The generalized Cattaneo equation for the description of anomalous transport processes

Abstract

The Cattaneo equation, which describes a diffusion process with a finite velocity of propagation, is generalized to describe anomalous transport. Three possible generalizations are proposed, each one supported by a different scheme: continuous time random walks, non-local transport theory, and delayed flux-force relation. The properties of these generalizations are studied in both the long-time and the short-time regimes. In the long-time limit, we recover the mean-square displacement which is characteristic for these anomalous processes. As expected, the short-time behaviour is modified in comparison to generalized diffusion equations.