Time Dependent Harmonic Oscillator via OM-HPM

In this study, we present a semi-analytical technique known as the Optimal and Modified Homotopy Perturbation Method (OM-HPM) for solving nonlinear oscillators with time-dependent mass. The work extends existing approaches, including the standard Homotopy Perturbation Method (HPM), by introducing an auxiliary linear operator that minimizes residual error and enhances the method’s efficiency for both singular and non-singular nonlinear ordinary differential equations. The model of a harmonic oscillator with exponentially decaying mass is investigated using this method, and its equation of motion is derived using the Lagrangian formulation. The OM-HPM technique is applied to solve the resulting second-order nonlinear differential equation, and solutions are presented in series form. The method significantly reduces computational cost through the use of Newton-Cotes quadrature. Analytical illustrations demonstrate that the effectiveness of OM-HPM in solving complex nonlinear oscillatory systems.