Power Series and Finite Element Methods for Solving Cahn-Hilliard Equation

Abstract

The Cahn-Hilliard equation is a nonlinear partial differential equation that describes spinodal decomposition, coarsening phenomena, and the dynamics of phase separation for ternary iron alloys. This article employs a power series technique and the finite element method to obtain analytical and numerical solutions of the Cahn-Hilliard equation, respectively. For the power series method, the nonlinear terms in the proposed problem are dealt with using the generalised Cauchy product of power series, which allows us to obtain an explicit recursion formula for the expansion function coefficient of the series solution. On the other hand, numerical solution to the Cahn-Hilliard equation is obtained using the finite element method that is based on the implicit time-stepping scheme and the sparse linear algebra technique. The obtained analytical and numerical solutions are compared with the exact solution to illustrate the accuracy and reliability of the proposed methods. The absolute errors obtained show that the proposed methods are accurate and reliable. Two and three dimensional graphs of the exact and approximate solutions are presented for comparison purposes.