Equações de difusão e o cálculo fracionário

Abstract

Some systems involving transport phenomena are not adequately explained by a usual diffusion equation, therefore, diffusion type equations with fractional operators are sometimes employed. In general, solutions for equations with fractional operators are given in terms of Mittag-Leffler and H Fox functions, and these are closely linked to fractional calculus. In this context, this paper deals with the study of diffusion phenomena and how the ideas within this field of study were developed. One of the main objectives of the study is to investigate systems that show anomalous diffusive behaviour, which are governed by differential equations with fractional operators. Thus, an introduction to this type of calculus was carried out, emphasizing some properties that were relevant to solve the problems addressed in this work. In addition to the fractional calculus tools, the continuous time random walk model is discussed. This topic aims to show how the distribution of steps and time lapse between these steps can influence the system's behaviour, a fact that leads it to be governed by distinct differential equations. The first case studied is a system with a geometric bond. Here, the particles in bulk are in a region where there is a sphere of radius R, so that they can be adsorbed and/or desorbed from the surface of this sphere. Such phenomena depend on the choice of the kernel of the integro-differential boundary condition associated with this surface. The other system studied consists of a model that considers a generalization of a first order kinetic equation. Here, the dispersion of one system into another is studied, where particles, through chemical or physical means, may be sorbed and/or desorbed. The last problem discussed is the transport of charged particles in an electrolytic cell in three dimensions. In this final part of the work, the objective is to calculate an expression for the electrical impedance of the system and to analyze the cases when the electrodes are perfectly blocking or whether they are subjected to the effects of adsorption and/or desorption. From the obtained expression, a comparison is made with the one-dimensional case in order to analyze whether different behaviors arise.