Inverse Analysis of a New Anomalous Diffusion Model Employing Maximum Likelihood and Bayesian Estimation

The classical diffusion equation models the behavior of several physical phenomena related to dispersion processes quite successfully; however, in some cases, this approach fails to represent the actual physical behavior. In most published works dealing with this situation, the well-known second order parabolic equation is assumed as the basic governing equation of the dispersion process, but the anomalous diffusion effect is modeled with the introduction of fractional derivatives or the imposition of a convenient variation of the diffusion coefficient with time or concentration. Alternatively, Bevilacqua and coauthors developed a new analytical formulation for the simulation of the phenomena of diffusion with retention. Its purpose is to reduce all diffusion processes with retention to a unifying phenomenon that can adequately simulate the retention effect. This model may have relevant applications in different areas, such as population spreading with partial hold up of the population to guarantee territorial domain, chemical reactions inducing adsorption processes, and multiphase flow through porous media. In the new formulation, a discrete approach is first formulated with regard to a control parameter that represents the fraction of particles allowed to diffuse, and the governing equation for the modeling of diffusion with retention in a continuum medium requires a fourth order differential term. In order to apply this new formulation to the modeling of practical problems, the newly introduced parameters need to be accurately determined through an inverse problem analysis. Hence, this chapter provides an overview of the inverse analysis of anomalous diffusion problems as modeled through this new formulation, and a summary is also presented on the inverse problem formulation and related solution through three different approaches: (1) the maximum likelihood estimation, (2) the Bayesian approach through the Maximum a Posteriori objective function, and (3) Markov Chain Monte Carlo methods.