Open Access

ARTICLE

A Unification of the Concepts of the Variational Iteration, Adomian Decomposition and Picard Iteration Methods; and a Local Variational Iteration Method

Xuechuan Wang¹, Satya N. Atluri²

1 Texas Tech University; Center for Advanced Research in the Engineering Sciences; Visiting Scholar from Northwestern Polytechnical University, China.
2 Texas Tech University; Center for Advanced Research in the Engineering Sciences.

Computer Modeling in Engineering & Sciences 2016, 111(6), 567-585. https://doi.org/10.3970/cmes.2016.111.567

Abstract

This paper compares the variational iteration method (VIM), the Adomian decomposition method (ADM) and the Picard iteration method (PIM) for solving a system of first order nonlinear ordinary differential equations (ODEs). A unification of the concepts underlying these three methods is attempted by considering a very general iterative algorithm for VIM. It is found that all the three methods can be regarded as special cases of using a very general matrix of Lagrange multipliers in the iterative algorithm of VIM. The global variational iteration method is briefly reviewed, and further recast into a Local VIM, which is much more convenient and capable of predicting long term complex dynamic responses of nonlinear systems even if they are chaotic.

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Keywords

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