Tau method implementation for approximating the solution to a two-phase change problem with temperature-dependent thermal coefficients

Abstract

A one dimensional two-phase Stefan problem is considered to model the solidification process of a semi-infinite material with power-type temperature-dependent thermal coefficients and a Dirichlet boundary condition at the fixed face. Through a similarity transformation, an equivalent system of ordinary differential equations is obtained, which will be shown to have a unique solution. Since the domain is unbounded, a novel condition is imposed to transform it into a finite domain, allowing the application of the Tau approximation method. This method is based on shifted Chebyshev operational matrix of differentiation. Some comparisons between exact and numerical solutions are shown in order to test the accuracy of the method.