Introducing Graded Meshes in the Numerical Approximation of Distributed-order Diffusion Equations

In this paper we deal with the numerical approximation of initial-boundary value problems to the diffusion equation with distributed order in time. As it is widely known, the solutions of fractional differential equations may present a singularity at t = 0 and therefore in these cases, standard finite difference schemes usually suffer a convergence order reduction with respect to time discretization. In order to overcome this, here we propose a finite difference scheme with a graded time mesh, constructed in such a way that the time step-size is smaller near the potential singular point. Numerical results are presented and compared with those obtained with finite difference schemes with uniform meshes.