Two different fractional Stefan problems which are convergent to the same classical Stefan problem

Abstract

Two fractional Stefan problems are considered by using Riemann-Liouville and Caputo derivatives of order $\alpha \in (0,1)$ such that in the limit case ($\alpha =1$) both problems coincide with the same classical Stefan problem. For the one and the other problem, explicit solutions in terms of the Wright functions are presented. We prove that these solutions are different even though they converge, when $\alpha \nearrow 1$, to the same classical solution. This result also shows that some limits are not commutative when fractional derivatives are used.