Anomalous diffusion and fractional diffusion equations

Abstract

In this work we investigate the anomalous diffusion equations, usually applied to describe the anomalous diffusion, which employ fractional derivatives for the time or the spatial variables. In particular, we obtain exact solutions by taking a generic initial condition into account and developing a perturbation theory to investigate complex situations. We also verify that the fractional derivatives, when applied to the time variable, lead us to a anomalous diffusion with second moment finite, i.e., <χ²> ∝ t^α (0 < α < 1, and α > 1, corresponding to sub and superdifusive behavior, respectively). By way of contrast, the fractional derivative applied to the spatial variable results in a anomalous diffusion where the second moment is not finite. These equations generalize the usual diffusion equation in order to incorporate several situations. We also employ the continuous time random walking formalism to investigate the implications obtained by using fractional derivatives in the diffusion equation