  In this paper we consider a one-dimensional one-phase Stefan problem
corresponding to the solidification process of a semi-infinite material with a
convective boundary condition at the fixed face. The exact solution of this
problem, available recently in the literature, enable us to test the accuracy
of the approximate solutions obtained by applying the classical technique of
the heat balance integral method and the refined integral method, assuming a
quadratic temperature profile in space. We develop variations of these methods
which turn out to be optimal in some cases. Throughout this paper, 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 estimate the errors committed by each approach for the
corresponding free boundary and temperature profiles.
