In the last decade, the analysis of time-space fractional Bloch–Torrey equation has been considered by different studies due to its applications in many fields. Since the analytical solution of this equation is difficult or impossible, numerical solutions can be helpful and sometimes are the only choice. Therefore, in this work, a numerical-based solution is shown by virtue of the Crank–Nicolson weighted shifted Grunwald difference method. The stability, as well as solvability of this method, are also investigated. It is shown that the method for time-space fractional Bloch–Torrey equation is of order \({\mathcal {O}}({\tau ^{2 - \alpha }},{h^2})\), where \(0<\alpha <1\). Also, \(\tau \) and h are the time step and space step, respectively. At the end, numerical applications are presented and the thrust of the present study is compared with other sophisticated schemes in the literature. The main advantage of the proposed scheme is that, it is more efficient in terms of accuracy and CPU time in comparing with the existing ones in open literature.

在过去的十年中，由于时空分数Bloch-Torrey方程在许多领域中的应用，已被不同的研究所考虑。由于此方程的解析解很难或不可能，因此数值解可能会有所帮助，有时是唯一的选择。因此，在这项工作中，借助Crank-Nicolson加权移位Grunwald差分法显示了一种基于数值的解决方案。还研究了该方法的稳定性和可溶性。结果表明，时空分数Bloch–Torrey方程的方法的阶数为\（{\ mathcal {O}}（{\ tau ^ {2-\ alpha}}，{h ^ 2}）\），其中\（0 <\ alpha <1 \）。此外，\（\ tau \）和h分别是时间步长和空间步长。最后，介绍了数值应用，并将本研究的主旨与文献中的其他复杂方案进行了比较。所提出的方案的主要优点是，与公开文献中的现有方法相比，它在准确性和CPU时间方面更为有效。