Nonlinear Integro-differential Equations by Differential Transform Method with Adomian Polynomials

Transcription

1 Math Sci Lett No - Mathematical Science Letters An International Journal Nonlinear Intero-dierential Equations b Dierential Transorm Method with Adomian Polnomials S H Behir General Required Courses Department Jeddah Communit Collee Kin Abdulaziz Universit Jeddah KSA Address: salah_behir@hotmailcom Received: Ma Revised: Jun ; Accepted: Jul Published online: Sep Abstract: A modiication o dierential transormation method is applied to nonlinear intero-dierential equations In this technique the nonlinear term is replaced b its Adomian polnomials or the inde hence the dependent variable components are replaced in the recurrence relation b their correspondin dierential transorm components o the same inde Thus the nonlinear intero-dierential equation can be easil solved with less computational wor or an analtic nonlinearit due to the properties available alorithms o the Adomian polnomials Numerical simulations o intero-dierential equations with dierent tpes o nonlinearit are treated the proposed technique has provided ood results Kewords: Dierential transorm method; nonlinear intero-dierential equations; Adomian polnomials Introduction Interal intero-dierential equations pla an important role in characterizin man social bioloical phsical enineerin problems; or more details see [-] reerences cited therein Nonlinear interal intero-dierential equations are usuall hard to solve analticall eact solutions are rather diicult to be obtained In literature nonlinear interal intero-dierential equations can be solved b man numerical methods such as the Leendre wavelets method [] the Haar unctions method [ ] the linearization method [] the inite dierence method [] the Tau method [ ] the hbrid Leendre polnomials bloc-pulse unctions [] the Adomian decomposition method [ ] the Talor polnomial method [-] the dierential transorm method [] The dierential transorm method DTM has been proved to be eicient or hlin nonlinear problems but the nonlinear unctions used in these studies are restricted to polnomials products with derivatives [-] or other tpes o nonlinearities the usual wa to calculate their transormed unctions as introduced b [] is to ep the nonlinear unction in an ininite power series then tae the dierential transorm o this series The problem with this approach is that the massive computational diiculties will arise in determinin the dierential transorm o nonlinear unction while worin with this ininite series Another approach or obtainin the dierential transorm o nonlinear terms is the alorithm in [] It is based on usin the properties o dierential transorm calculus to develop a canonical equation Then this equation is solved or the required dierential transorm o nonlinear term But as seen in the simple eamples in section in [] the alorithm requires a sequence o dierentiation alebraic manipulations computations o dierential transorm or other unctions which is more diicult or the case o composite nonlinearities In this wor we introduce a comprehensive more eicient approach or usin the DTM to solve nonlinear intero-dierential equations; the idea is based on the methodolo in [] The nonlinear unction is replaced b its Adomian polnomials then the dependent variable components are replaced b their correspondin dierential transorm component o the same inde This technique beneited the

2 S H Behir: Nonlinear Intero-dierential Equations properties o the Adomian polnomials the eicient alorithm to enerate them quicl as in the wor [-] Dierential Transorm Method The basic deinition the undamental theorems o the dierential transormation its applicabilit or various inds o dierential interal equations are iven in [-] A review o dierential transormation is presented here The transormation o the -th derivative o a unction in one variable is as ollows! d d the inverse transormation is deined b In this wor we use lower case letter or the oriinal unctions upper case letter st or the transormed unctions The ollowin theorems can be deduced rom equations Theorem I h then H Theorem I c then c where c is a constant Theorem I n then!! n Theorem I h then H Theorem I m then m where m m m Theorem I dt t then Theorem I dt t t then Theorem I dt t h t h then H H The ollowin relation is quite useul in the solution o redholm interals; it can be obtained rom Theorem equation i e

3 H Behir: Nonlinear Intero-dierential Equations b t t dt b a a The Modiied Dierential Transorm Method In this section we introduced a reliable eicient alorithm to calculate the dierential transorm o a nonlinear unction This nonlinear unction can be decomposed as A n n where A n n are the Adomian polnomials determined ormall as ollows [ ] A n n d i n! n d i i The Adomian polnomials o are introduced as A A A! A! A A!!!!!! so on Hence the dierential transorm components o are computed b utilizin their properties the can be written in the ollowin orm or = G G G G!!

