Explicit solutions for the Solomon-Wilson-Alexiades's mushy zone model with convective or heat flux boundary conditions

We complete the Solomon-Wilson-Alexiades's mushy zone model (Letters Heat Mass Transfer, 9 (1982), 319-324) for the one-phase Lam\'e-Clapeyron-Stefan problem. We obtain explicit solutions when a convective or heat flux boundary condition is imposed on the fixed face for a semi-infinite material. We also obtain the necessary and sufficient condition on data in order to get these explicit solutions. Moreover, when these conditions are satisfied the two problems are equivalents to the same problem with a temperature boundary condition on the fixed face and therefore an inequality for the coefficient which characterized one of the two free interfaces is also obtained.