Anomalous diffusion: Models, their analysis, and interpretation

Abstract

In this chapter, modeling of anomalous diffusion processes in terms of differential equations of an arbitrary (not necessarily integer)ord er is discussed. We start with micro-modeling and first deduce a probabilistic interpretation of normal and anomalous diffusion from basic random walk models. The fractional differential equations are then derived asymptotically in the Fourier-Laplace domain from random walk models and generalized master equations, in the same way as the standard diffusion equation is obtained from a Brownian motion model. The obtained equations and their generalizations are analyzed both with the help of the Laplace-Fourier transforms (the Cauchy problems)and the spectral method (initial-boundary-value problems). In particular, the maximum principle, well known for elliptic and parabolic type PDEs, is extended to initial-boundary-value problems for the generalized diffusion equation of fractional order.