Mathematical Computatio March 015, Volume, Issue 1, PP.1 6 The Adomia Polyomials ad the New Modified Decompositio Method for BVPs of oliear ODEs Jusheg Dua # School of Scieces, Shaghai Istitute of Techology, Shaghai 0118, P.R. Chia # Email: duajs@sit.edu.c Astract I this paper we cosider the ew algorithm for the Adomia polyomials ad the ew modified decompositio method for solvig oudary value prolems of oliear ordiary differetial equatios. I the ew method, the recursio scheme does ot ivolve udetermied coefficiets. Thus we avoid the complicatios resultig from the ecessity of evaluatig such udetermied coefficiets at each stage of approximatio. Furthermore, the recursio scheme ca emed a covergece parameter to efficietly calculate the sequece of the aalytical approximate solutios. Keywords: Adomia Decompositio Method; Adomia Polyomials; Boudary Value Prolem; Ordiary Differetial Equatio 1 INTRODUCTION The Adomia decompositio method (ADM) [1-6] is a powerful tool for solvig liear or oliear fuctioal equatios. The method give aalytic approximatios y a recursive maer. Applyig the ADM to the oudary value prolems (BVPs) for ordiary differetial equatios (ODEs) ca avoid usig the Gree fuctio cocept, which greatly facilitates aalytic approximatios ad umerical computatios. There are several differet resolutio techiques ased o the ADM for solvig BVPs for oliear ODEs, such as [, 7, 8] the doule decompositio method ad the Dua-Rach modified decompositio method [9]. The doule decompositio method decomposes the solutios, the oliearities ad the udetermied coefficiets ito series efore desigig the recursio scheme for the solutio compoets. The Dua-Rach modified decompositio method excludes all udetermied coefficiets whe computig successive solutio compoets. We ote that parametrized recursio scheme ca e used to achieve simple-to-itegrate series, fast rate of covergece ad exteded regio of covergece [9-11]. We remar that the covergece of the Adomia series has already ee prove y several ivestigators [5, 1, 1]. For example, Adelrazec ad Peliovsy [1] have pulished a rigorous proof of covergece for the ADM uder the aegis of the Cauchy-Kovalevsaya theorem for iitial value prolems. A ey cocept is that the Adomia decompositio series is a computatioally advatageous rearragemet of the Baach-space aalog of the Taylor expasio series aout the iitial solutio compoet fuctio, which permits solutio y recursio. Furthermore covergece of the ADM is ot limited to cases whe oly the fixed-poit theorem applies, which is far too restrictive for most physical applicatios. Differet classes ad geeralizatio of the Adomia polyomials were preseted i [, 5, 1-16]. New applicatios ad umerical methods ased o the ADM were developed i [17-0]. ADOMIAN POLYNOMIALS The decompositio method decomposes the solutio ux ( ) ad the oliearity Nu ito series ux ( )= u( x), Nu= A, (1) =0 =0 where A = A ( u ( x), u ( x),, u ( x)) are the Adomia polyomials 0 1-1 -
We list the first five Adomia polyomials for the oliearity Nu = f ( u ), 1 d A = N( ), 0. u ()! d =0 =0 u1 A = f( u ), A = f( u ) u, A = f( u ) u f( u ), 0 0 1 0 1 0 0! A = f( u ) u f( u ) uu f( u ) u!, 0 0 1 0 1 u u u 1 () u A = f ( u ) u f ( u ) ( ) ( ) 1. 0 0 uu f u f u 1 0 0!!! Several algorithms [1, 5, 6, 1-7] for symolic programmig have sice ee devised to geerate the Adomia polyomials quicly ad to high orders. New, more efficiet algorithms ad surouties i MATHEMATICA for rapid computer-geeratio of the Adomia polyomials are provided i [10, 15, 8-0]. Here we list the Corollary algorithm i [0] as follows. where the coefficiets 0 0 0 =1 - - A = f u, A = C f u, 1, () C are defied recursively as 1 C = u, 1, 1 1 C = 1,. j u C j1 1 j j =0 We remar that this algorithm does ot ivolve the differetiatio operator for the coefficiets C, ut oly requires the elemetary operatios of additio ad multiplicatio, ad is thus emietly coveiet for computer algera systems such as MATHEMATICA, MAPLE or MATLAB. THE NEW MODIFIED DECOMPOSITION METHOD We display the ew modified decompositio method y cosiderig the BVP for the secod-order oliear ODE, Lu = Nu g( x), a < x <, (5) where L = d dx, ( ) solutio of the BVP exists uiquely. ua ( )=, u ( )=, (6) g x is a prescried cotiuous