Diffusive transport is one of the most important transport mechanisms found in nature. At a microscopic level, diffusion is the result of the random motion of individual particles, and the use of Laplacian operators to model it rests on the key assumption that this random motion is an stochastic Gaussian process. However, a growing number of works have shown the prevalence of anomalous diffusion in which the mean square variance grows faster (in the case of superdiffusion) or slower (in the case of subdiffusion) than in a Gaussian diffusion process. In this project we explore the use of fractional derivative operators to model non-Gaussian, anomalous diffusive transport. Fractional calculus is a natural mathematical generalization of standard calculus, and in recent years has been applied to a growing class of problems in the applied sciences. Fractional diffusion incorporates in a natural way anomalous diffusion. Also, when viewed as an integral operator with a broad (algebraic decaying) propagator, fractional diffusion provides a natural description of non-local transport.
Numerical methods for the solution of partial differential equations of fractional order
Front dynamics in reaction-diffusion systems with Levy flights: a fractional diffusion approach
Non-diffusive transport in plasma turbulence: a fractional diffusion approach