A new analytic method with a convergence-control parameter for solving nonlinear problems

TL;DR

This paper introduces a novel analytic method with a convergence-control parameter that enhances the convergence region and rate for solving nonlinear problems, demonstrated through various physical models.

Contribution

The paper proposes a new analytic method with a convergence-control parameter, generalizing the Adomian decomposition method and improving convergence control for nonlinear problems.

Findings

Enlarged convergence region and rate with proper parameter choice

Effective application to diverse physical models

Potential to improve existing analytic and numerical techniques

Abstract

In this paper, a new analytic method with a convergence-control parameter $c$ is first proposed. The parameter $c$ is used to adjust and control the convergence region and rate of the resulting series solution. It turns out that the convergence region and rate can be greatly enlarged by choosing a proper value of $c$. Furthermore, a numerical approach for finding the optimal value of the convergence-control parameter is given. At the same time, it is found that the traditional Adomian decomposition method is only a special case of the new method. The effectiveness and applicability of the new technique are demonstrated by several physical models including nonlinear heat transfer problems, nano-electromechanical systems, diffusion and dissipation phenomena, and dispersive waves. Moreover, the ideas proposed in this paper may offer us possibilities to greatly improve current analytic and numerical techniques.