Finite difference methods for degenerate diffusion equations and fractional diffusion equations

This thesis focuses on the study of three mathematical models consisting of degenerate diffusion equations and fractional diffusion equations arising from different study needs. The first one is taken from the study of self-organized criticality phenomena arising from recent papers by Barbu. About the second one, that is connected to the obstacle problem, we analyze a nonlinear degenerate parabolic problem whose diffusion coefficient is the Heaviside function of the distance of the solution itself from a given target function. More precisely, we show that this model behaves as an evolutive variational inequality having the target as an obstacle: under suitable hypotheses, starting from an initial state above the target the solution evolves in time towards an asymptotic solution, eventually getting in contact with part of the target itself. At last, referring to the third one, that is a time-fractional type model (a Caputo time fractional degenerate diffusion equation) that can find a wider use, for example, from biology to mechanics, to superslow diffusion in porous media, till financial type phenomena, we prove to be equivalent to the fractional parabolic obstacle problem, showing that its solution evolves for any α ∈ (0, 1) to the same stationary state, the solution of the classic elliptic obstacle problem. The only thing which changes with α is the convergence speed. For all these models we provide numerical simulations.