An efficient method for solving strongly nonlinear oscillators: combination of the multi-step homotopy analysis and spectral method

In this study, a new efficient semi-analytical method is introduced to give approximate solutions of strongly nonlinear oscillators. The proposed method is based on combination of two different methods, the multi-step homotopy analysis method and spectral method, called the multi-step spectral homotopy analysis method (MSHAM). In this method, firstly, we propose a new spectral homotopy analysis method and then, we apply it on smaller subintervals and join the resulting solutions that obtained from new spectral homotopy analysis method at the end points of the subintervals. Several significant strongly nonlinear oscillators are tested with the new scheme to demonstrate its accuracy and easy implementation.