The essence of the homotopy analysis method

TL;DR

This paper clarifies the core mechanism of the homotopy analysis method by linking its auxiliary parameter to Taylor series expansions at different points, enhancing understanding of its convergence control.

Contribution

It reveals the true nature of the auxiliary parameter in the homotopy analysis method, showing it corresponds to Taylor expansion points, thus solving a key conceptual mystery.

Findings

The generalized Taylor expansion is equivalent to a Taylor series at a different point.

A relationship between the auxiliary parameter and the expansion point is established.

The method's results are validated through the Blasius equation example.

Abstract

The generalized Taylor expansion including a secret auxiliary parameter $h$ which can control and adjust the convergence region of the series is the foundation of the homotopy analysis method proposed by Liao. The secret of $h$ can't be understood in the frame of the homotopy analysis method. This is a serious shortcoming of Liao's method. We solve the problem. Through a detailed study of a simple example, we show that the generalized Taylor expansion is just the usual Taylor's expansion at different point $t_1$. We prove that there is a relationship between $h$ and $t_1$, which reveals the meaning of $h$ and the essence of the homotopy analysis method. As an important example, we study the series solution of the Blasius equation. Using the series expansion method at different points, we obtain the same result with liao's solution given by the homotopy analysis method.