Transient anomalous diffusion with Prabhakar-type memory.

ABSTRACT

In this paper, we derive the general properties of anomalous diffusion and non-exponential relaxation from the Fokker-Planck equation with the memory function related to the Prabhakar integral operator. The operator is a generalization of the Riemann-Liouville fractional integral and permits one to study transient anomalous diffusion processes with two-scale features. The aim of this work is to find a probabilistic description of the anomalous diffusion from the Fokker-Planck equation, more precisely from the memory function. The temporal behavior of such phenomena exhibits changes in time scaling exponents of the mean-squared displacement through time domain-a more general picture of the anomalous diffusion observed in nature.