Improved Finite Difference Results 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 intego-differential equation where we may examine the weakly singular nature of this convolution kernel.\iffalse The order of fractional derivative, $\alpha$, is tied to finite difference methods and is of great interest.\fi We examine this new equation and utilize a numerical scheme that is derived in parallel to the L1-method for the time variable and a usual fourth order approximation in the spatial variable. The main method derived in this paper has a rate of convergence of $O(k^{2}+h4)$ for $u(x,t) \in C^{6(\Omega)\times} C^2[0,T]$, which improves previous estimates by a factor of $k^{\alpha}$. We also present a novel alternative method for a first order approximation in time, which allows us to relax our regularity assumption to $u(x,t) \in C^{6(\Omega)\times} C^1[0,T]$, while exhibiting order of convergence slightly less than $O(k^{1+\alpha})$ 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.