The Spectral Method for the Cahn-Hilliard Equation with Concentration-Dependent Mobility

In this paper, we apply the spectral method to approximate the solutions of Cahn-Hilliard equation, which is a typical class of nonlinear fourth-order diffusion equations. Diffusion phenomena is widespread in the nature. Therefore, the study of the diffusion equation caught wide concern. Cahn-Hilliard equation was proposed by Cahn and Hilliard in 1958 as a mathematical model describing the diffusion phenomena of phase transition in thermodynamics. Later, such equations were suggested as mathematical models of physical problems in many fields such as competition and exclusion of biological groups 1 , moving process of river basin 2 , and diffusion of oil film over a solid surface 3 . Due to the important application in chemistry, material science, and other fields, there were many investigations on the Cahn-Hilliard equations, and abundant results are already brought about. The systematic study of Cahn-Hilliard equations started from the 1980s. It was Elliott and Zheng 4 who first study the following so-called standard Cahn-Hilliard equation with constant mobility: