Problema da difusão em meio poroso via cálculo fracionário

Abstract

Throughout history, renowned researchers have dedicated themselves and their studies to investigate, understand and describe nonlinear diffusion processes, using analytical, numerical and computational tools. In this work, we present a problem of nonlinear anomalous diffusion in porous media and aim to present Lie symmetry transformation groups as an alternative to solving the diffusion problem. Initially, a study was carried out on fractional calculus and its main operators, where we point out definitions, theorems, properties and applications. Fractional calculus allows us to describe natural characteristics more precisely, thus obtaining a greater amount of information linked to nonlocal operators, then called the memory effect. An analysis of anomalous fractional diffusion in porous media was carried out, where the problem of nonlinear fractional diffusion in time-space was addressed. Another important point presented here is the Lie point transformation groups, which we use as an alternative to the diffusion problem, since Lie symmetries prove to be a very important tool as it allows the transformation of a PDE into an ODE. We apply Lie symmetries to the discovery of diffusion in fractional porous media in terms of Riesz derivatives, including the Weyl derivative. And we demonstrate that the results can be understood for the fractional derivative in terms of the function È. We emphasize that the study of anomalous diffusion has been increasingly deepened and its concept applied in several fields such as diffusion in plasmas, diffusion in turbulent fluids, fluid transport in porous media, diffusion in fractals, anomalous diffusion on liquid surfaces and analysis of heartbeat histograms in healthy individuals, among other physical systems. Keywords: Anomalous Diffusion, Fractional Calculus, Fractional Diffusion in Porous Media, Lie Points