  We develop a finite difference approximation of order $\alpha$ for the
$\alpha$-fractional derivative. The weights of the approximation scheme have
the same rate-matrix type properties as the popular Gr\"unwald scheme. In
particular, approximate solutions to fractional diffusion equations preserve
positivity. Furthermore, for the approximation of the solution to the skewed
fractional heat equation on a bounded domain the new approximation scheme keeps
its order $\alpha$ whereas the order of the Gr\"unwald scheme reduces to order
$\alpha-1$, contradicting the convergence rate results by Meerschaert and
Tadjeran.
