A Comparison of AdomiansDecomposition Method and Picard Iterations Method in Solving Nonlinear Differential Equations
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1 Global Joural of Scece Froer Research Mahemacs a Decso Sceces Volume Issue 7 Verso. Jue Te : Double Bl Peer Revewe Ieraoal Research Joural Publsher: Global Jourals Ic. (USA Ole ISSN: & Pr ISSN: A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos B Ibjola E.A & Aegboegu B.J Ek Sae Uvers Ngera Absrac - I hs aerolear ffereal equaos are solve hrough Aoma ecomoso meho(adm a he resuls are comare wh hose of Pcar eraos meho. I s oe ha ADM akes he form of a coverge seres wh easl comuable comoes. The ADM meho s able o solve large class of olear equaos effecvel more easl a accurael; a hus he meho has bee wel alcable o solve a class of equaos sceces a egeerg. Kewors : Pcar erao Aoma ecomoso Nolear ffereal equao Voleral egral equao of frs k. GJSFR-F Classfcao MSC : 34L3. A Comarso of AomasDecomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos Srcl as er he comlace a regulaos of :. Ibjola E.A & Aegboegu B.J. Ths s a research/revew aer srbue uer he erms of he Creave Commos Arbuo-Nocommercal 3. Uore Lcese h://creavecommos.org/lceses/b-c/3./ ermg all o commercal use srbuo a rerouco a meum rove he orgal work s roerl ce.
![Irouco Large classes of lear ffereal equaos ca be solve b he Aoma 35 a Pcar erao meho [ 9 ].](/docs-images/93/111281584/images/2-3.jpg "The ADM was frs 6 o a ecomoso meho [ ] comare wh he Pcar meho b Rach [ 4] a Bellomo a Sarafa [ ] umber of examles.")
![golberg[ 7] showe ha he Aoma aroach o llear ffereal equaos was equvale o he classcal meho of successve aroxmaos (Pcar meho.](/docs-images/93/111281584/images/2-4.jpg "Howeverwe shall show ha he equvalece oes o hol for olear ffereal equaos. We coser he geeralze frs orer olear ffereal equaos he form: ' = f( R f : Rx R R. Wh al coo ( = R.")
2 Ref. A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos α σ Ibjola E.A & Aegboegu B.J Jue 37. Ibjola E.A a Aegboegu B.J o AomaDecomosoo Meho (ADM for Numercal Soluo of Orar Dffereal Equao. Avaces Naural Ale Sceces (3: U.S Absrac - I hs aerolear ffereal equaos are solve hrough Aoma ecomoso meho(adm a he resuls are comare wh hose of Pcar eraos meho. I s oe ha ADM akes he form of a coverge seres wh easl comuable comoes. The ADM meho s able o solve large class of olear equaos effecvel more easl a accurael; a hus he meho has bee wel alcable o solve a class of equaos sceces a egeerg. Kewors : Pcar erao Aoma ecomoso Nolear ffereal equao Voleral egral equao of frs k. I. Irouco Large classes of lear ffereal equaos ca be solve b he Aoma 35 a Pcar erao meho [ 9 ]. The ADM was frs 6 o a ecomoso meho [ ] comare wh he Pcar meho b Rach [ 4] a Bellomo a Sarafa [ ] umber of examles.golberg[ 7] showe ha he Aoma aroach o llear ffereal equaos was equvale o he classcal meho of successve aroxmaos (Pcar meho. Howeverwe shall show ha he equvalece oes o hol for olear ffereal equaos. We coser he geeralze frs orer olear ffereal equaos he form: ' = f( R f : Rx R R. Wh al coo ( = R. We shall rocee o scuss he basc coces of he wo mehos. II. Dervao of The Mehos I revewg he basc mehoolog we coser a absrac ssem of ffereal equao (. a assume ha f( s olear a aalc ear = =. I s equvale o solve he al value roblem (. a he Volera egral equao of frs k ( f ( s ( s s = = To se u he Aoma meho Rach [4] wre (. a oeraor form: L =. f (.3 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I Auhor α σ: Dearme of Mahemacal Sceces Ek Sae Uvers Ngera. E-mal : bolucrow@ahoo.com Global Jourals Ic. (US
3 A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos Jue 38 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I Where L =.He ale L (a oe fol egral o (.3 a mose he al coo o oba: ( = + L f( Aoma cosere he soluo ( A wre he olear f( he seres form: ( = = + as he seres of fuco f( = A (.5 = The eeece of A o a ma be o-olomal. Formall A s obae b = = A = f( =! Where s a formal arameer. Fucos A are olomals ( whch are referre o as he Aoma olomals. The frs few Aoma for = are lse b Zhu a Chag [8] as follows: A = f( A= f( ' ' '' A = f ( + f ( ' '' 3 ''' A3 = 3f ( + f ( + f ( 6 where rmes eoe he aral ervaves wh resec o. I was show b Cherruaul a Abbaou [3] ha he Aoma olomals efe b he exlc formulae:.4.6 A are k A = f (.7 k k= k! + + k = Khelfa a Cherruaul [8] rove a bou for Aoma olomals b Where ( k su ( J ( + A M ( +! + f M for a gve me erval J R. B subsug (.4 a (.5 o (. gves a recursve equao for + ( :.8 erms of = + A ( s ( s ( s ( s s =.9 The Pcar s meho s use for he roof of exsece a uqueess of soluos of a ssem of ffereal equaos. The meho sars wh aalss of Volerra s egral equao (.. Assume ha f( sasfes a local Lschz coo a ball arou Ref. 8. Y. Zhu Q. Chag a S. Wu A ew algorhm for calculag Aoma Polomals Al. Mah. Comu (5. 3. N. Hmou K. Abbaou a Y. Cherruaul Shor ew resuls o Aoma meho Kberees (3. Global Jourals Ic. (US
