Time Fractional Diffusion Equations and Analytical Solvable Models

The anomalous diffusion of a particle that moves in complex environments is analytically studied by means of the time fractional diffusion equation. The influence on the dynamics of a random moving particle caused by a uniform external field is taken into account. We extract analytical solutions in terms either of the Mittag-Leffler functions or of the M-Wright function for the probability distribution, for the velocity autocorrelation function as well as for the mean and the mean square displacement. Discussion of the applicability of the model to real systems is made in order to provide new insight of the medium from the analysis of the motion of a particle embedded in it.