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

TL;DR

This paper compares two fractional Stefan problems using different derivatives, showing they have distinct solutions that converge to the same classical problem as the fractional order approaches one, revealing non-commutative limits.

Contribution

It introduces and analyzes two fractional Stefan problems with Riemann-Liouville and Caputo derivatives, highlighting their differences and convergence behavior.

Findings

Solutions are explicitly expressed via Wright functions.

The two fractional problems have different solutions for lpha < 1.

Both solutions converge to the same classical solution as lpha approaches 1.

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.