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A Maple package of new ADM-Padé approximate solution for nonlinear problems. (English) Zbl 1216.65085

Summary: Based on the new definition of four classes of Adomian polynomials proposed by R. C. Rach [Kybernetes 37, No. 7, 910–955 (2008; Zbl 1176.33023)], a Maple package for a new Adomian-Padé approximate method for solving nonlinear problems is presented. This package combines the merits of the Adomian decomposition method and the diagonal Padé technique, and can give more accurate solutions of nonlinear problems with strong nonlinearity. Besides, the package is user-friendly and efficient; one only needs to input the initial conditions, governing equation and four optional parameters, then our package will output the analytic approximate solution within a few seconds. The equation is decomposed into three parts, namely, the linear term \(R\), the nonlinear term \(NN\) and the source function \(g\), which are all in functional form. Several graphs generated from the above solutions are displayed and demonstrate a favorable comparison. In this paper, several different types of examples are given to illustrate the validity and promising flexibility of the package. This package provides us with a convenient and useful tool for dealing with nonlinear problems; also, its electronic version is free to download over the journal website.

MSC:

65L05 Numerical methods for initial value problems involving ordinary differential equations

Citations:

Zbl 1176.33023

Software:

Maple; NAPA
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Adomian, G., Solving Frontier Problems of Physics: The Decomposition Method (1994), Kluwer Academic: Kluwer Academic Dordrecht · Zbl 0802.65122
[2] Adomian, G., Solution of the Thomas-Fermi equation, Appl. Math. Lett., 11, 131-133 (1998) · Zbl 0947.34501
[3] Adomian, G., Nonlinear Stochastic Operator Equations (1986), Academic: Academic Orlando · Zbl 0614.35013
[4] Adomian, G., Stochastic Systems (1983), Academic Press: Academic Press New York · Zbl 0504.60067
[5] Adomian, G., Nonlinear Stochastic Systems Theory and Applications to Physics (1989), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0659.93003
[6] Rach, R., A new definition of the Adomian polynomials, Kybernetes, 37, 910-955 (2008) · Zbl 1176.33023
[7] Chen, W.; Lu, Z., An algorithm for Adomian decomposition method, Appl. Math. Comput., 159, 221-235 (2004) · Zbl 1062.65059
[8] Babolian, E.; Javadi, Sh., New method for calculating Adomian polynomials, Appl. Math. Comput., 153, 253-259 (2004) · Zbl 1055.65068
[9] Gu, H.; Li, Z., A modified Adomian method for system of nonlinear differential equations, Appl. Math. Comput., 187, 748-755 (2007) · Zbl 1121.65082
[10] Duan, J., Recurrence triangle for Adomian polynomials, Appl. Math. Comput., 216, 1235-1241 (2010) · Zbl 1190.65031
[11] Cherruault, Y.; Adomian, G., Decomposition method: a new proof of convergence, Math. Comput. Model., 18, 103-106 (1993) · Zbl 0805.65057
[12] Boyd, J., Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain, Comput. Phys., 11, 299-303 (1997)
[13] Yang, P.; Chen, Y.; Li, Z., Adomian decomposition method and Padé approximants for solving the Blaszak-Marciniak lattice, Chin. Phys. B, 17, 3953-3964 (2008)
[14] Chu, H.; Liu, Y., The new ADM-Padé technique for the generalized Emden-Fowler equations, Modern Phys. Lett. B, 24, 1237-1254 (2010) · Zbl 1407.65073
[15] Adomian, G.; Rach, R., Modified Adomian polynomials, Math. Comput. Model., 24, 39-46 (1996) · Zbl 0874.65051
[16] Rach, R., A convenient computational form for the Adomian polynomials, J. Math. Anal. Appl, 102, 415-419 (1984) · Zbl 0552.60061
[17] Wazwaz, A. M., Adomian decomposition method for a reliable treatment of the Emden-Fowler equation, Appl. Math. Comput., 161, 543-560 (2005) · Zbl 1061.65064
[18] Saha Ray, S.; Bera, R. K., An approximate solution of a nonlinear fractional differential equation by Adomian decomposition method, Appl. Math. Comput., 167, 561-571 (2005) · Zbl 1082.65562
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