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

Abstract

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 firstorder solvability condition. The approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

A novel technique to obtain a time-delayed vibration control analytical solution with simulation of He’s formula

“…El-Dib 49 has also suggested an efficient approach for solving a VDP-Duffing jerk oscillator with fractional derivatives. On the other hand, there are important contributions in the literature for both controlling and analytical solutions of nonlinear systems ; for more details , see.Ref 50–57…”

An instrumental insight for a periodic solution of a fractal Mathieu–Duffing equation

“…As mentioned before, the first-order periodic solution in the sub-harmonic resonance case V ≈ 2ω 0 is carried out by using ( 26) and (31) into the expansion (18), letting ρ → 1, to get…”

“…7 The stability zone analysis of the damped Mathieu model was studied by many researchers, such as Taylor and Narendra, 8 Gunderson et al, 9 Turyn, 10 Pradeep and Shrivastava, 11 Allievi and Soudack, 12 and Barrios et al 13 The same study was also carried out in the presence of a time delay effect in Refs. [14][15][16][17] Moreover, El-Dib and Elgazery 18 proposed an analytical solution for the damped Mathieu model by using a modulating homotopy perturbation technique. Recently, Li-He's method developed as a novel modified homotopy perturbation technique to deal with this type of problems.…”

On the oscillations in a nonextensive complex plasma by improved differential transformation method: An application to a damped Duffing equation

“…Also, modified homotopy perturbation method was also used successfully to solve damped third-order oscillator that exhibits the nonlinearity of the Duffing equation. 40,41,42 Thus, in future work, we can use these techniques for studying the proprieties of nonlinear oscillations in different plasma models.…”