The similarity method and explicit solutions for the fractional space one-phase Stefan problems

Abstract

In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space one-phase Stefan problem in terms of the three parametric Mittag-Leffler function E α,m,l (z). We consider Dirichlet and Newmann conditions at the fixed face, involving Caputo fractional space derivatives of order 0 < α < 1. We recover the solution for the classical one-phase Stefan problem when the order of the Caputo derivatives approaches one.

Special Solutions to the Space Fractional Diffusion Problem

“…The argument we use is based on the use of the maximum principle for (1.1). This idea was used first in the proof of a similar result in [13]. We provide our own and extended version of the argument.…”

“…We could justify positivity of E on the grounds of the theory of Mittag-Leffler functions, (actually we do this in the appendix). However, we would like to stress that our proof of positivity of E is entirely based on a PDE tool, which the maximum principle, we use this idea after [13].…”