Abstract
Stefan problems characterized by moving or free boundary often due to change of phase do appear in many important applications of diverse fields. Among these, we review some recent developments in geophysical situations such as the plate techtonics and magma solitons. The various analytical approaches developed for coping with the moving boundary (Stefan) problems are reviewed and outlined. We then introduce the reader to the theory of a new series expansion method based on the Lagrange-Biirmann expansions which can be applied to the Stefan problems. Unlike classical series solutions, the new solutions demonstrate a markedly improved convergence having validity often over the entire physical time domain.
