In this paper, we derive a fourth order approximation for the generalized fractional derivative that is characterized by a scale function z(t) and a weight function w(t). Combining the new approximation with compact finite difference method, we develop a numerical scheme for a generalized fractional diffusion problem. The stability and convergence of the numerical scheme are proved by the energy method, and it is shown that the temporal and spatial convergence orders are both 4. Several numerical experiments are provided to illustrate the efficiency of our scheme.

中文翻译：

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广义分数阶导数的新逼近及其在广义分数阶扩散方程中的应用

在本文中，我们导出了广义分数阶导数的四阶近似，其特征是比例函数z（t）和权重函数w（t）。结合新的近似与紧致的有限差分方法，我们开发了一个广义分数阶扩散问题的数值方案。通过能量方法证明了数值方案的稳定性和收敛性，并证明了时间和空间收敛阶数均为4。提供了一些数值实验，说明了该方案的有效性。