Front propagation in anomalous diffusive media governed by time-fractional diffusion

In this paper, a multi-dimensional model is proposed to study the propagation of random fronts in media in which anomalous diffusion takes place. The front position is obtained as the weighted mean of fronts calculated by means of the level set method, using as weight-function the probability density function which characterizes the anomalous diffusion process. Since anomalous diffusion is assumed to be governed by a time-fractional diffusion equation, its fundamental solution is the required probability density function. It is shown that this fundamental solution can be expressed in the multi-dimensional case in terms of the well-known MM-Wright/Mainardi function, as in the one-dimensional case. Making use of this representation for the practical purpose of numerical evaluation, the propagation of random fronts in two-dimensional subdiffusive media is discussed and investigated.