The Comparison Adomian Decomposition Method and Differential Quadrature Method for Solving Some Nonlinear Partial Diferential Equations

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1 merica Joral of pplied Mathematics 5; (): 9-94 Pblished olie pril 5, 5 ( doi: 648/jajam5 ISSN: -4 (Prit); ISSN: -6X (Olie) The Compariso domia Decompositio Method ad Differetial Qadratre Method for Solig Some Noliear Partial Diferetial Eqatios Zahra dabi Firoozjae, llahbakhsh yazdai Departmet of Mathematics, Faclty of Mathematical Scieces, Uiersity of Mazadara, abolsar, Ira address: zahraadabi@gmailcom (Z Firoozjae) To cite this article: Zahra dabi Firoozjae, llahbakhsh yazdai The Compariso domia Decompositio Method ad Differetial Qadratre Method for Solig Some Noliear Partial Diferetial Eqatios merica Joral of pplied Mathematics Vol, No, 5, pp 9-94 doi: 648/jajam5 bstract: Noliear partial diferetial eqatios are a class of partial diferetial eqatios haig may importat ses i egieerig ad scieces I this work we display a compariso betwee domia Decompositio Method (DM) ad Differetial Qadratre Method (DQM) for solig some oliear partial diferetial eqatios We fod the existece of exact soltios for those models The merical reslts show the efficiecy ad accracy of this method Keywords: domia Decompositio Method, Differetial Qadratre Method, Noliear Partial Diferetial Itrodctio Noliear partial differetial eqatios ca be fod i wide ariety scietific ad egieerig applicatios May importat mathematical models ca be expressed i terms of oliear partial differetial eqatios The most geeral form of oliear partial differetial eqatio is gie by: F(,t,x, y, x, y,t) with iitial ad bodary coditios (x, y,) φ (x, y), x, y Ω,Ω R (x, y, t) f (x, y,t), x, y Ω (a) (b) (c) where Ω is the soltio regio ad Ω is the bodary of Ω I recet years, mch research has bee focsed o the merical soltio of oliear partial eqatios by sig merical methods ad deelopig these methods [,] I the merical methods, which are commoly sed for solig these kid of eqatios large size or difficlt of comptatios is appeared ad sally the rod-off error cases the loss of accracy The domia decompositio method which eeds less comptatio was employed to sole may problems [,4] Therefore, we applied the domia decompositio method to sole some models of oliear partial eqatio, this stdy reeals that the domia decompositio method is ery efficiet for oliear models, ad it reslts gie eidece that high accracy ca be achieed The domia Decompositio Method The domia decompositio method (DM) [5] is a wellkow systematic method for prac- tical soltio of liear or oliear ad determiistic or stochastic operator eqatios, icldig ordiary diferetial eqatios (ODEs), partial diferetial eqatios (PDEs), itegral eqatios, itegrodiferetial eqatios, etc The DM is a powerfl techiqe, which proides effciet- gorithms for aalytic approximate soltios ad meric simlatios for real-world applicatios i the applied scieces ad egieerig It permits s to sole both oliear iitial ale problems(ivps) ad bodary ale problems (VPs) [6, 7] withot physical restrictie assmptios sch as reqired by liearizatio, pertrbatio, ad hoc assmptios, gessig the iitial term or a set of basis fctios, ad so forth Frthermore the DM does ot reqire the se of Gree's fctios, which wold complicate sch aalytic calclatios sice Gree's fctios are ot easily determied i most cases The accracy of the aalytic approximate soltios obtaied ca be erifed by direct sbstittio datages of the DM oer Picard's iterated method were demostrated i [8] More adatages of the DM oer the ariatioal iteratio method were preseted i [9, ] key o- tio is the domia polyomials, which are tailored to the particlar oliearity to sole oliear

