A New Hybrid in the Nonlinear Part of Adomian Decomposition Method for Initial Value Problem of Ordinary Differential Equation
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1 Joural of Matematics Researc; Vol No ; ISSN - E-ISSN - Publised b Caadia Ceter of Sciece ad Educatio A New Hbrid i te Noliear Part of Adomia Decompositio Metod for Iitial Value Problem of Ordiar Differetial Equatio Adeee F J Igobi D K & Ibijola E A Departmet of Mats/ComputerFederal Uiversit of Petroleum Resources Effurru Nigeria Departmet of Matematics/Statistics Federal Uiversit Wukari Nigeria Departmet of Matematics Ekiti State UiversitAdo-Ekiti Nigeria Correspodece: Igobi D K Departmet of Matematics/Statistics Federal Uiversit Wukari Nigeria dodiigobi@aoocom Received: Ma Accepted: Jue Olie Publised: Februar doi:/jmrvp URL: ttp://ddoiorg//jmrvp Abstract I tis paper a ew tecique of Hbrid of Adomia polomial is developed b te substitutio of a Oe-step metod of Talor s series approimatio of orders I ad II ito te oliear part of Adomia decompositio metod Tis ield a ew Talor series of Adomia Decompositio Metod i te oliear part Te applicatios of te modified ew brid polomial to some classical problems give results tat approimate te aaltical solutios of te problems wit miimal errors Keword: Adomia decompositio metod Talor series approimatio ew brid of Adomia polomial ordiar differetial equatio Itroductio Recetl a great deal of iterest as bee focused o te applicatio of Adomia s decompositio metod to solve a wide variet of liear ad oliear problems Adomia Tis metod geerates solutios i te form of a series wose terms are determied b a recursive relatiosip usig te Adomia polomials Researcers wo ave used te Adomia s decompositio metod ave frequetl eumerated o te advatages tat it offers Sice it was first preseted i te eigties Adomia decompositio metod as led to several modificatios i a attempt to improve te accurac or epad te applicatio of te origial metod Oe of suc recet modificatio was preseted i Adomia & Rac ad a improved versio of it tat coverges sligtl faster was preseted i Rac But ispite of te various tpes of modified Adomia polomial available te algoritm of te origial Adomia polomial is more simple ad coveiet i applicatio Recetl a simple Matematical program to compute te t term of te Adomia polomial was developed i Abbaoui & Cerrault ad Wazwaz were Pade approimats were emploed to obtai modified decompositio to geerate a improved solutio wit miimal error Several oter researc work o Adomia decompositio metods aboud i Adomia ; Adomia & Rac ; Almazmum Hedi & Bakoda Te modificatios of Adomia decompositio metod arise from difficulties ecouter i solvig certai classical problems of differetial equatios Tis usuall ivolves sligt cage i te Adomia polomial wic is aimed at improvig te covergece of te series solutio Te coverget cocept of te Adomia decompositio metod is addressed i Abbaoui & Cerrault ; Adomia & Rac ; Amed ad its applicatio to te iitial value problem of ordiar differetial equatio is discussed i Aguelov & Lubuma ; Wazwaz Terefore te aim of tis work is to geerate a ew Hbrid of Adomia polomial i te Adomia decompositio metod ad te appl it to some classical equatios ad compare te result wit teir respective aaltical solutios as sow i te tables ad D grap Geeral Adomia Decompositio Metod Cosiderig a geeral oliear differetial equatio preseted i Adomia & Cerruault as F f
2 were F is a oliear differetial operator operator form f ad are fuctios of Rewritig te equatio i te L R N f L R are ivertible ad o-ivertible liear operator of F respectivel ad N is a oliear operator represetig te oliear term i F Applig te iverse operator L te equatio implies L L L f L R L N B te defiitio of F as a differetial operator ad L is liear te give iitial coditio L L ield a equatio of te form L is a itegrator ad wit a g L R L N were g is te fuctio geerated b itegratig f usig defied iitial coditios Emploig te Adomia decompositio metod o a ifiite series is obtaied wit te oliear term defied as A is te Adomia polomials wic is determied b a N A b A d! d N Equatio ca be epress i terms of a ad as wit a recursive relatiosip L R L A g L R L A Usig equatio wit te correspodig relatio of te series solutio is determie immediatel Te Modified New Hbrid of Adomia polomial Cosiderig te recurret relatio of te o-liear part of equatio as L A ad te substitutio of te o-local approimatio defied iaguelov & Lubuma ; Aguelov & Lubuma ; Aguelov & Lubuma as ito L
