A Finite Difference Spectral-Collocation Method for Fractional Reaction-Diffusion Systems

This presentation deals with the numerical solution of a reaction-diffusion problems, where the time derivative is of fractional order. Since the fractional derivative of a function depends on its past history, these systems can successfully model evolutionary problems with memory, as for example electrochemical processes, porous or fractured media, viscoelastic materials, bioengineering applications. On the side of numerical simulation, the research mainly focused on suitable extensions of methods for PDE. This approach often produced low accuracy and/or high computational methods, due to the lack of smoothness of the analytical solution and to the longrange history dependence of the fractional derivative. Here we consider a finite difference scheme along space, to discretize the integer-order spatial derivatives, while we adopt a spectral collocation method through time. A suitable choice of the function basis produces an exponential convergence though time at a low computational cost, since the spectral method avoids the step-by-step methods