On the approximate solutions to a damped harmonic oscillator with higher-order nonlinearities and its application to plasma physics: semi-analytical solution and moving boundary method

Abstract

In this paper, a strongly nonlinear oscillator equation with higher-order nonlinear (cubic) restoring force, namely damped Duffing equation with higher-order nonlinearities, has been solved and analyzed analytically and numerically using some effectiveness and high-accuracy methods. Firstly, the physical oscillator system consisting of a spring–mass moves under cubic resorting force with constant spring is considered for getting damped Duffing equation. After that a semi-analytical solution is obtained in terms of the Jacobian elliptic function cn. Also, the numerical approximate solution using Runge–Kutta-4th (RK4) procedure is carried out and compared to the semi-analytical solution. It is found that the agreement between both solutions is very good especially if the coefficient of the cubic nonlinear term becomes larger than the coefficient of the linear one. However, we made some improvements to the semi-analytical solution using the numerical moving boundary method to get higher accuracy solution. Furthermore, the error analysis is estimated for both the semi-analytical solution, improved semi-analytical solution, and RK4 solution. The method that is used for getting the semi-analytical solution is elementary, i.e., allows to obtain only integrable case without using any Lie group theory. In another meaning, our approach gives exact/analytical solution for the integrable case such as undamped Duffing equation. Moreover, the t-intercepts and amplitude points are estimated precisely. Furthermore, the plasma oscillations are discussed in the framework of damped Duffing equation. Firstly, we reduced the fluid basic equations of pair-ion plasmas with Maxwellian electrons using reductive perturbation technique to a modified korteweg de-Vries Burgers (mKdVB) equation, and by using traveling wave transformation the mKdVB equation is converted to the damped Duffing equation. After that we compared between all mentioned solutions using plasma parameters.