Table of Content

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
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.

Keywords

Variational iteration method, Adomian decomposition method, Picard iteration method, asymptotic technique, nonlinear dynamical system.

Cite This Article

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. CMES-Computer Modeling in Engineering & Sciences, 111(6), 567–585.



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.
  • 807

    View

  • 741

    Download

  • 0

    Like

Share Link

WeChat scan