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

TL;DR

This paper derives self-similarity solutions for a fractional space one-phase Stefan problem using Mittag-Leffler functions, extending classical solutions to fractional derivatives with boundary conditions.

Contribution

It introduces explicit solutions for fractional Stefan problems with Dirichlet and Neumann conditions using special functions, generalizing classical models.

Findings

Solutions expressed in terms of Mittag-Leffler functions

Recovers classical Stefan problem solutions as fractional order approaches one

Provides explicit formulas for fractional boundary conditions

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-Leffer function $E_{\alpha,m;l}(z)$. We consider Dirichlet and Newmann conditions at the fixed face, involving Caputo fractional space derivatives of order $0 < \alpha < 1$. We recover the solution for the classical one-phase Stefan problem when the order of the Caputo derivatives approaches one.