The Third-Order Solution of AB-Equation Using the Homotopy Perturbation Method

Abstract

This paper discusses the solution of AB equation using the homotopy perturbation method. The AB equation is a wave equation that improves the KdV wave equation. Unlike the KdV wave equation whose dispersion relation is non-exact, this AB equation has an exact dispersion relation. The AB equation is solved using the homotopy perturbation method up to the third order. The generated waves are monochromatic. The results show that in the first-order solution, the relation between the frequency and the wave number, referred to as the dispersion relation was obtained. In the second-order solution, we found that the frequency of the wave was two times larger than in the first-order solution. Nevertheless, it had less amplitude than in the first-order solution. Finally, the homotopy perturbation method also provided a good solution because of the absence of the need to specify the embedding parameters used.