The modified homotopy perturbation method with an auxiliary term for the nonlinear oscillator with discontinuity

Recently, a modified homotopy perturbation method with an auxiliary term was proposed for obtaining an analytical approximate solution of a nonlinear equation. The auxiliary term involves a tuning parameter that gives the modified homotopy perturbation method an additional degree-of-freedom to construct the approximate solution. In this paper, we present a systematic approach for choosing the auxiliary parameter and furthermore show that an optimal value of the parameter yields significant improvement in the approximate solution compared to the standard homotopy perturbation method. We consider the nonlinear oscillator with a discontinuity to illustrate the proposed approach.