Periodic Solutions of Strongly Nonlinear Oscillators Using He's Frequency Formulation

In this paper, we address several scientific and technological challenges with a novel He's non-perturbative approach (NPA), it simplifying processing time compared to traditional methods. The proposed approach transforms nonlinear ordinary differential equations (ODEs) into linear ones, analogous to simple harmonic motion, and producing a new frequency. Studying the periodic solutions leads to enhanced design, performance, reliability, and efficiency across these fields. This new approach is based mainly on the He's frequency formulation (HFF). This method yields highly accurate outcomes, surpassing well-known approximate methodologies, as validated through numerical comparisons in the Mathematical Software (MS). The congruence between numerical solution tests and theoretical predictions further supports our findings. While classical perturbation methods rely on Taylor expansions to simplify restoring forces, the NPA also enables stability analysis. Consequently, for analyzing approximations of highly non-linear oscillators in MS, the NPA serves as a more reliable tool.