Abstract The classical Stefan problem is one of the most studied free boundary problems of evolution type. Recently, there has been interest in treating the corresponding free boundary problem with nonlocal diffusion. We start the paper by reviewing the main properties of the classical problem that are of interest to us. Then we introduce the fractional Stefan problem and develop the basic theory. After that we center our attention on selfsimilar solutions, their properties and consequences. We first discuss the results of the one-phase fractional Stefan problem, which have recently been studied by the authors. Finally, we address the theory of the two-phase fractional Stefan problem, which contains the main original contributions of this paper. Rigorous numerical studies support our results and claims.

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关于两相分数阶Stefan问题

摘要 经典Stefan问题是进化型自由边界问题中研究最多的问题之一。最近，人们对用非局部扩散处理相应的自由边界问题感兴趣。我们通过回顾我们感兴趣的经典问题的主要属性来开始这篇论文。然后我们介绍分数 Stefan 问题并发展基本理论。之后，我们将注意力集中在自相似解、它们的性质和后果上。我们首先讨论作者最近研究的单相分数阶 Stefan 问题的结果。最后，我们讨论了两相分数阶 Stefan 问题的理论，其中包含本文的主要原始贡献。严格的数值研究支持我们的结果和主张。