Exact and approximate solutions to the Stefan problem in ellipsoidal coordinates

In this paper, we present exact and approximate solutions of the Stefan problems in ellipsoidal coordinates. We consider two models of electrical contact heating for melting process. The first problem describes the contact heating for liquid and solid zones based on the two-phase Stefan problem, where time $t$ is present as a parameter. Contact heating including softening processes are described by a mathematical model based on the three-phase Stefan problem for the ellipsoidal heat equation. Numerical results are presented and discussed.