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

Abstract

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 approach. Using a new analytical technique, namely, the modification of He's a 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 He's fractal derivative approach. The analytic approximate solution has significant match with the results of the numerical simulation as the fractal parameter is very closer to unity, which proves the reliability of the method. Stability behavior is discussed and illustrated graphically. These new powerful analytical tools are developed in an attempt to obtain effective analytical tools, valid for any fractal nonlinear problems.

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

“…The fractal operator obeys the modification of He’s definition of the fractal derivative 51 . This modification was established by El-Dib and Elgazery 56,57 and El-Dib et al 58 and was to transform the oscillator in the fractal space into the continuous space. It has the formwhere the parameter S represents the porosity of the medium and depends on the fractal order α.…”