An exact solution of a limit case Stefan problem governed by a fractional diffusion equation

An anomalous diffusion version of a limit Stefan melting problem is posed. In this problem, the governing equation includes a fractional time derivative of order 0 < β ⩽ 1 and a fractional space derivative for the flux of order 0 < α ⩽ 1. Solution of this fractional Stefan problem predicts that the melt front advance as image. This result is consistent with fractional diffusion theory and through appropriate choice of the order of the time and space derivatives, is able to recover both sub-diffusion and super-diffusion behaviors for the melt front advance.