In this paper, high‐order numerical methods for time‐Caputo and space‐Riesz fractional Bloch‐Torrey equations in one‐ and two‐dimensional space are constructed, where the second‐order backward fractional difference operator and the sixth‐order fractional‐compact difference operator are applied to approximate the time and space fractional derivatives, respectively. The stability and convergence of the methods are analyzed and it is shown that the convergence orders are higher than the earlier work. Finally, some numerical experiments are presented to demonstrate the effectiveness of the methods and confirm our theoretical results.

中文翻译：

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时间Caputo和空间Riesz分数Bloch-Torrey方程的数值算法

本文构造了一维和二维空间中时间Caputo和空间Riesz分数Bloch-Torrey方程的高阶数值方法，其中二阶后向分数差算子和六阶分数紧致差分算子分别用于近似时间和空间分数导数。分析了方法的稳定性和收敛性，结果表明收敛阶数高于早期的工作。最后，通过一些数值实验证明了该方法的有效性并证实了我们的理论结果。