Abstract
Novel analytical and numerical solutions to the (un)damped Helmholtz-Duffing (H-D) equation for arbitrary initial conditions are derived. Both the analytical (for undamped case) and approximate analytical (for damped case) solutions, are obtained in the form of Weierstrass elliptic function. Also, the soliton solution to the undamped H-D equation is obtained in detail. The semi-analytical solution to the damped H-D equation is compared to the fourth-order Runge-Kutta (RK4) numerical solution. The obtained solution shows an excellent agreement with the numerical simulations but sometimes (according to the values of initial conditions) not on all time interval. Thus, the moving boundary method is utilized to improve the semi-analytical solution. It is found that the improved solution gives good results with high accuracy in the whole time domain. As a realistic application, the obtained solutions are applied to the study of nonlinear oscillations in an electronegative non-Maxwellian dusty plasma. Finally, we conclude that our novel solutions could help us to understand the dynamics of various nonlinear oscillations in engineering and in different branches of sciences such as oscillations in different plasma models.