2 merica Joral of pplied Mathematics 5; (): operator eqatios The priciple of the domia decompositio method (DM) whe applied to a geeral oliear eqatio is i the followig form (): ierse operator L, with hece as; L L R N g () L T () () dt Eqatio () ca be (g) L ( R) L ( N) () The decompositio method represets the soltio of eqatio () as the followig ifiite series: (4) The oliear operator N Ψ() is decomposed as: where []: N (5) are domia s polyomials, which are defied as d [ ψ ( ],,,, d! λ i λ i λ (6) i Sbstittig eqatios (4) ad (5) ito eqatio (), we hae ( ( )) ( ) (7) L R L Coseqetly, it ca be writte as: φ L ( g) L R L L R L ( ( )) ( ) ( ( )) ( ) ( ( )) ( ) L R L where φ is the iitial coditio, Hece all the terms of are calclated ad the geeral soltio obtaied accordig to DM as The coerget of this series has bee proed i [] Howeer, for some problems [] this series ca t be determied, so we se a approximatio of the soltio from trcated series (8) Problem M with lim lim (9) Let s cosider the Problem t ith the iitial coditio U M M M ( ),, 4 x x x t Eq () has the exact soltio []: I this problem we hae () ( x,), x () x (, t) x tah(t) () N ψ (( ) ( ), g( x, t) x, x R, L ad φ ( x,) t y sig Eq (6), we obtai x x x ( ) x x x a x x x x 4 ( ) 4 x x x x x y sig Eq (8), we hae x t 5 x t 5 x t 7 7 x t 5 6 x t x t () (4)

3 9 Zahra dabi Firoozjae ad llahbakhsh yazdai: The Compariso domia Decompositio Method ad Differetial Qadratre Method for Solig Some Noliear Partial Diferetial Eqatios From Eq(4) we hae ( x, t) x [ t t t t t t ] (5) which gies the exact soltio () Problem Cosider the oliear system of eqatios with the iitial coditios t x y t x y (6) ( x, y,) ( x, y,) x y (7) ( x y) ( x, y, t) ( x, y, t) ( t) I this problem Eqs (6) ca be writte as: ( ) ( ) L N L N where L(), N ψ (, ) t x y adn ψ (, ) x y y sig Eq (4) the soltios ca be writte as: ( x, y, t) (,, ) x y t ( x, y, t) ( x, y, t) The associated decompositio scheme is gie by ψ,,, ψ ( x, y,), L ( (, )) ( x, y,), L ( (, )) We decompose ad respectiely, Where ad (8) (9) () () ψ ad ψ accordig to the series are calclated by the domia s polyomials which are defied i Eq(6) the we obtai x y x y x y () x y x y x y Similarly: x y x y x y () x y x y x y y sig Eq (8) we hae From Eq(4) we hae ( x, y, t) ( x, y, t) x y x y ( x y)( t) ( x y)( t) x y ( )( t) ( x y)( t) ( x y)( t) ( x y)( t) ( x y)( t) ( x y)( t) ( x y)[ t ( t) ( t) ( t) ] which gies the exact soltio (8) 4 Differetial Qadratre Method (4) (5) The differetial qadratre method (DQM) is a merical techiqe sed to sole the iitial ad bodary ale problems The DQM compared with the other merical method sch as the fiite differece methods (FDM) ad fiite elemet methods (FEM), ad showig excellet merical reslts, it eeds oly applyig a few grid poits i

4 merica Joral of pplied Mathematics 5; (): order to get high-precise soltios, a good coergece ad it reqires oly less comptatioal workload [,4] This method was proposed by ellma i the early 7s [5,6] The, the techiqe has bee sccessfl employed i a ariety of problems i egieerig ad physical scieces hece attracted may researchers attetio i recet years l-saif ad Zh [7], sig the differetial qadratre method to sole the copled icompressible Naier Stokes eqatio ad heat eqatio ad showig that accrate merical reslts ca be obtaied by the DQM sig oly a few grid poit ad reqires less storage ad comptatioal effort compared to the coetioal low-order fiite differece method I aother work, l-saif ad Zh [8], sig the mixed differetial qadratre method(mdqm) for solig the copled two-dimesioal icompressible Naier - Stokes eqatio ad heat eqatio The reslts show that the ew method is more accrate ad has better coergece tha the traditioal DQM The prpose of this paper is to itrodce ad applicatio the differetial qadratre method to solig steady state two-dimesioal coectiodiffsio eqatio The reslts demostrated that high accrate merical soltio by sig oly a few grid poits ad reqires less storage ad comptatioal effort compared to the some merical methods wealthy from some researchers i the precedet stdies Discssio Table Compariso of DM ad DQM soltios for problem I smmary, the DM is a powerfl ad e±ciet techiqe for the soltio of oliear ordiary, partial ad fractioal diferetial eqatios It proides the aalyst with a easily comptable, readily eriable ad rapidly coerget seqece of aalytic approximate fctios for the soltio DM DQM t X U 5 U * 9 854e e- 5 6e- 6578e e- 7687e e e e e e- 67e e-5 4e e e e e- 4565e e e-6 Table Compariso of DM ad DQM soltios for problem DM DQM t y X U 5 U * 5 e-4 4e e e-4 7e-5 45e e-4 7e-5 45e e-4 7e-5 45e e-4 5e-5 754e e- 5546e- 6e e- 445e-5 e e- 4e-5 e e- 4e-5 e e- 7e-5 75e e- 4567e- 7685e e- 7e-5 64e e- 5457e- 64e e- 454e- 64e e- 4e- 4569e- Coclsio I this paper, we hae applied the domia decompositio method for solig three problems of oliear partial eqatios We demostrated that the decompositio procedre is qite efficiet to determie the exact soltios Howeer, the method gies a simple powerfl tool for obtaiig the soltios withot a eed for large size of comptatios It is also worth otig that the adatage of this method sometimes displays a fast coergece of the soltios I additio, the merical reslts which obtaied by this method idicate a high degree of accracy The prpose of this paper is to itrodce ad applicatio the differetial qadratre method to solig steady state two-

