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On the two-phase fractional Stefan problem. (English) Zbl 1434.80006

Summary: 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.

MSC:

80A22 Stefan problems, phase changes, etc.
35D30 Weak solutions to PDEs
35K15 Initial value problems for second-order parabolic equations
35K65 Degenerate parabolic equations
35R09 Integro-partial differential equations
35R11 Fractional partial differential equations
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
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