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

In this paper we obtain self-similarity solutions for a one-phase one-dimensional fractional space Stefan problem in terms of the three parametric Mittag-Leffler function Eα,m,l(z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\alpha ,m,l}(z)$$\end{document}. We consider Dirichlet and Neumann conditions at the fixed face, involving Caputo fractional space derivatives of order 0<α<1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha <1$$\end{document}. We recover the solution for the classical one-phase Stefan problem when the order of the Caputo derivatives approaches one.