Solution to the 1D Stefan problem using the unified transform method

Abstract

In this paper the one-dimensional two-phase Stefan problem is studied analytically leading to a system of non-linear Volterra-integral-equations describing the heat distribution in each phase. For this the unified transform method has been employed which provides a method via a global relation, by which these problems can be solved using integral representations. To do this, the underlying partial differential equation is rewritten into a certain divergence form, which enables to treat the boundary values as part of the integrals. Classical analytical methods fail in the case of the Stefan problem due to the moving interface. From the resulting non-linear integro-differential equations the one for the position of the phase change can be solved in a first step. This is done numerically using a fix-point iteration and spline interpolation. Once obtained, the temperature distribution in both phases is generated from their integral representation.