Study of autonomous conservative oscillator using an improved perturbation method

TL;DR

This paper introduces an improved homotopy perturbation method (LH) that enhances the accuracy of analytical solutions for autonomous conservative oscillators by incorporating frequency and auxiliary parameter expansions, outperforming previous methods.

Contribution

The paper develops an improved homotopy perturbation method with frequency and auxiliary parameter expansion, providing more accurate solutions for autonomous conservative oscillators.

Findings

LH method yields frequency and displacement with errors one or two orders lower than AT.

Laplace transform simplifies calculations in the improved method.

LH provides highly accurate results compared to existing approximation methods.

Abstract

In a recent article \cite{manimegalai2019}, Aboodh transform based homotopy perturbation method ($AT$) has been found to produce approximate analytical solutions in a simple way but with better accuracy in comparison to those obtained from some of the established approximation methods \cite{mehdipour2010application,nofal2013analytical} for some physically relevant anharmonic oscillators such as autonomous conservative oscillator (ACO). In the present article, expansion of frequency ($\omega$) and an auxiliary parameter ($h$) are incorporated in the framework of the homotopy perturbation method (HPM) to improve the accuracy by retaining its simplicity. Laplace transform is used to make the calculation simpler. This improved HPM ($LH$) is simple but provides highly accurate results for ACO in comparison to those obtained from $AT$. The error in the values of frequency and displacement calculated using the $LH$ is found to be one or two order of magnitude less than those obtained from $AT$ for the considered parameter sets.