[Paper Review] Perturbation approaches and Taylor series

Francisco M. Fernández|ArXiv.org|Oct 1, 2009

Fractional Differential Equations Solutions7 references3 citations

TL;DR

This paper critiques the recent trend in mathematical physics of applying advanced perturbation methods—such as homotopy perturbation (HPM), homotopy analysis (HAM), and Adomian decomposition (ADM)—to unphysical models with known exact solutions. It demonstrates that these methods redundantly reproduce textbook Taylor series expansions, and shows that the same results are obtained more directly and exactly via standard Taylor series methods, exposing the superfluous complexity of the newer approaches.

ABSTRACT

We comment on the new trend in mathematical physics that consists of obtaining Taylor series for fabricated linear and nonlinear unphysical models by means of homotopy perturbation method (HPM), homotopy analysis method (HAM) and Adomian decomposition method (ADM). As an illustrative example we choose a recent application of the HPM to a dynamic system of anisotropic elasticity.

Motivation & Objective

• To challenge the perceived novelty and utility of recent applications of HPM, HAM, and ADM to fabricated unphysical models.
• To demonstrate that the solutions obtained via these advanced perturbation methods are equivalent to straightforward Taylor series expansions.
• To highlight that the use of complex methods for problems with known exact solutions undermines methodological rigor and efficiency.
• To expose the lack of physical relevance in models that yield unbounded solutions over time.
• To argue that the trend reflects a shift toward methodological complexity over conceptual or physical insight.

Proposed method

• The paper applies standard Taylor series expansion to the same dynamic system of anisotropic elasticity studied by Koçak and Yıldırım using HPM.
• It derives a recursive formula for the coefficients of the Taylor series: $ \mathbf{u}_{j+2} = \frac{1}{(j+1)(j+2)} \mathbf{\rho}^{-1}(\hat{L}\mathbf{u}_j + \mathbf{f}_j) $, which generates the solution term by term.
• The method is applied to the illustrative example from Koçak and Yıldırım’s paper, where $ \mathbf{u}_0 = \mathbf{f}_0 = 0 $, $ \hat{L}\mathbf{u}_1 = -\mathbf{f}_1 $, leading to $ \mathbf{u}(\mathbf{x},t) = t\mathbf{u}_1(\mathbf{x}) $.
• The paper compares the HPM results from the cited work with the exact Taylor series solution, revealing inconsistencies in the HPM implementation.
• It identifies a specific error in the first-order correction of the HPM, where $ u^{(1)} = \Theta(t)t\delta(x) $ fails to satisfy the differential equation $ \partial^2 u^{(1)}/\partial t^2 = \delta(t)\delta(x) $, due to a missing factor of two.
• The analysis concludes that the standard Taylor series method provides exact, closed-form solutions without the need for iterative perturbation schemes.

Experimental results

Research questions

• RQ1 Do HPM, HAM, and ADM provide novel or more efficient solutions for problems with known exact Taylor series expansions?
• RQ2 Why do researchers apply complex perturbation methods to unphysical models with trivial exact solutions?
• RQ3 What is the source of the discrepancy between HPM results and the exact Taylor series solution in the case of Koçak and Yıldırım’s model?
• RQ4 How does the use of unphysical models with unbounded growth affect the perceived validity of these methods?
• RQ5 Why is there a growing trend in publishing such methods despite their redundancy and methodological flaws?

Key findings

• The Taylor series approach reproduces all results from the HPM application in Koçak and Yıldırım’s paper exactly and more directly.
• The HPM solution for the illustrative example yields $ \mathbf{u}(\mathbf{x},t) = t\mathbf{u}_1(\mathbf{x}) $, which is identical to the exact Taylor series solution.
• A critical error in the HPM implementation—specifically, a missing factor of two in the first-order correction—prevents the method from recovering the exact solution even at second order.
• The paper confirms that the HPM results are mathematically equivalent to Taylor series expansions, but the implementation is flawed, leading to incorrect intermediate terms.
• The models studied, such as those with unbounded solutions over time, lack physical relevance and do not represent realistic physical systems.
• The broader trend of applying advanced methods to trivial problems with known solutions undermines methodological rigor and suggests a decline in scientific novelty and timeliness.