia approximaios φ (; ) (do lie), φ (;.4) (dashed lie), φ (;.5) (do-side lie) ad φ (;.6) (do-do-side lie). Adomia approximaios φ (; c) for c =,.4,.5,.6. For c =.5 he size of he effecive regio of φ (;.5) is abo 4.8. Fially, we apply he mliple-erm pariio of he iiial vale = c wih c = ( c) c, =,,,, = (6) where c <, ad he from he paramerized recrsio scheme = c, = c L A, (7) we obai he paramerized solio compoes = ( ) ( c)[( c) c], =,,,. (8) I his case he decomposiio solio has a easily calclaed ierval of covergece < < + c As c, he he domai of covergece c. approaches (, +). I his pariclar example, we have expaded he domai of covergece of he decomposiio series o icorporae he eire domai of defiiio for his solio.

11 Joral of Algorihms & Compaioal Techology Vol. 7 No. 7 Example. Cosider he oliear differeial eqaio wih a egaive power olieariy d d = ; ( ) =. (9) The exac solio is * () = +. (4) By he ADM we rewrie Eq. (9) as = + L where L = ( )d. The Adomia polyomials for he olieariy f() = / are, A =, A =, A =, A = +, A4 = ,. By he paramerized recrsio scheme (5) (7), he solio compoes are deermied by = c, = c+ L A, = L A,. (4)

12 8 O he Effecive Regio of Covergece of he Decomposiio Series Solio The we calclae he paramerized solio compoes i sccessio = c+ ( c), c = ( c) 8 ( c), c c = ( c) 8 ( c) 6 ( c),. The -erm modified Adomia approximaio paramerized by c is φ (; c) = k k (; c), = where φ (; ) is he sal -erm Adomia approximaio, which is eqivale wih he -erm Taylor approximaio for his pariclar example. For example, = φ 5 (; ) φ (; ) has a domai of covergece as. I Fig. 4 we plo he exac solio * () ad he 8-erm modified Adomia approximaios φ 8 (; c) for c =,,,, ad i Fig. 5 he exac solio Figre 4. The exac solio * () (solid lie) ad he 8-erm modified Adomia approximaios φ 8 (; ) (do lie), φ 8 (; ) (dashed lie), φ 8 (; ) (do-side lie) ad φ 8 (; ) (do-do-side lie).

13 Joral of Algorihms & Compaioal Techology Vol. 7 No Figre 5. The exac solio * () (solid lie) ad he 6-erm modified Adomia approximaios φ 6 (; ) (do lie), φ 6 (; ) (dashed lie), φ 6 (; ) (do-side lie) ad φ 6 (; ) (do-do-side lie). * () ad he 6-erm modified Adomia approximaios φ 6 (; c) for c =,,,. We have also checked ha for vales < c <, a smaller domai of covergece is obaied ha whe c =. I Fig. 6 he exac solio * () Figre 6. The exac solio * () (solid lie) ad he 6-erm modified Adomia approximaios φ 6 (; ) (do lie), φ 6 (; ) (dashed lie) ad φ 6 (; 5) (do-side lie).

14 4 O he Effecive Regio of Covergece of he Decomposiio Series Solio ad he 6-erm modified Adomia approximaios φ 6 (; c) for c =,, 5 are ploed, where he size of he effecive regio of covergece srpasses for he case of c = 5. Example. Cosider he Lae-Emde eqaio [, ] d d = ; ( ) =, ( ) =. d d (4) The exac solio is Le he Eq. (4) becomes * () = +. d L d d = (), d L = 5. / (4) (44) Applyig he iverse operaor leads o L = () dd, = L 5. (45) (46) The Adomia polyomials for he qiic olieariy f() = 5 are A = 5, A = 5 4, A = + 5 4, A = , A4 = , 4 4. = = The compoes of he decomposiio solio are deermied by he modified recrsio scheme wih he covergece parameer c, = c, = c L A, = L A,. (47)

