  In this paper we consider two different Stefan problems for a semi-infinite
material for the non classical heat equation with a source which depends on the
heat flux at the fixed face x = 0. One of them (with constant temperature on x
= 0) was studied in [4] where it was found a unique exact solution of
similarity type and the other (with a convective boundary condition at the
fixed face) is presented in this work. Due to the complexity of the exact
solution it is of interest to obtain some kind of approximate solution. For the
above reason, the exact solution of each problem is compared with approximate
solutions obtained by applying the heat balance integral method and the refined
heat balance integral method, assuming a quadratic temperature profile in
space. In all cases, a dimensionless analysis is carried out by using the
parameters: Stefan number (Ste) and the generalized Biot number (Bi). In
addition it is studied the case when Bi goes to infinity, recovering the
approximate solutions when a Dirichlet condition is imposed at the fixed face.
Some numerical simulations are provided in order to verify the accuracy of the
approximate methods
