Solution of Fourth Order Diffusion Equations and Analysis Using the Second Moment

Abstract

The classical concept of diffusion characterized by Fick’s law is well suited for describing a wide class of practical problems of interest. Nevertheless, it has been observed that it is not enough to properly represent other relevant applications of practical interest. When in a system of particles their spreading is slower or faster than predicted by the classical diffusion model, such a phenomenon is referred to as anomalous diffusion. Time fractional, space fractional and even space-time fractional equations are widely used to model phenomena such as solute transport in porous media, financial modelling and cancer tumor behavior. Considering the effects of partial and temporary retention in dispersion processes a new analytical formulation was derived to simulate anomalous diffusion. The new approach leads to a fourth-order partial differential equation (PDE) and assumes the existence of two concomitant fluxes. This work investigates the behavior of the bi-flux approach in one dimensional (1D) medium evaluating the mean square displacement for different cases in order to classify the diffusion process in normal, sub-diffusive or super-diffusive.