fuctio ad Nu is a aalytic oliearity. We assume the For the BVP i Eqs. (5) ad (6), we tae the iverse liear operator as 1 L to oth sides of Eq. (5) yields Let x = i Eq. (7) ad the solve for u ( a), the () 1 x x L ()= () dxdx. Applyig the operator 1 1 ux ( ) ua ( ) u( a)( xa)= LNu Lgx ( ). (7) 1 1 u ( ) ua ( ) [ L Nu] [ L gx ( )] x= x= u( a)=, a 1 x where [ L ()] = () dxdx. x= Sustitutig Eq. (8) ito Eq. (7) yields a a u( ) u( a) xa 1 1 1 xa 1 u( x)= u( a) ( xa) [ L g] L g L Nu [ L Nu]. (9) x= x= a a a Thus the right had side of Eq. (9) does ot cotai the udetermied coefficiet u ( a). Next, we decompose the solutio ad the oliearity a a (8)
ux ( )= u( x), Nu= A, (10) =0 =0 From Eq. (9), the solutio compoets are determied y the modified recursio scheme u ( ) ua ( ) x a 1 1 u = u( a) ( xa) [ L g] L g, 0 x= a a 1 x a 1 u = L A [ L A ], 0. (1) 1 x= a 1 The -term approximate solutio is deoted as ( )= x =0u. We ca also desig other recursio schemes icludig the parametrized recursio scheme [9-11]. Example 1. Cosider the oliear BVP The exact solutio is [1] where satisfies sec =, to 6 sigificat figures, = 1.606. We decompose the solutio ux ( ) ad the oliearity e u as (11) u( x)= e u,0 x 1, (1) u(0)=0, u (1)=0. (1) * (x1) u ( x)=l( sec ) l(), (15) u =0 =0 ux ( )= u( x), e = A, where the Adomia polyomials for f ( u)= e u are u 0 u 0 u u 0 1 u u 0 1 A = e, A = e u, A = e ( u ), A = e, 0 1 1 uu u 1 6 u u 0 1 1 u A = e,. u u uu u 1 1 Accordig to the aove procedure, the solutio compoets are determied y the modified recursio scheme u =0, u = L A x[ L A ], =1,,. 1 1 0 1 1 x=1 By computatio we have 5 6 x x x x x x x x x x u =, u =, u =,. 1 1 160 1 96 60 180 0.01 0.001 ME 10 10 5 5 6 7 FIG. 1: LOGARITHMIC PLOTS OF MAXIMAL ERRORS ME VERSUS ( =,,,5,6,7 ). - -
1 To examie the covergece of the -term approximatio ( )= x =0u, we cosider the maximal errors * ME = max ( x) u ( x), (16) 0x1 which are computed y usig MATHEMATICA. I Fig. 1 we display the logarithmic plots of the maximal errors ME versus for = to 7. The data poits lie almost o a straight lie, which meas that the maximal errors decrease approximately at a expoetial rate. Example. Cosider the oliear BVP =6,0 1, (17) u u x u(0) = 1, u (1) = 1/. (18) * The exact solutio is u ( x)=(1 x). Accordig to our procedure, we have 1 1 u =1 x6l u 6 x[ L u ], (19) x=1 1 x x where L ()= () dxdx. The Adomia polyomials for the oliearity 0 0 u u are = =0 f ( u)= u with the decompositio We have checed that the recursio scheme A = u, A = u u, A = u u u,, A = u u. 0 0 1 0 1 0 1 u =1 x, 0 1 1 u =6L A 6 x[ L A ], 0, 1 x=1 yields a diverget series. But if we use the parametrized recursio scheme u = c1 x, 0 =0 c 1 1 u = 6L A 6 x[ L A ], 0, 1 1 x=1 coverget approximate solutios ca e otaied for appropriate values of c. For example, we tae c = 0. ad * calculate the solutio compoets u. The maximal errors ME = max 0 x 1 ( ) ( ) x u x, for = 1 through 16, are listed i Tale 1, where the sequece { ME } decreases mootoically. TABLE 1: FOR c = 0., MAXIMAL ERRORS ME FOR =1 THROUGH 16. 1 5 6 7 8 ME 0. 0. 0.1 0.05 0.05 0.015 0.0065 0.0015 9 10 11 1 1 1 15 16 ME 0.00156 0.000781 0.00091 0.000195 0.000098 0.00009 0.0000 0.00001 CONCLUSIONS We have preseted the ew modified decompositio method for solvig BVPs of oliear ODEs. The recursio scheme ca emed a covergece parameter to efficietly calculate the sequece of the aalytical approximate solutios. I the ew method, the udetermied coefficiets are iserted due to the ature of the oudary coditios. Thus we avoid the complicatios resultig from the ecessity of evaluatig such udetermied coefficiets at each stage of approximatio. The overall efficiecy of our ew modificatio of the ADM is further ehaced y the ew - -
