Method of the Integral Error Functions for the Solution of the One- and Two-phase Stefan Problems and its Application

Abstract

The analytical solutions of the one- and two-phase Stefan problems are found in the formof series
containing linear combinations of the integral error functions which satisfies a priori the heat equation. The
unknown coefficients are defined from the initial and boundary conditions by the comparison of the same
power terms of the series using the Faa di Bruno formula. The convergence of the series for the temperature
and for the free boundary is proved. The approximate solution is found using the replacement of series
by the corresponding finite sums and the collocation method. The presented test examples confirm a good
approximation of such approach. This method is applied for the solution of the Stefan problem describing
the dynamics of the interaction of the electrical arc with electrodes and corresponding erosion.