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

Sound & Vibration 2022年第1期56卷 21-36页

作者： yusry o.el-dib Nasser S.elgazery Department of Mathematics Faculty of EducationAin Shams UniversityRoxyCairoEgypt

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 *** a simple example,the nonlinear damping Mathieu equation has been *** this investigation,two nonlinear solvability conditions are *** of them was imposed in the first-order homotopy perturbation and used to study the stability behavior at resonance and non-resonance *** 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 *** approach assumed here is so significant for solving many parametric nonlinear equations that arise within the engineering and nonlinear science.

关键词： Damped Mathieu Equation parametric nonlinear oscillator resonance instability homotopy perturbation method(HPM)

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Stability analysis of a time-delayed Van der Pol–Helmholtz–Duffing oscillator in fractal space with a non-perturbative approach

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Communications in Theoretical Physics 2024年第4期76卷 22-31页

作者： yusry o.el-dib Department of Mathematics Faculty of EducationAin Shams UniversityRoxyCairoEgypt

The time-delayed fractal Van der Pol–Helmholtz–Duffing(VPHD)oscillator is the subject of this paper,which explores its mechanisms and highlights its stability *** time-delayed technologies are currently garnering significant attention,the focus of this research remains crucially relevant.A non-perturbative approach is employed to refine and set the stage for the system under *** innovative methodologies introduced yield an equivalent linear differential equation,mirroring the inherent nonlinearities of the ***,the incorporation of quadratic nonlinearity into the frequency formula represents a cutting-edge *** analytical solution's validity is corroborated using a numerical *** conditions are ascertained through the residual Galerkin ***,it is observed that the delay parameter,in the context of the fractal system,reverses its stabilizing influence,impacting both the amplitude of delayed velocity and the *** analytical solution's precision is underscored by its close alignment with numerical ***,the study reveals that fractal characteristics emulate damping *** its applicability across diverse nonlinear dynamical systems,this non-perturbative approach emerges as a promising avenue for future research.

关键词： Van der Pol-Helmholtz-Duffing oscillator fractal space time-delay non-perturbative approach stability analysis el-dib's frequency formula

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An efficient approach to solving fractional Van der Pol-Duffing jerk oscillator

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Communications in Theoretical Physics 2022年第10期74卷 50-59页

作者： yusry o.el-dib Department of Mathematics Faculty of EducationAin Shams UniversityRoxyCairoEgypt

The motive behind the current work is to perform the solution of the Van der Pol–Duffing jerk oscillator,involving fractional-order by the simplest *** effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative *** approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order.A detailed solving process is given for the obtained oscillator with the traditional system.

关键词： Van der Pol oscillator Duffing jerk oscillator fractional oscillator non-perturbative approach numerical simulation

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An innovative technique to solve a fractal damping Duffing-jerk oscillator

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Communications in Theoretical Physics 2023年第5期75卷 3-11页

作者： yusry o.el-dib Nasser S elgazery Youmna M Khattab Haifa A Alyousef Department of Mathematics Faculty of EducationAin Shams UniversityRoxyCairoEgypt Department of Physics College of SciencePrincess Nourah bint Abdulrahman UniversityPO Box 84428Riyadh 11671Saudi Arabia

The idea of the present article is to look into the nonlinear dynamics and vibration of a damping Duffing-jerk oscillator in fractal space exhibiting the non-perturbative *** a new analytical technique,namely,the modification of a He’s fractal derivative that converts the fractal derivative to the traditional derivative in continuous space,this study provides an effective and easy-to-apply procedure that is dependent on the He’s fractal derivative *** analytic approximate solution has a significant match with the results of the numerical simulation as the fractal parameter is very closer to unity,which proves the reliability of the *** behavior is discussed and illustrated *** new powerful analytical tools are developed in an attempt to obtain effective analytical tools,valid for any fractal nonlinear problems.

关键词： modification of He’s fractal derivative fractal duffing-jerk vibration stability analysis non-perturbative approach damping nonlinear fractal system

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