Two Stefan problems for a non-classical heat equation with nonlinear thermal coefficients

Abstract

The mathematical analysis of two one-phase unidimensional and non-classical Stefan problems with nonlinear thermal coefficients is obtained. Two related cases are considered, one of them has a temperature condition on the fixed face $x=0$ and the other one has a flux condition of the type $-q_{0}/\sqrt{t }$ $ ( q_{0}>0 ) .$ In the first case, the source function depends on the heat flux at the fixed face $x=0,$ and in the other case it depends on the temperature at the fixed face $x=0. $ In both cases, we obtain sufficient conditions in order to have the existence of an explicit solution of a similarity type, which is given by using a double fixed point.