3 is obtaied were A B Talor series epasio!! ' f f f ad rewritig equatio i terms of te ew brid of Adomia polomial is tus! f L Numerical Implemetatio of Equatio I tis sectio some iitial value problems are cosidered to sow te efficiec of te modified equatio Problem Cosider te iitial value problem of te form Equatio as aaltical solutio Solvig usig te modified brid equatio te followig terms of te series is obtaied Ad O Usig te terms of to obtaied O
4 Table Comparism of results of te New Hbrid ADM wit te aaltical solutio at = X New Hbrid ADM Aaltical solutio Error Table sows te performace of te New-Hbrid ADM wit te Aaltical solutio solutio At = sows a better comparative iteratio but wit o value at = call poit of sigularit Diagram - - ADOMIAN ACTUAL Tis sows reflectio of te table above wit te poit of sigularit = clearl sow PROBLEM Cosider te iitial value problem of te form Equatio as aaltical solutio
5 Solvig usig te modified brid equatio te followig terms of te series is obtaied Ad O Usig te terms of to obtaied O
6 Table Comparism of results of te New Hbrid ADM wit te aaltical solutio at = New Hbrid ADM Aaltical solutio Error Table sows te performace of te New-Hbrid ADM wit te Aaltical solutio ad value at a eigborood of = Diagram ADOMIAN ACTUAL Te figure sows a reflectio of te table above Problem Cosider te Abelia psical problem of []
7 Solvig usig te modified brid equatio te results is preseted i table as Table Comparism of results of te Abelia psical problem at = wit te results i Omor New Hbrid ADM Omor Result Error - - Table sows te performace of te New-Hbrid ADM wit te Omor at = Diagram HYBRID ADM OMARKJ Te figure sows a reflectio of te table above Coclusio Te Talor s series approimatio of orders I ad II is a good matematical tool i te cocept of te modified New Hbrid Adomia polomial from te Adomia Decompositio Metod Te applicatio of te ew Hbrid Adomia polomial i solvig some iitial values problems - ield results tat approimate te aaltical solutios at step cage of = wit miimal errors Tis is see i results comparisms of tables - ad diagrams - wic cofirmed te suitabilit of te New Hbrid for umerical approimatio
8 Refereces Adomia G Solvig Frotier problem of Psics Te Decompositio Metod Spriger New York Abbaoui K J & Cerrault Y Covergece of Adomia s Metod Applied to Differetial Equatios Iteratioal Joural of Computer/Matematics Applicatio - Adomia G & Rac C Modified Adomia Polomial Matematical ad Computer Modelig Joural - ttp://ddoiorg//s-- Adomia G & Cerruault Y Abbaoui K A Nopertubative Aaltical Solutio of Immue Respose wit Time-delas ad possible Geeralizatio Mat Comput Modellig - ttp://ddoiorg//s-- Amed A H M Covergece Aalsis ad Numerical Approimatio MSc Tesis Almazmum M Hedi F A & Bakoda H A Recet modificatio of Adomia Decompositio Metod for Iitial Value problem i Ordiar Differetial Equatios America joural of Computatioal Matematics - ttp://ddoiorg//ajcm Aguelov R & Lubuma J M S Nostadard Fiite Differece Metod b Nolocal Approimatio Keote address at te Sout Africa Matematical Societ Pretoria Sout Africa - October Notices of SA Mat Soc - Aguelov R & Lubuma J M S Cotributios of te Matematics of te Nostadard Fiite Metod ad Applicatios Num Mat Partial Differetial Equatios - ttp://ddoiorg//um Aguelov R & Lubuma J M S Nostadard Fiite Differece Metod b Nolocal Approimatio Matematics ad Computers i Simulatio Joural - ttp://ddoiorg//s-- Omor K T Adomia Depositio metod for solvig Abelia Differetial equatios Joural of Applied Sciece - ttp://ddoiorg//jas Rac C A New Defiitio of te Adomia Polomial Kbermeters - Wazwaz A Aaltical Approimatios ad Pade approimatios for Volterras Populatio Model Appl Mat Computer - ttp://ddoiorg//s-- Wazwaz A A Compariso betwee Adomia decompositio metod ad Talor Series Metod i te Series Solutios Appl Mat Computer - ttp://ddoiorg//s-- Coprigts Coprigt for tis article is retaied b te autors wit first publicatio rigts grated to te joural Tis is a ope-access article distributed uder te terms ad coditios of te Creative Commos Attributio licese ttp://creativecommosorg/liceses/b//
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