The multi-step homotopy analysis method: A powerful scheme for handling non-linear oscillators

This paper presents approximate analytical solutions for nonlinear oscillators using the multi-step homotopy analysis method (MSHAM). The proposed scheme is only a simple modification of the homotopy analysis method, in which it is treated as an algorithm in a sequence of small intervals (i.e. time step) for finding accurate approximate solutions to the corresponding problems. Several illustrative examples are given to demonstrate the effectiveness of the present method. Figurative comparisons between the MSHAM and the classical fourth-order Runge-Kutta method (RK4) reveal that this modified method is very effective and convenient.