4 S H Behir: Nonlinear Intero-dierential Equations G! G!!!! so on! The advantae o usin this alorithm compared to the alorithm suested in [] or computin dierential transormation o nonlinear unctions is this alorithm deals directl with nonlinear unction o the problem in h in its orm without an dierentiation alebraic manipulation no need to compute the dierential transorm o other unctions to obtain the required one Applications Numerical Results In this section we implement the proposed method on some dierent eamples with dierent tpes o nonlinearit All alebraic computations are eecuted usin MATHEMATICA sotware pacae Eample Consider the nonlinear Volterra intero-dierential equation cos with the initial conditions sin t dt t = = The dierential transormation o equation the initial conditions are! cos!! m m m m G m m m sin G m m! where G are the dierential transorm Adomian polnomials o the nonlinear unction = - = = Usin the relations in the Adomian polnomials or this nonlinear unction are G G

5 H Behir: Nonlinear Intero-dierential Equations G G G Utilizin the recurrence relation the transormed initial conditions the Adomian polnomials are evaluated Hence usin the inverse transormation in equation the ollowin series solution up to O can be obtained O!!!!!!!! or suicientl lare number o terms the closed orm o the solution is =sin +cos which is the eact solution Table shows the absolute relative error Abs rel err obtained or three various numbers o terms at some test points Table : Numerical comparison o results in Eample Abs rel err Terms Abs rel err Terms Abs rel err Terms E- E- E- E- E- E- E- E- Eample Consider the nonlinear Volterra intero-dierential equation t tln t dt with the initial conditions = = Application o the dierential transorm to equation ives G where G are the Adomian polnomials o the nonlinear unction = ln Substitute = = into equation one can et the ollowin relations Also or equation can be written as

6 S H Behir: Nonlinear Intero-dierential Equations G The initial conditions in are transormed b usin to Usin the relations in the Adomian polnomials or the nonlinear unction o this eample are G G G G G G G Utilizin the above relations - one can easil solve or Usin the inverse transormation rule in equation the ollowin series solution or equation up to O is O or suicientl lare number o terms the closed orm o the solution is e which is the eact solution Table shows the absolute relative error Abs rel err obtained or three various numbers o terms at some test points Table : Numerical comparison o results in Eample Abs rel err Terms Abs rel err Terms Abs rel err Terms E- E- E- E- E- E- E- Eample Consider the nonlinear Volterra intero-dierential equation with the initial condition e tan t t e dt

7 H Behir: Nonlinear Intero-dierential Equations The dierential transormation o equation the initial condition are! where G are the Adomian polnomials o the nonlinear unction G e tan I we substitute into equation we can et hence The ollowin sstem or is obtained rom! G G! G!!! G G tan G can be obtained b usin the relations in or the unction e as G tan e G G G

8 S H Behir: Nonlinear Intero-dierential Equations G On solvin the above sstems or the series solution o equation up to O is iven b O or suicientl lare number o terms the closed orm o the solution is tan which is the eact solution Table shows the absolute relative error Abs rel err obtained or three various numbers o terms at some test points Table : Numerical comparison o results in Eample Abs rel err Terms Abs rel err Terms Abs rel err Terms E- E- E- E- E- E- E- E- E- Eample Let us consider the nonlinear Volterra intero-dierential equation t dt t with the initial conditions The dierential transormation o equation the initial condition are! m m m m m G m! m m where G are the Adomian polnomials o the nonlinear unction G can be obtained b usin the relations in as G G G G

9 H Behir: Nonlinear Intero-dierential Equations G On solvin the above sstems - or the series solution o equation up to O b is iven O or suicientl lare number o terms the closed orm o the solution is tan which is the eact solution Table shows the absolute relative error Abs rel err obtained or three various numbers o terms at some test points Table ; Numerical comparison o results in Eample Abs rel err Terms Abs rel err Terms Abs rel err Terms E- E- E- E- E- E- E- E- E- E- Eample Let us consider the nonlinear redholm intero-dierential equation [] t t t dt with the initial conditions The dierential transormation o equation its initial conditions are!!!! where t t dt t t dt To obtain substitute into equation utilizin the transormation hence Substitute into equation one can et the ollowin relations

10 S H Behir: Nonlinear Intero-dierential Equations The ollowin recurrence relation can be obtained rom equation!!!! Utilizin relation or it can be shown that the ollowin equalities hold or N G G N where N is a suitabl lare inteer that represents the number o terms to be chosen Adomian polnomials o the nonlinear unction G G G G G G G as ollow G are the Tain into account relations in one can solve equations in b tain N to obtain the ollowin results Substitutin these values o into equations hence solvin or the epansion or the unnown unction can be obtained that is which is the eact solution Eample Lastl the ollowin nonlinear sstem o Volterra intero-dierential equations is considered [ ] u v v u [ u [ u t v t v t] dt t] dt

11 H Behir: Nonlinear Intero-dierential Equations with the initial conditions u u v v The dierential transormation o the sstem is iven b! G G G U!! G G V U! where G G u u v v equation as ollows G are the dierential transorms o the nonlinear unctions v v respectivel The initial conditions in equations are transormed usin U U V V Utilizin the relations in we obtained G V G V V G V V G V V G V V G V V G V V G U G U U V V V V V V V V V V V G U U U V V V V G U U U U G U U U U U G U U U U U U

