Anomalous diffusion and fractional advection-diffusion equation

Anomalous diffusion happens often in nature and society systems. In this paper, we develop a non-local method with temporal and spatial correlations to introduce a fractional order advection-diffusion equation based on the usually used local 2-nd order advection-dispersion equation. In this equation, the diffusion is a fractional order derivative of time and space. And then, we extend the classical Fick's law for standard diffusion to a general fractional Fick's law. The fractional Fick's law shows that the current is related to the concentrations all over the space, also depends on the previous history and the initial condition. The solution of this fractional order advection-dispersion equation is fractional Levy probability distribution density. And the mean square displacement is a nonlinear function of time.