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(h ² 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.