12 S H Behir: Nonlinear Intero-dierential Equations G U U U G V U U U U G V V G V V V G V V V V G V V V V V G V V V V V V G V V V V V V V Usin the recurrence relations in sstem the transormed initial conditions in the relations in equations - one can easil evaluate U V Hence utilizin the inverse rule in the series solution o the sstem up to O is iven b u O v O or suicientl lare number o terms the closed orm o the solution is u e v e which is the eact solution Table shows the absolute relative error Abs rel err obtained or three various numbers o terms at some test points Table : Numerical comparison o results in Eample Abs rel err Terms Abs rel err Terms Abs rel err Terms E- E- E- E- E- E- E- Conclusion In this wor we presented a new approach or applin the modiied dierential transorm method or solvin nonlinear intero-dierential equations The dierential transorm o the nonlinear term is replaced in the recurrence relation b its Adomian polnomial o inde Hence the dependent variable components are replaced b their correspondin dierential transorms o the same inde The considered test eamples include Volterra redholm coupled sstem o intero-dierential equations with dierent tpes o nonlinearit rom these eamples the presented technique enerated numerical results is eective in solvin nonlinear intero-dierential equations Reerences [] P K Kthe P Puri Computational Methods or Linear Interal Equations Universit o New Orleans New Orleans [] A M Wazwaz A comparison stud between the modiied decomposition method traditional method Appl Math Comput -

13 H Behir: Nonlinear Intero-dierential Equations [] M T Rashed Numerical solution o unctional dierential interal intero-dierential equations Appl Numer Math - [] M Razzahi S ousei Leendre wavelets method or nonlinear Volterra-redholm interal equations Math Comput Simul - [] K Malenejed Mirzaee Numerical solution o intero-dierential equations b usin rationalized Haar unctions method Kbernetes Int J Sst Math - [] M H Reihani Z Abadi Rationalized Haar unctions method or solvin redholm Volterra interal equations J Comput Appl Math - [] P Darania E Abadian A V Osoi Linearization method or solvin nonlinear interal equations Math Probl En doi: /MPE// [] J Zhao R M Corless Compact inite dierence method or intero-dierential equations Appl Math Comput - [] S Abbasb A Taati Numerical solution o the sstem o nonlinear Volterra intero-dierential equations with nonlinear dierential part b the operational Tau method error estimation J Comput Appl Math - [] G Ebadi M Rahimi-Ardabili S Shahmorad Numerical solution o the nonlinear Volterra intero-dierential equations b the Tau method Appl Math Comput - [] K Malenejad B Basirat E Hashemizadeh Hbrid Leendre polnomials Bloc-pulse unctions approach or nonlinear Volterra- redholm intero-dierential equations Comput Math Appl - [] A M Wazwaz The combined Laplace transorm-adomian decomposition method or hlin nonlinear Volterra intero-dierential equations Appl Math Comput - [] M A Arahi Sh S Behzadi Solvin nonlinear Volterra-redholm intero-dierential equations usin the modiied Adomian decomposition method Comput Meth Appl Math - [] P Darania K Ivaz Numerical solution o nonlinear Volterra-redholm intero-dierential equations Appl Math Comput - [] K Malenejad Mohmoudi Talor Polnomial solution o hih-order nonlinear Volterra-redholm intero-dierential equations Appl Math Comput - [] S alcinbas Talor polnomial solution o nonlinear Volterra-redholm interal equations Appl Math Comput - [] A Borhaniar R Abazari Dierential transorm method or a class o nonlinear intero-dierential equations with derivative tpe ernel Canad J Comput Math Natural Sciences En Mid - [] Ariolu I Ozol Solution o boundar value problems or intero-dierential equations b usin dierential transorm method Appl Math Comput - [] Ariolu I Ozol Solution o interal intero-dierential equation sstems b usin dierential transorm method Comput Math Appl - [] Z M Odibat Dierential transorm method or solvin Volterra interal equation with separable ernels Math Comput Model - [] J Biazar M Eslami Dierential transorm method or sstems o Volterra interal equations o the second ind comparison with homotop perturbation method Int J Phs Sci - [] J K Zhou Dierential Transormation Its Applications or Electrical Circuits Huazhon Universit Press Wuhan China [] S H Chan I L Chan A new alorithm or calculatin one-dimensional dierential transorm o nonlinear unctions Appl Math Comput - [] A Elsaid ractional dierential transorm method combined with Adomian polnomials Appl Math Comput - [] J S Duan Convenient analtic recurrence alorithm or Adomian polnomials Appl Math Comput - [] J S Duan Recurrence trianle or Adomian polnomials Appl Math Comput - [] J S Duan An eicient alorithm or the multivivariable Adomian polnomials Appl Math Comput - [] G Adomian Solvin rontier problems o phsics:the decomposition method Kluwer Academic Publishers MA [] J Biazar H Ghazvini M Eslami He s homotop perturbation method or sstems o intero-dierential equations Chaos Solitons ractals -