Application of a modified He’s homotopy perturbation method to obtain higher-order approximations to a nonlinear oscillator with discontinuities

A modified He’s homotopy perturbation method is used to calculate the periodic solutions of a nonlinear oscillator with discontinuities for which the elastic force term is proportional to sgn(xx). 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. We find that this modified homotopy perturbation method 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 period of less than 1.6% for all values of oscillation amplitude, while this relative error is 0.65% for the second iteration and 0.24% when the third-order approximation is considered. Comparison of the result obtained using this method with the exact ones reveals that this modified method is very effective and convenient.