Models of the neutral-fractional anomalous diffusion and their analysis

Anomalous diffusion can be roughly characterized by the property that the diffusive particles do not follow the Gaussian statistics. When the mean squared displacement of the particles behaves in time like a power function, time- and/or space-fractional derivatives were shown to be useful in modeling of such diffusion processes. In this paper, a special class of anomalous diffusion processes, the so called neutral-fractional diffusion, is considered. The starting point is a stochastic formulation of the model in terms of the continuous time random walk processes. The neutral-fractional diffusion equation in form of a partial differential equation with the fractional derivatives of the same order both in time and in space is then derived from the master equation for a special choice of the probability density functions. An explicit form of the fundamental solution of the Cauchy problem for the neutral-fractional diffusion equation is presented. Its properties are studied and illustrated by plots.