  The solution of a Caputo time fractional diffusion equation of order
$0<\alpha<1$ is expressed in terms of the solution of a corresponding integer
order diffusion equation. We demonstrate a linear time mapping between these
solutions that allows for accelerated computation of the solution of the
fractional order problem. In the context of an $N$-point finite difference time
discretisation, the mapping allows for an improvement in time computational
complexity from $O\left(N^2\right)$ to $O\left(N^\alpha\right)$, given a
precomputation of $O\left(N^{1+\alpha}\ln N\right)$. The mapping is applied
successfully to the least-squares fitting of a fractional advection diffusion
model for the current in a time-of-flight experiment, resulting in a
computational speed up in the range of one to three orders of magnitude for
realistic problem sizes.
