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