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

Abstract

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 method. An effective procedure has been introduced for executing the fractional-order by utilizing a new method without the perturbative approach. The approach depends on converting the fractional nonlinear oscillator to a linear oscillator with an integer order. The sincerity of the established results is justified by providing a suitable example of nonlinear oscillators. The obtained analytical solutions are supported by the numerical simulation results. The method applied here does not require hard mathematical work, which makes it distinct from other methods. The simplicity of the present approach provides extra advantages for fractional-order solutions. This method enriches the analysis with more details in new dimensions.

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

“…Recently, for an electromechanical system in nano-scale, He et al 48 have extended He’s frequency formula for a faster calculation of the pull voltage. 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…”

Successive approximate solutions for nonlinear oscillation and improvement of the solution accuracy with efficient non-perturbative technique

“…Inserting equation (24) into equations ( 22) or (23), the least square of the displacement is found to be…”

“…22 He's frequency formulation has been modified to cover the fractional nonlinear oscillator as given in Ref. 23. Also, this approach was applied to the parametric Gaylord's oscillator with a discussion of the resonance response with the non-perturbative approach.…”

Properties of complex damping Helmholtz–Duffing oscillator arising in fluid mechanics

“…34 Further, He's frequency formula has successful established the frequency for fractional nonlinear oscillation. 35,36 Based on the non-perturbative method, we present an improved deep approach to solving complex oscillations. The present approach can lead to obtaining a remarkably rigorous solution.…”

“…34 Further, He’s frequency formula has successful established the frequency for fractional nonlinear oscillation. 35,36…”