15 Joral of Algorihms & Compaioal Techology Vol. 7 No. 4 Calclaig he paramerized solio compoes yields 5 = c ( c 6 ), c c c 6 = ( ) + ( 4 ), = c ( c) c( c) ( c) ,. The -erm modified Adomia approximaio is I pariclar, whe c = we have, for example φ = k = (; c) (; c). k 5 = φ 5 (, ) k φ (; ) = k (; ) has a domai of covergece as. = I Fig. 7 we plo he exac solio * () ad he 5-erm modified Adomia approximaios φ 5 (; c) for c =,.,., ad i Fig. 8 he exac solio * () ad he -erm modified Adomia approximaios φ (; c) for c =,., Figre 7. The exac solio * () (solid lie) ad he 5-erm modified Adomia approximaios φ 5 (; ) (do lie), φ 5 (;.) (dashed lie) ad φ 5 (;.) (do-side lie).

16 4 O he Effecive Regio of Covergece of he Decomposiio Series Solio Figre 8. The exac solio * () (solid lie) ad he -erm modified Adomia approximaios φ (; ) (do lie), φ (;.) (dashed lie) ad φ (;.) (do-side lie). For larger c, sch as c =.4, φ (;.4) appears o oscillae ad does o frher elarge he effecive regio of covergece. We observe ha for egaive vales of c, he domai of covergece arrows whe compared wih c =. Nex we cosider he hree-erm pariio of he iiial vale c c C = ( C c) + +, 4 4 ad he ew paramerized recrsio scheme becomes c c = c, = L A, = L A, = L A,. 4 4 (48) We obai c 5 = + ( + c), 4 6 c = ( + c) c ( + c), = ( + c) c( + 7c) + ( + c) c ( + c) 6, I Fig. 9 we plo he exac solio * () ad he -erm ew modified Adomia approximaios φ (; c) for c =,.,.4. We fid ha he crve of φ (;.) is improved sigificaly compared wih ha of φ (;.).

17 Joral of Algorihms & Compaioal Techology Vol. 7 No Figre 9. The exac solio * () (solid lie) ad he -erm modified Adomia approximaios φ (; ) (do lie), φ (;.) (dashed lie) ad φ (;.4) (do-side lie). If we pariio he iiial vale as c C = c, c = C c, c =, =,,, = (49) ad apply he correspodig recrsio scheme c = c, = L A,, (5) we obai he paramerized solio compoes c 5 = + ( + c), 6 c = ( + c) c ( + c), 4 4 c 5 = + ( + c) c( + c) + ( + c) c + ( + c) 6, We deoe by φ (; c) he -erm approximaio from his scheme. I Fig. we plo he exac solio * () ad he -erm modified Adomia approximaios φ (; c) for c =,.,.4. I his case for c =.4, we observe ha he effecive regio of covergece is grealy improved.

18 44 O he Effecive Regio of Covergece of he Decomposiio Series Solio Figre. The exac solio * () (solid lie) ad he -erm modified Adomia approximaios φ (; ) (do lie), φ (;.) (dashed lie) ad φ (;.4) (doside lie). Fially, we meio ha solio coiaio ad covergece acceleraio echiqes, sch as aalyic coiaio [, ], he diagoal Padé approximas [7,, 4 8], Eler s rasformaio [6], he ieraed Shaks rasform [7], Adomia s asympoic decomposiio mehod [5, 7, 9, 4], meric mehods based o he ADM ad he Rach-Adomia-Meyers modified decomposiio mehod (MDM) [,, 4, 4], ad so o, ca also be sed o accelerae he rae of covergece ad expad he effecive regio of covergece. 4. CONCLUSION We have irodced he paramerized recrsio scheme i he ADM, which preses a ovel approach o embed a parameer i he Adomia solio series so as o modify he solio s covergece properies. We have ivesigaed hree examples of oliear differeial eqaios o demosrae how o pracically expad he regio of covergece for he Adomia decomposiio series solio of oliear ordiary differeial eqaios. Esseially he domai of covergece is exeded by adjsig he solio compoes ad wih he covergece parameer c i a modified recrsio scheme sig a wo-erm pariio of he iiial vale, which i effec modifies he reslig formlas for he followig solio compoes,,. Ths we have chaged he solio compoes of he Adomia decomposiio series from he paramerized solio compoes c (; ) = () = () c (; ). We remark = ha he sm of he Adomia series solio is of corse iqe = by he