4 A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos = a = : δ : f( f( K. Ref. 9. I. K. Youssef Pcar erao algorhm combe wh Gauss-Seel echque for al value roblems Ale mahemacs a comuao (7. Where K s lshz cosa a Le ( = a efe a recurrece relao s a orm R ( + f s s s ( = + ( ( =. If s small eough he ew aroxmao ( + ( δ for all a (. s a coraco he sese ha Where Q = K < so ha <. k belogs o he same ball [ f( ss ( f( ss ( ] s Qsu ( (. B he Baach fxe o heorem here exss a uque soluo ( C([ ] Bδ ( where Bδ ( s a oe ball R ceere a wh raus δ. The comuaos of Pcar erave algorhm were reore recel b Youssef [9]. III. Alcaos a Resuls Sce he roblems gve he arcle a referece [9] o he comuaos a aalss of Pcar s erave scheme are releva for our am we reuse some of he roblems. Problem Coser a scalar frs orer ODE where. Ths ffereal equao has exac soluo; ( = =.3 ( = ( ( Followg he Aoma meho we wre (.3 as comue he Aoma olomals from f( = he form: A = A = ( A = + ( ( 3 3 A3 = + ( ( = + ( s s a Jue 39 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I Global Jourals Ic. (US
5 A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos Usg (.9 we eerme few erms of he Aoma seres: ( = ( = ( = Jue 4 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I ( = 3 ( = 4 ( 3! ( ! 3 4 From ( ( (6 7 + ( = = ! 4! Exag (.3 a ower seres of we ca see ha heaoma meho recovers he ower seres soluo ( = ( ( Usg he Pcar eraos meho ( (6 7 + = ! 4! ( ( + ( = + ( ( s s We oba successve aroxmaos he form: ( = ( = + ( ( + = (3 ( = + ( s + I s oe from ( ha Pcar erao mx u owers of whch make ( beg ffere from he h aral sum of he ower seres.if = he Noes ( = = ( = + = ( = = Global Jourals Ic. (US
6 A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos Noes Sce ( s a aral sum of he ower seres (.3 we coclue ha he Aoma meho beer aroxmaes he exac ower seres soluo comare o he Pcar meho. Problem Coser he olear ffereal equao wh he exac soluo ( = + ah( ( = =.5 B he ADM we wre (.5 he egral form ( = + ( ( ( comue he Aoma olomals for A = A = A = A3 = 3 ( 3 + A4 = 4 ( = he form; f( Usg (.9 we eerme few erms of he seres; Thus s s s 3 5 = ; = ; = ; 3 = ; 4 = ; 5 = ( = + + = + ah( 3 5 B usg Pcar meho we wre (.5 he egral form ( ( ( + = + ( s ( ( s ( = = ( = + = + = + = ( (3 = + + = a a we oba he successve aroxmaos he form; Jue 4 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I Global Jourals Ic. (US
7 A Comarso of Aoma s Decomoso Meho a Pcar Ieraos Meho Solvg Nolear Dffereal Equaos (3 We ca equall see from ( a ha Pcar meho mx u owers of ( whch also make beg ffere from he h aral sum of he seres. Jue 4 Global Joural of Scece Froer Research F Volume XII Issue VII V erso I IV. Cocluso a Dscusso I hs aer ADM has bee successfull ale o fg he soluos of olear ODE. The obae resuls are comare wh hose of Pcar eraos meho. I s oe ha he Pcar meho mxes u owers of he aral sum for he exac soluos whle he Aoma seres s equvale o he ower seres me a Aoma meho requres aalc of f( whch s more resrcve ha he Lschz coo requre for he Pcar meho.the resuls show ha ADM s a owerful mahemacal ool for solvg olear ffereal equaos a herefore ca be wel ale he fel of scece a egeerg. Refereces Référeces Referecas. Ibjola E.A a Aegboegu B.J o AomaDecomosoo Meho (ADM for Numercal Soluo of Orar Dffereal Equao. Avaces Naural Ale Sceces (3: U.S. Aegboegu B.J a Ibjola E.A o he Theor a Alcao of Aoma Decomoso Meho for Numercal Soluo of Seco oher Orar Dffereal Equao. The Pacfc Joural of Scece a Techolog Volume 9 (: November N. Hmou K. Abbaou a Y. Cherruaul Shor ew resuls o Aoma meho Kberees (3. 4. R. Rach O he Aoma ecomoso meho a comarsos wh Pcar s meho J. Mah. Aal. Al ( K. All-Khale Numercal aroxmao for oulao growh moels Ale mahemacs a comuao (5. 6. N. Bellomo a D. Sarafa O Aomas ecomoso meehos a some comarsos wh car oerave scheme J. Mah. Aal. Al ( M. A. Golberg. A oe o he ecomoso meho for oeraor equao Al. Mah.Comu ( Y. Zhu Q. Chag a S. Wu A ew algorhm for calculag Aoma Polomals Al. Mah. Comu (5. 9. I. K. Youssef Pcar erao algorhm combe wh Gauss-Seel echque for al value roblems Ale mahemacs a comuao (7. Noes Global Jourals Ic. (US
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