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 α + h 4) for u(x, t) ∈ C α ([0, T]; C 6 (Ω)), 0 < α < 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) ∈ C 1 ([0, T]; C 6 (Ω)), 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.
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