Abstract
In the present work, a new method for solving a strong nonlinear oscillator equation of the form ẍ + F(x) = 0, where F(−x) = −F(x), is carried out. This method consists of approximating function F(x) by means of a suitable Chebyshev polynomial: F(x) ≈ P(x) = px + qx3 + rx5, and then, the original oscillator is replaced by the cubic–quintic Duffing equation ẍ + px + qx3 + rx5 = 0 with arbitrary initial conditions, which admits the exact solution in terms of elliptic functions. The efficacy of the present method is demonstrated through the fluid multi-ion plasma equations and a generalized pendulum problem. For the generalized pendulum problem, the governing motion is directly reduced to the cubic–quintic Duffing oscillator with the help of the Chebyshev polynomial, and the approximate analytical and exact solutions are obtained. In addition, the comparison between our solutions and the Runge–Kutta numerical solution is examined. Moreover, the periodic time formula of the oscillations for both the approximate analytical solution and the exact solution is deduced, and the comparison between them is implemented. With respect to the plasma application, the fluid plasma equations of its particles are reduced to the Extended Korteweg–de Vries (EKdV) equation utilizing a reductive perturbation method. Then, we proved for the first time that any undamped polynomial oscillator of the nth degree can be reduced to a (2n − 1)th odd parity Duffing. Accordingly and after applying the previous theory to the EKdV equation, it was converted to the cubic–quintic Duffing equation. Finally, we can deduce that our new solutions and theory help us to understand and investigate many nonlinear phenomena in various branches of science.