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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

Texas Tech University; Center for Advanced Research in the Engineering Sciences; Visiting Scholar from Northwestern Polytechnical University, China.
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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APA Style
Wang, X., Atluri, S.N. (2016). A unification of the concepts of the variational iteration, adomian decomposition and picard iteration methods; and a local variational iteration method. Computer Modeling in Engineering & Sciences, 111(6), 567-585. https://doi.org/10.3970/cmes.2016.111.567
Vancouver Style
Wang X, Atluri SN. A unification of the concepts of the variational iteration, adomian decomposition and picard iteration methods; and a local variational iteration method. Comput Model Eng Sci. 2016;111(6):567-585 https://doi.org/10.3970/cmes.2016.111.567
IEEE Style
X. Wang and S.N. Atluri, “A Unification of the Concepts of the Variational Iteration, Adomian Decomposition and Picard Iteration Methods; and a Local Variational Iteration Method,” Comput. Model. Eng. Sci., vol. 111, no. 6, pp. 567-585, 2016. https://doi.org/10.3970/cmes.2016.111.567



cc Copyright © 2016 The Author(s). Published by Tech Science Press.
This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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