An approximate analytical solution of the Allen-Cahn equation using homotopy perturbation method and homotopy analysis method

Abstract

In this paper, an approximate analytical solution of the bistable Allen-Cahn equation is given. The Allen-Cahn equation is a mathematical model to study the phase separation process in binary alloys and emerged as a convection-diffusion equation in fluid dynamics or reaction-diffusion equation in material sciences. A phase transition occurs at the interface when one material changes its composition or structure. The homotopy perturbation method and homotopy analysis method are used for finding the approximate solution. These methods don't need the use of any transformation, discretization, unrealistic restriction and assumption. The error estimates are computed by comparing with a numerical method, and a good agreement is observed.