A Cα finite difference method for the Caputo time-fractional diffusion equation

We begin with a treatment of the Caputo time-fractional diffusion equation, by using the Laplace transform, to obtain a Volterra integro-differential equation. We derive and utilize a numerical scheme that is derived in parallel to the L1-method for the time variable and a standard fourth-order approximation in the spatial variable. The main method derived in this article has a rate of convergence of O(k(alpha) + h(4)) for u(x, t) is an element of C-alpha([0, T]; C-6(omega)), 0 < alpha < 1, which improves previous regularity assumptions that require C-2[0, T] regularity in the time variable. We also present a novel alternative method for a first-order approximation in time, under a regularity assumption of u(x, t) is an element of C-1([0, T]; C-6(omega)), while exhibiting order of convergence slightly more than O(k) in time. This allows for a much wider class of functions to be analyzed which was previously not possible under the L1-method. We present numerical examples demonstrating these results and discuss future improvements and implications by using these techniques.