algorithms ad surouties [10, 8-0] for geeratig the Adomia polyomials quicly ad to high orders. ACKNOWLEDGEMENTS This wor was supported y the Natural Sciece Foudatio of Shaghai (No.1ZR10800) ad the Iovatio Program of Shaghai Muicipal Educatio Commissio (No.1ZZ161). REFERENCES [1] G. Adomia, R. Rach. Iversio of oliear stochastic operators. J. Math. Aal. Appl. 91 (198) 9 6 [] G. Adomia. Stochastic Systems. New Yor: Academic, 198 [] G. Adomia. Noliear Stochastic Systems Theory ad Applicatios to Physics. Dordrecht: Kluwer, 1989 [] G. Adomia. Solvig Frotier Prolems of Physics: The Decompositio Method. Dordrecht: Kluwer, 199 [5] R. Rach. A ew defiitio of the Adomia polyomials. Kyeretes 7 (008) 910 955 [6] A. M. Wazwaz. Partial Differetial Equatios ad Solitary Waves Theory. Beijig: Higher Educatio, 009 [7] G. Adomia, R. Rach. Aalytic solutio of oliear oudary-value prolems i several dimesios y decompositio. J. Math. Aal. Appl. 17 (199) 118 17 [8] G. Adomia, R. Rach. A ew algorithm for matchig oudary coditios i decompositio solutios. Appl. Math. Comput. 58 (199) 61 68 [9] J. S. Dua, R. Rach. A ew modificatio of the Adomia decompositio method for solvig oudary value prolems for higher order oliear differetial equatios. Appl. Math. Comput. 18 (011) 090 118 [10] J. S. Dua. Recurrece triagle for Adomia polyomials. Appl. Math. Comput. 16 (010) 15 11 [11] J. S. Dua, R. Rach, ad Z. Wag. O the effective regio of covergece of the decompositio series solutio. J. Algorithms Comput. Tech. 7 (01) 7 7 [1] Y. Cherruault. Covergece of Adomia s method. Kyeretes 18 (1989) 1 8 [1] A. Adelrazec, D. Peliovsy. Covergece of the Adomia decompositio method for iitial-value prolems. Numer. Methods Partial Differetial Equatios 7 (011) 79 766 [1] G. Adomia, R. Rach. Modified Adomia polyomials. Math. Comput. Modellig (1996) 9 6 [15] J. S. Dua. New recurrece algorithms for the oclassic Adomia polyomials. Comput. Math. Appl. 6 (011) 961 977 [16] J. S. Dua. New ideas for decomposig oliearities i differetial equatios. Appl. Math. Comput. 18 (011) 177 178 [17] C. Li, Y. Wag. Numerical algorithm ased o Adomia decompositio for fractioal differetial equatios. Comput. Math. Appl. 57 (009) 167 1681 [18] J. S. Dua, R. Rach. New higher-order umerical oe-step methods ased o the Adomia ad the modified decompositio methods. Appl. Math. Comput. 18 (011) 810 88 [19] R. Rach, J. S. Dua. Near-field ad far-field approximatios y the Adomia ad asymptotic decompositio methods. Appl. Math. Comput. 17 (011) 5910 59 [0] X. Y. Qi, Y. P. Su, Approximate aalytical solutios for a mathematical model of a tuular paced-ed catalytic reactor usig a Adomia decompositio method. Appl. Math. Comput. 18 (011) 1990 1996 [1] R. Rach. A coveiet computatioal form for the Adomia polyomials. J. Math. Aal. Appl. 10 (198) 15 19 [] A. M. Wazwaz. A ew algorithm for calculatig Adomia polyomials for oliear operators. Appl. Math. Comput. 111 (000) 5 69 [] F. Adelwahid. A mathematical model of Adomia polyomials. Appl. Math. Comput. 11 (00) 7 5 [] W. Che, Z. Lu. A algorithm for Adomia decompositio method. Appl. Math. Comput. 159 (00) 1 5 [5] Y. Zhu, Q. Chag, ad S. Wu. A ew algorithm for calculatig Adomia polyomials. Appl. Math. Comput. 169 (005) 0 16 [6] J. Biazar, M. Ilie, ad A. Khoshear. A improvemet to a alterate algorithm for computig Adomia polyomials i special cases. Appl. Math. Comput. 17 (006) 58 59 [7] M. Azreg-Aiou. A developed ew algorithm for evaluatig Adomia polyomials. CMES-Comput. Model. Eg. Sci. (009) 1 18 [8] J. S. Dua, A. P. Guo. Reduced polyomials ad their geeratio i Adomia decompositio methods. CMES-Comput. Model. Eg. Sci. 60 (010) 19 150-5 -
[9] J. S. Dua. A efficiet algorithm for the multivariale Adomia polyomials. Appl. Math. Comput. 17 (010) 56 67 [0] J. S. Dua. Coveiet aalytic recurrece algorithms for the Adomia polyomials. Appl. Math. Comput. 17 (011) 67 68 [1] M. R. Scott, W. H. Vadeveder. A compariso of several ivariat imeddig algorithms for the solutio of two-poit oudary-value prolems. Appl. Math. Comput. 1 (1975) 187 18 AUTHOR Jusheg Dua is a professor i the School of Scieces at the Shaghai Istitute of Techology i Shaghai, PR Chia. He was or i Hohhot, Ier Mogolia, PR Chia, i 1965. He received his BS degree ad MS degree i Mathematics from Ier Mogolia Uiversity, i 198 ad 1996, respectively, ad received his Ph. D. degree i Mathematics from Shadog Uiversity i 00. He has oth authored ad co-authored 90 papers i applied ad computatioal mathematics ad has cotriuted otaly to theoretical advaces i the oliear aalysis, icludig the aalytical approximate solutios with high accuracy ad the high-ordered umerical solutios for the oliear differetial equatios ad the fractioal differetial equatios. - 6 -