A finite difference method for an anomalous sub-diffusion equation, theory and applications

The numerical solution for a class of fractional sub-diffusion equations is studied. For the time discretization, we use a generalized Crank-Nicolson method combined with the second central finite difference (FD) for the spatial discretization which will then define a fully discrete implicit FD scheme. An error of order O(h² max (1, log k^-1) + k^2+α) has been shown where h and k denote the maximum space and time steps, respectively. A non-uniform time step is employed to compensate for the singular behaviour of the exact solution at t = 0. Our theoretical results are numerically validated in a series of test problems.