Application of a modified He’s homotopy perturbation method to obtain higher-order approximations of an x1/3 force nonlinear oscillator

Abstract

A modified He’s homotopy perturbation method (HHPM) is used to calculate the periodic solutions of a conservative nonlinear oscillator for which the elastic force term is proportional to x1/3. The He’s homotopy perturbation method is modified by truncating the infinite series corresponding to the first-order approximate solution before introducing this solution in the second-order linear differential equation, and so on. We find this modified HHPM works very well for the whole range of initial amplitudes, and the excellent agreement of the approximate frequencies and periodic solutions with the exact ones has been demonstrated and discussed. Only one iteration leads to high accuracy of the solutions with a maximal relative error for the approximate frequency of less than 0.6% for small and large values of oscillation amplitude, while this relative error is 0.17% for the second iteration and as low as 0.024% when the third approximation is considered. Comparison of the result obtained using this method with those obtained by different harmonic balance methods reveals that the former is very effective and convenient.