Some Novel Analytical Approximations to the (Un)damped Duffing–Mathieu Oscillators

Some novel exact solutions and approximations to the damped Duffing–Mathieu-type oscillator with cubic nonlinearity are obtained. This work is divided into two parts: in the first part, some exact solutions to both damped and undamped Mathieu oscillators are obtained. These solutions are expressed in terms of the Mathieu functions of the first kind. In the second part, the equation of motion to the damped Duffing–Mathieu equation (dDME) is solved using some effective and highly accurate approaches. In the first approach, the nonintegrable dDME with cubic nonlinearity is reduced to the integrable dDME with linear term having undermined optimal parameter (maybe called reduced method). Using a suitable technique, we can determine the value of the optimal parameter and then an analytical approximation is obtained in terms of the Mathieu functions. In the second approach, the ansatz method is employed for deriving an analytical approximation in terms of trigonometric functions. In the third approach, the homotopy perturbation technique with the extended Krylov–Bogoliubov–Mitropolskii (HKBM) method is applied to find an analytical approximation to the dDME. Furthermore, the dDME is solved numerically using the Runge–Kutta (RK) numerical method. The comparison between the analytical and numerical approximations is carried out. All obtained approximations can help a large number of researchers interested in studying the nonlinear oscillations and waves in plasma physics and many other fields because many evolution equations related to the nonlinear waves and oscillations in a plasma can be reduced to the family of Mathieu-type equation, Duffing-type equation, etc.