Extensões da equação de difusão : soluções e aplicações

Abstract

The present work is dedicated to investigate extensions of diffusion equations by incorporating spatial and time fractional derivatives, in the presence of external forces and nonlocal terms. We start by studying the continuous time random walk approach and the Langevin equation in order to understand the usual diffusion equation and the consequences obtained when it is extended. Following, we present important properties concerned to the fractional calculus which will be used in the others chapters. After, the one dimensional fractional diffusion equations are analyzed. Next, we investigate the effects obtained when nonlocal terms are incorporated. The influence of the boundary conditions on the spreading of the system also investigated. For equations with fractional derivatives or nonlocal terms in the absence of external forces or not, we obtain analytical solutions using the time-dependent formalism of the Green functions and the solutions exhibit an anomalous dispersion. Finally, we present our conclusions.