  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
$O(k^{2}+h^4)$ 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.
