Moving boundary problems governed by anomalous diffusion

Abstract

Anomalous diffusion can be characterized by a mean-squared displacement x 2 (t) that is proportional to t a where a = 1. A class of one-dimensional moving boundary problems is investigated that involves one or more regions governed by anomalous diffusion, specifically subdiffusion (a < 1). A novel numerical method is developed to handle the moving interface as well as the singular history kernel of subdiffusion. Two moving boundary problems are solved: the first involves a subdiffusion region to the one side of an interface and a classical diffusion region to the other. The interface will display non-monotone behaviour. The subdiffusion region will always initially advance until a given time, after which it will always recede. The second problem involves subdiffusion regions to both sides of an interface. The interface here also reverses direction after a given time, with the more subdiffusive region initially advancing and then receding.

A Note on Models for Anomalous Phase-Change Processes

“…Some advances in this direction were recently given in [13]. Looking for approximate solutions (both numerical and analytical) is an active area of research, [3,6,7,16,17,25,33,38,40].…”

“…Free boundary problems involving fractional heat equation (2.4) and fractional Stefan condition (2.5) were considered in several works at dealing with anomalous diffusive-like processes, e.g. [2,3,6,7,16,17,19,25,33,[38][39][40]. A remarkable contribution from Voller, Falcini, and Garra was to identify a definition for the heat flux and a suitable classical global balance leading to a transparent generalization of the classical model to a fractional one.…”

“…Timefractional conservation equations were also considered in [5], where the relation with non-local transport theory with memory effects is discussed. In this way, we provide a new approach to derive some fractional problems that are of interest in pure and applied fields, [2,3,6,7,13,16,17,19,25,33,[38][39][40].…”

An Integral Relationship for a Fractional One-Phase Stefan Problem

“…Remark 3. Similar problems for the fractional derivative in the Caputo sense were studied in [2,18,19,20,21,27,28,29]. For example, the formulation given in [19] is:…”

An approximate solution to a moving boundary problem with space–time fractional derivative in fluvio-deltaic sedimentation process

“…Voller [13] presented fractional (non-integer) form of a limit Stefan problem using Caputo derivatives for both space and time, and discussed exact solution of the problem. Recently, some researchers [14][15][16][17] also discussed various mathematical models governed with different fractional derivatives for both the space and time.…”