O cálculo fracionário aplicado a fenômenos difusivos

Abstract

In this work we have as main theme the diffusive processes in which the mean squared displacement of the particles is not proportional to the time, that is, we investigate the anomalous diffusion processes. We approach the fundamental contents relevant to this subject, such as the first observations carried out by Robert Brown in 1827, which led to the development of diffusive theory and the diffusion equation, found later by Albert Einstein. Subsequently, we present briefly the fractional calculus, with the objective of showing the efficiency of this theory for the description of anomalous diffusion phenomena. A brief review of some useful mathematical methods to find the solution of partial differential equations of the diffusion type is also included and, in this context, it is requested the Fourier and Laplace transforms and the Green function technique. Then, we apply the concepts of diffusion and make use of calculation techniques to analytically solve a diffusive system whose geometry is in the form of a comb (comb - model). This system is primarily modeled for the case in which the substances present are not related to reaction terms and the only influence on the system is given by the restrictive geometry, characteristic of the comb model. Then, we add the reaction terms, which can be related to the irreversible (unilateral) or reversible (bilateral) reaction. We show that to obtain the solution of the equations for the comb model with the reaction terms, it is necessary to use some functions from the fractional calculus formalism, such as Fox’s H function and Mittag-Leffler function. We use the necessary formalisms to perform an analysis on the behavior of the mean squared displacement for each direction in cases of irreversible and reversible reaction. The behavior found for irreversible reaction is highly correlated with the signs of the reaction terms. This means that for equal signs, the behavior of the system is in one type, which is modified when opposite signs are considered. For both cases, the occurrence of anomalous diffusion is identified. For the system containing reversible terms, the dynamic is the same for short and long times. However, the same system undergoes a transition process at intermediate times, an effect related to the reaction terms present in the equation.