The analytical solutions of the one- and two-phase Stefan problems are found
in the form of series containing linear combinations of the integral error
functions which satisfy a priori the heat equation. The unknown coefficients
are defined from the initial and boundary conditions by the comparison of the
like 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.