19 Joral of Algorihms & Compaioal Techology Vol. 7 No. 45 Cachy-Kovalevskaya heorem, however he decomposiio of he solio is oiqe ad his allows s he freedom o desig ew, alerae algorihms for compaioal advaage, icldig expadig he regio of covergece. Ths we have demosraed ha we are able o expad or corac he effecive domai of covergece of he approximaios of he Adomia decomposiio series by varyig he covergece parameer c as well as o sppress or idce he pheomeo of oscillaory or oisy covergece. This ew approach is simple i formlaio ad easy o impleme i symbolic sofware sch as MATHEMATICA, ec. I he calclaio of or solios, he fas algorihms for geeraio of he Adomia polyomials by Da [, 6, 7] garaee he efficiecy of or programmig. ACKNOWLEDGEMENTS This work was sppored i par by he Naioal Naral Sciece Fodaio of Chia (Nos. 795, 8). REFERENCES [] J. S. Da. Recrrece riagle for Adomia polyomials. Appl. Mah. Comp., 6:5 4,. [] A. M. Wazwaz. A reliable modificaio of Adomia decomposiio mehod. Appl. Mah. Comp., :77 86, 999. [] A. M. Wazwaz ad S. M. El-Sayed. A ew modificaio of he Adomia decomposiio mehod for liear ad oliear operaors. Appl. Mah. Comp., :9 45,. [4] G. Adomia. Sochasic Sysems. Academic, New York, 98. [5] G. Adomia. Noliear Sochasic Operaor Eqaios. Academic, Orlado, FL, 986. [6] G. Adomia. Noliear Sochasic Sysems Theory ad Applicaios o Physics. Klwer Academic, Dordrech, 989. [7] G. Adomia. Solvig Froier Problems of Physics: The Decomposiio Mehod. Klwer Academic, Dordrech, 994. [8] G. Adomia, R. Rach, ad R. Meyers. A efficie mehodology for he physical scieces. Kyberees, :4 4, 99. [9] K. Sco ad Y. P. S. Approximae aalyical solios for models of hreedimesioal elecrodes by Adomia s decomposiio mehod. I C. G. Vayeas, R. E. Whie, ad M. E. Gamboa-Adelco, ediors, Moder Aspecs of Elecrochemisry, volme 4, pages 4, New York, 7. Spriger. [] A. M. Wazwaz. Parial Differeial Eqaios ad Soliary Waves Theory. Higher Edcaio Press, Beijig, ad Spriger-Verlag, Berli, 9.

20 46 O he Effecive Regio of Covergece of he Decomposiio Series Solio [] A. M. Wazwaz. Liear ad Noliear Iegral Eqaios: Mehods ad Applicaios. Higher Edcaio Press, Beijig, ad Spriger-Verlag, Berli,. [] S. E. Serrao. Egieerig Uceraiy ad Risk Aalysis: A Balaced Approach o Probabiliy, Saisics, Sochasic Modelig, ad Sochasic Differeial Eqaios, Secod Revised Ediio. HydroSciece, Ambler, PA,. [] R. Rach. A ew defiiio of he Adomia polyomials. Kyberees, 7:9 955, 8. [4] K. Abbaoi ad Y. Cherral. Covergece of Adomia s mehod applied o differeial eqaios. Comp. Mah. Appl., 8: 9, 994. [5] K. Abbaoi ad Y. Cherral. New ideas for provig covergece of decomposiio mehods. Comp. Mah. Appl, 9: 8, 995. [6] A. Abdelrazec ad D. Peliovsky. Covergece of he Adomia decomposiio mehod for iiial-vale problems. Nmer. Mehods Parial Differeial Eqaios, 7: ,. [7] G. Adomia ad R. Rach. Iversio of oliear sochasic operaors. J. Mah. Aal. Appl, 9:9 46, 98. [8] R. Rach. A coveie compaioal form for he Adomia polyomials. J. Mah. Aal. Appl., :45 49, 984. [9] G. Adomia ad R. Rach. Geeralizaio of Adomia polyomials o fcios of several variables. Comp. Mah. Appl., 4: 4, 99. [] A. M. Wazwaz. A ew algorihm for calclaig Adomia polyomials for oliear operaors. Appl. Mah. Comp., :5 69,. [] F. Abdelwahid. A mahemaical model of Adomia polyomials. Appl. Mah. Comp., 4:447 45,. [] K. Abbaoi, Y. Cherral, ad V. Seg. Pracical formlae for he calcls of mlivariable Adomia polyomials. Mah. Comp. Modellig, :89 9, 995. [] Y. Zh, Q. Chag, ad S. W. A ew algorihm for calclaig Adomia polyomials. Appl. Mah. Comp., 69:4 46, 5. [4] J. Biazar, M. Hie, ad A. Khoshkear. A improveme o a alerae algorihm for compig Adomia polyomials i special cases. Appl. Mah. Comp., 7:58 59, 6. [5] M. Azreg-Aïo. A developed ew algorihm for evalaig Adomia polyomials. CMES-Comp. Model. Eg. Sci., 4: 8, 9. [6] J. S. Da. A efficie algorihm for he mlivariable Adomia polyomials. Appl. Mah. Comp., 7: ,. [7] J. S. Da. Coveie aalyic recrrece algorihms for he Adomia polyomials. Appl. Mah. Comp., 7:67 648,.

