Open Access

ARTICLE

A New Modified Adomian Decomposition Method for Higher-Order Nonlinear Dynamical Systems

Jun-Sheng Duan^1,2, Randolph Rach³, Abdul-Majid Wazwaz⁴

1 School of Sciences, Shanghai Institute of Technology, Shanghai 201418, PR China.
Corresponding author. Email: duanjssdu@sina.com; duanjs@sit.edu.cn
2 School of Mathematics and Information Sciences, Zhaoqing University, Zhaoqing, Guang Dong 526061, P.R. China
3 316 S. Maple St., Hartford, MI 49057-1225, USA. Email: tapstrike@gmail.com
4 Department of Mathematics, Saint Xavier University, Chicago,IL 60655, USA. Email: wazwaz@sxu.edu

Computer Modeling in Engineering & Sciences 2013, 94(1), 77-118. https://doi.org/10.3970/cmes.2013.094.077

Abstract

In this paper, we propose a new modification of the Adomian decomposition method for solution of higher-order nonlinear initial value problems with variable system coefficients and solutions of systems of coupled nonlinear initial value problems. We consider various algorithms for the Adomian decomposition series and the series of Adomian polynomials to calculate the solutions of canonical first- and second-order nonlinear initial value problems in order to derive a systematic algorithm for the general case of higher-order nonlinear initial value problems and systems of coupled higher-order nonlinear initial value problems. Our new modified recursion scheme is designed to decelerate the Adomian decomposition series so as to always calculate the solution’s Taylor expansion series using easy-to-integrate terms. The corresponding nonlinear recurrence relations for the solution coefficients are deduced. Next we consider convergence acceleration and error analysis for the sequence of solution approximations. Multistage decomposition and numeric algorithms are designed and we debut efficient MATHEMATICA routines PSSOL and NSOL that implement our new algorithms. Finally we investigate several expository examples in order to demonstrate the rapid convergence of our new approach..

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Keywords

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