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.

翻译：我们为 $\ alpha$ - 折射衍生物开发了一定的差差差近似值 $\ alpha$ 。 近似法的权重与流行的 Gr\\"unwald 方案具有相同的速率矩阵属性。 特别是, 分数扩散方程式的近似解决方案保留了假设性。 此外, 对于封闭域的倾斜分热方程的解决方案的近似值, 新的近似法维持着它的定值 $\ alpha$, 而 Gr\\"unwald 方案的顺序则降低到 $\ alpha-1, 与 Meerschaert 和 Tadjeran 的趋同率结果相矛盾。