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We develop a finite difference approximation of order $\alpha$ for the$\alpha$-fractional derivative. The weights of the approximation scheme havethe same rate-matrix type properties as the popular Gr\"unwald scheme. Inparticular, approximate solutions to fractional diffusion equations preservepositivity. Furthermore, for the approximation of the solution to the skewedfractional heat equation on a bounded domain the new approximation scheme keepsits order $\alpha$ whereas the order of the Gr\"unwald scheme reduces to order$\alpha-1$, contradicting the convergence rate results by Meerschaert andTadjeran.

A Higher Order Resolvent-positive Finite Difference Approximation for Fractional Derivatives