An innovative efficient approach to solving damped Mathieu-Duffing equation with the non-perturbative technique

The effectiveness of the non-perturbative technique is further proven in this paper for the parametric nonlinear oscillatory system. The method's efficiency and practicality in obtaining the frequency amplitude of parametric nonlinear issues have been effectively shown. The obtained approximate solutions are not based on a series expansion. Our interest in this work is to begin to obtain approximate solutions without restraints to a small amplitude of the parametric coefficients far from the perturbation methods. As is explicitly shown, the accuracy obtained is independent of the value of the parametric coefficients. In addition, the analysis is extended to establish accurate solutions for the large amplitude of nonlinear oscillation. The most important property is a quick estimate of the frequency-amplitude relationship to obtain successive approximate solutions for the parametric nonlinear oscillation. The stability behavior and the bifurcation are explored. It has been proven that the current technique is quite accurate. The method is uncomplicated in its fundamentals, simple to apply, appropriate, and provides very good numerical accuracy. It is also useful as a mathematical tool for dealing with nonlinear parametric problems because it avoids any mathematical intricacy.