ABSTRACT

There is numerical, theoretical, and experimental  evidence of non-Gaussian, non-local diffusive transport in fluids, plasmas, and biological systems.  These results are incompatible with models based on Laplacian operators, and there is the need to develop alternative descriptions. Here we explore the idea of constructing  transport models using fractional diffusion operators. Fractional diffusion operators are integro-differential operators with algebraic decaying kernels and as such they are a natural tool for describing non-local transport processes. Also, these operators are intimately connected with non-Gaussian stochastic processes (e.g. anomalous diffusion caused by Levy flights). We present analytical and numerical solutions of the fractional diffusion equation.  Also, we discuss the role of fractional diffusion on front propagation in reaction-diffusion systems. In particular, we present numerical and analytical results showing  exponential front propagation, and universal power law decay of fronts in the fractional Fisher-Kolmogorov equation.