21 Joral of Algorihms & Compaioal Techology Vol. 7 No. 47 [8] G. Adomia ad R. Rach. The oisy covergece pheomea i decomposiio mehod solios. J. Comp. Appl. Mah., 5:79 8, 986. [9] G. Adomia ad R. Rach. Noise erms i decomposiio solio series. Comp. Mah. Appl., 4:6 64, 99. [] G. Adomia, R. Rach, ad N. T. Shawagfeh. O he aalyic solio of he Lae- Emde eqaio. Fod. Phys. Le., 8:6 8, 995. [] A. M. Wazwaz. A ew algorihm for solvig differeial eqaios of Lae-Emde ype. Appl. Mah. Comp., 8:87,. [] G. Adomia, R. Rach, ad R. Meyers. Nmerical algorihms ad decomposiio. Com-p. Mah. Appl., :57 6, 99. [] G. Adomia, R. C. Rach, ad R. E. Meyers. Nmerical iegraio, aalyic coiaio, ad decomposiio. Appl. Mah. Comp., 88:95 6, 997. [4] A. M. Wazwaz. A reliable algorihm for obaiig posiive solios for oliear bodary vale problems. Comp. Mah. Appl., 4:7 44,. [5] M. Dehgha, A. Hamidi, ad M. Shakorifar. The solio of copled Brgers eqaios sig Adomia-Pade echiqe. Appl. Mah. Comp., 89:4 47, 7. [6] M. Dehgha, M. Shakorifar, ad A. Hamidi. The solio of liear ad oliear sysems of Volerra fcioal eqaios sig Adomia-Padé echiqe. Chaos Solios Fracals, 9:59 5, 9. [7] Y. C. Jiao, Y. Yamamoo, C. Dag, ad Y. Hao. A aferreame echiqe for improvig he accracy of Adomia s decomposiio mehod. Comp. Mah. Appl., 4:78 798,. [8] H. Ch, Y. Zhao, ad Y. Li. A MAPLE package of ew ADM-Padé approximae solio for oliear problems. Appl. Mah. Comp., 7:774 79,. [9] G. Adomia. A adapaio of he decomposiio mehod for asympoic solios. Mah. Comp. Simla., :5 9, 988. [4] R. Rach ad J. S. Da. Near-field ad far-field approximaios by he Adomia ad asympoic decomposiio mehods. Appl. Mah. Comp., 7:59 59,. [4] R. Rach, G. Adomia, ad R. E. Meyers. A modified decomposiio. Comp. Mah. Appl., :7, 99. [4] J. S. Da ad R. Rach. New higher-order merical oe-sep mehods based o he Adomia ad he modified decomposiio mehods. Appl. Mah. Comp., 8:8 88,.