5 94 Zahra dabi Firoozjae ad llahbakhsh yazdai: The Compariso domia Decompositio Method ad Differetial Qadratre Method for Solig Some Noliear Partial Diferetial Eqatios dimesioal coectio-diffsio eqatio The reslts demostrated that high accrate merical soltio by sig oly a few grid poits ad reqires less storage ad comptatioal effort compared to the some merical methods wealthy from some researchers i the precedet stdies Refereces [] l-saif SJ, (7), Nmerical stdy for coectio motio stability of the compressibletwo-dimesioal flid flow by differetial qadratre method, Jasrah Researches(Scieces), Vol, No p- [] Leeqe RJ, (6), Fiite differece methods for differetial eqatios Math 585 witers Qarter, Uiersity of Washigto ersio of Jaary [] Celik E, ayram M Yelogl T, (6), Soltio of differetial algebra Eqatios by domia decompositio method, Iter J Pre ad ppl Math Scieces, Vol, No, p 9- [4] Jaidi M ad Golbabai, (7), domia decompositio method for approximatig the soltio of parabolic eqatios, J ppl Math Scieces, Vol, No5, p 9-5 [5] SE Serrao () Egieerig Ucertaity ad Risk alysis: alaced pproach to Probability, Statistics, Stochas-btic Modelig, ad Stochastic Diferetial Eqatios, Secod Reised Editio, HydroSciece, mbler, P [6] PY Tsai ad CK Che () approximate aalytic soltio of the oliear Riccati diferetial eqatio, J Frak Ist [7] M Wazwaz () Liear ad Noliear Itegral Eqatios: Methods ad pplicatios, Higher Edcatio Press, eijig, ad Spriger-Verlag, erli [8] JS Da ad R Rach () ew modi catio of the domia decompositio method for solig bodary ale problems for higher order oliear diferetial eqatios, ppl Math Compt 8,49-48 [9] M Wazwaz ad R Rach, () Compariso of the domia decompositio method ad the ariatioal iteratio method for solig the Lae-Emde eqatios of the rst ad secod kids, Kyberetes 4,5-8 [] M Wazwaz () reliable stdy for extesios of the rat problem with bodary coditios, Math Methods ppl Sci 5, [] Seg V bbaoi K Cherralt Y, (996), domia s polyomial for oliear operators, J Math Compt Modelig, Vol 4, No, p59-65 [] ellma R, Kashef G ad Casti J, (97), Differetial qadratre: techiqe for the rapid soltio of oliear partial differetial eqatios, Jcompt Phys, Vol,No, p 4-5 [] H Hossaizadeh, G froziad Yazdai () pplicatio of domia Decompositio Method for Solig Implsie Differetial Eqatios The Joral of Mathematics ad Compter Sciece Vol No4,67-68 [4] ert C W ad Malik M (4)" Differetial qadratre method i comptatioal mechaics ", reiew : ppl Mech Re, 49, -7 [5] Sh C, Che W ad D H(4)" Free ibratio aalysis of criliear qadrilateral plates by the DQ method", J compt Phys, 6, [6] l Saif, SJ ad Zh, ZY ()'' Differetial qadratre method for solig the copledicompressible Naier-Stockes eqatios ad heat eqatio", Proc 4th It Cofer o Noliear Mech, Shaghai, [7] li N H (4) " Fiite elemet methods to sole twodimesioal trasport eqatio sig modified Galerki schemes ", M Sc thesis, asrah Uiersity, Iraq [8] l-saif, S J ad Zh ZY ()" pplicatio of Mixed Differetial qadratre method for solig the copled towdimesioal icompressible Naier Stockes eqatio ad heat eqatio ", J of Shaghai Uiersity,7(4), 4-5