Analytic solution of the Stefan problem in finite mediums

Abstract

Abstract. The classical Stefan problem is considered in this paper for finite mediums with Dirichlet boundary conditions. Analytic solutions for the temperature distributions and the location of the moving interface are obtained by using Lie group theory and the superposition principle. The existence of analytically exact solutions is established by proving the convergence of the solution.1. Introduction. The Stefan problem is very interesting because of its wide applications in science and engineering and its nonlinearity, which poses an immense challenge to those who seek analytically exact solutions. In casting industries, the melting and solidification of metals are studied extensively to understand the formation of microstructures. The thawing of soils, the formation of ice, and the cooling of large masses of igneous rock are important in geology. Melting and solidification are also encountered in welding, in crystallization, and in the formation of alloys. Recently, lasers are being increasingly used for cutting, drilling, welding, and developing novel metastable alloys, which involve phase changes and moving boundaries. The application of the Stefan problem to study the formation of metastable alloys during laser-aided materials processing can be found in [1][2][3]. Tao studied the Stefan