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Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method

Yusry O. El-Dib, Nasser S. Elgazery*
Department of Mathematics, Faculty of Education, Ain Shams University, Roxy, Cairo, Egypt
* Corresponding Author: Nasser S. Elgazery. Email:
(This article belongs to this Special Issue: Analytical Methods for Nonlinear Vibration & Active Control in Nano/Micro Devices and Systems)

Sound & Vibration 2022, 56(1), 21-36.

Received 06 September 2020; Accepted 25 June 2021; Issue published 10 January 2022


In this article, the main objective is to employ the homotopy perturbation method (HPM) as an alternative to classical perturbation methods for solving nonlinear equations having periodic coefficients. As a simple example, the nonlinear damping Mathieu equation has been investigated. In this investigation, two nonlinear solvability conditions are imposed. One of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance cases. The next level of the perturbation approaches another solvability condition and is applied to obtain the unknowns become clear in the solution for the first-order solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.


Damped Mathieu Equation; parametric nonlinear oscillator; resonance instability; homotopy perturbation method (HPM)

Cite This Article

El-Dib, Y. O., Elgazery, N. S. (2022). Damped Mathieu Equation with a Modulation Property of the Homotopy Perturbation Method. Sound & Vibration, 56(1), 21–36.


This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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