Abstract
In this paper, we present a numerical solution of a two-phase fractional Stefan problem with time derivative described in the Caputo sense. In the proposed algorithm, we use a special case of front-fixing method supplemented by the iterative procedure, which allows us to determine the position of the moving boundary. The presented method is an extension of a front-fixing method for the one-phase problem to the two-phase case. The novelty of the method is a new discretization of the partial differential equation dedicated to the second phase, which is carried out by introducing a new spatial variable immobilizing the moving boundary. Then, the partial differential equation is transformed to an equivalent integro-differential equation, which is discretized on a homogeneous mesh of nodes with a constant spatial and time step. A new convergence criterion is also proposed in the iterative algorithm determining the location of the moving boundary. The motivation for the development of the method is that the analytical solution of the considered problem is impossible to calculate in some cases, as can be seen in the figures in the paper. Moreover, the change of the boundary conditions makes obtaining a closed analytical solution very problematic. Therefore, creating new numerical methods is very valuable. In the final part, we also present some examples illustrating the comparison of the analytical solution with the results received by the proposed numerical method.
Highlights
Differential equations are an excellent tool for describing problems with technical applications [1]
The fractional Stefan problem is a natural generalization of the classical Stefan problem and is related to the anomalous mass or heat transfer usually occurs in materials with strong heterogeneities across a range of length scales
The numerical scheme presented in the paper allows to solve three partial differential Equations (13), (14) and (18) supplemented with a set of boundary and initial conditions
Summary
A special class of differential equations are the so-called moving boundary problems which are one of the most important area within partial differential equations. This particular kind of boundary-value problem was originally intended to describe the solid–liquid phase change process, and refers to such phenomena as solute transport, molecular diffusion or controlled drug release [2]. The fractional Stefan problem is a natural generalization of the classical Stefan problem and is related to the anomalous mass or heat transfer usually occurs in materials with strong heterogeneities across a range of length scales. The movement of molecules in such materials is described by the law
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