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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.


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.


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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