In this paper, we develop a new approximation for the generalized fractional derivative, which is characterized by a scale function and a weight function. The new approximation is then used in the numerical treatment of a class of generalized fractional sub-diffusion equations. The theoretical aspects of solvability, stability and convergence are established rigorously in maximum norm by discrete energy methodology. Due to the new approximation, the theoretical temporal convergence order of the numerical scheme improves those of earlier work. To confirm, four examples are presented to illustrate the accuracy of the proposed scheme and to compare with other methods in the literature.

在本文中，我们为广义分数导数开发了一个新的近似值，其特征是比例函数和权重函数。然后，将新的近似值用于一​​类广义分数次扩散方程的数值处理。可溶性，稳定性和收敛性的理论方面是通过离散能量方法在最大范数下严格建立的。由于采用了新的近似方法，数值方案的理论时间收敛阶数改进了先前的工作。为了证实这一点，给出了四个例子来说明所提出方案的准确性，并与文献中的其他方法进行比较。