The generalized Taylor series approach is not equivalent to the homotopy analysis method

TL;DR

This paper clarifies that the generalized Taylor series approach is only equivalent to the homotopy analysis method in specific cases, emphasizing the broader applicability and robustness of the homotopy analysis method with non-polynomial basis functions.

Contribution

The paper demonstrates that the generalized Taylor series is not generally equivalent to the homotopy analysis method and highlights the latter's ability to handle non-analytic solutions and global approximations.

Findings

Generalized Taylor series is limited to local solutions with finite convergence radius.

Homotopy analysis method can approximate solutions over infinite domains.

Non-polynomial basis functions enhance the robustness of homotopy solutions.

Abstract

In recent work on the area of approximation methods for the solution of nonlinear differential equations, it has been suggested that the so-called generalized Taylor series approach is equivalent to the homotopy analysis method. In the present paper, we demonstrate that such a view is only valid in very special cases, and in general the homotopy analysis method is far more robust. In particular, the equivalence is only valid when the solution is represented as a power series in the independent variable. As has been shown many times, alternative basis functions can greatly improve the error properties of homotopy solutions, and when the base functions are not polynomials or power functions, we no longer have that the generalized Taylor series approach is equivalent to the homotopy analysis method. We demonstrate this by consideration of an example where the generalizes Taylor series must always have a finite radius of convergence (and hence limited applicability), while the homotopy solution is valid over the entire infinite domain. We then give a second example for which the exact solution is not analytic, and hence it will not agree with the generalized Taylor series over the domain. We conclude that the generalized Taylor series approach is not equivalent to the homotopy analysis method. In particular, the generalized Taylor series can at best recover local information about solutions, whereas the homotopy analysis method can recover the solutions globally if appropriate base functions are selected. Such results have important implications for how series are calculated when approximating solutions to nonlinear differential equations.