Two equivalent Stefan's problems for the Time Fractional Diffusion Equation

Two Stefan's problems for the diffusion fractional equation are solved, where the fractional derivative of order $\al \in (0,1)$ is taken in the Caputo's sense. The first one has a constant condition on $x = 0$ and the second presents a flux condition $T_x (0, t) = \frac {q} {t ^ {\al/2}}$. An equivalence between these problems is proved and the convergence to the classical solutions is analysed when $\al \nearrow$ 1 recovering the heat equation with its respective Stefan's condition.