Comparison of a general series expansion method and the homotopy analysis method

TL;DR

This paper introduces a simple general series expansion method for solving nonlinear differential equations, demonstrating its equivalence to the homotopy analysis method and clarifying the role of its parameters.

Contribution

The paper proposes a straightforward series expansion approach that simplifies understanding and relates directly to the homotopy analysis method, revealing the significance of its parameters.

Findings

The new method is simpler and clearer than the homotopy analysis method.

It explains the role of the homotopy parameter h using the parameter t_0.

A detailed example demonstrates the method's effectiveness.

Abstract

A simple analytic tool namely the general series expansion method is proposed to find the solutions for nonlinear differential equations. By choosing a set of suitable basis functions $\{e_n(t,t_0)\}_{n=0}^{+\infty}$ such that the solution to the equation can be expressed by $u(t)=\sum_{n=0}^{+\infty}c_ne_n(t,t_0)$. In general, $t_0$ can control and adjust the convergence region of the series solution such that our method has the same effect as the homotopy analysis method proposed by Liao, but our method is more simple and clear. As a result, we show that the secret parameter $h$ in the homotopy analysis methods can be explained by using our parameter $t_0$. Therefore, our method reveals a key secret in the homotopy analysis method. For the purpose of comparison with the homotopy analysis method, a typical example is studied in detail.