We investigate the suitability of a fractional Laplacian in the construction of a three-dimensional (3D) multi-term time–space fractional Bloch–Torrey equation. Instead of using the Riesz fractional derivative as in most previous research, we consider the fractional Laplacian in space and multi-term Caputo fractional derivative in time. A vertex-centred finite volume method is firstly derived to discretise this 3D fractional dynamic system in combination with the matrix transfer technique and the weighted and shifted Grünwald–Letnikov formula. Since computing the fractional power of the matrix representation $m(-\Delta )$ of the Laplacian operator results in a highly dense matrix, we use matrix function approximations developed in a Krylov subspace framework and exploit the sparsity of $m(-\Delta )$ to reduce the computational overhead. Then the Lanczos method is employed to approximate vector products of matrix functions. A preconditioner is applied to accelerate the convergence process, which shows excellent performance and reduces the memory considerably. In particular, we introduce block matrix functions to deal with the coupled system. Furthermore, the analytical solution of the two-term time–space fractional diffusion equation is derived and expressed by the multinomial Mittag-Leffler function. The feasibility and effectiveness of the proposed numerical techniques are verified by some numerical examples. Compared with two-dimensional fractional models or single-term time fractional dynamic systems, this multi-term time–space fractional model provides more flexibility for the characterisation of anomalous diffusion in biological tissues.

我们研究了分数拉普拉斯算子在构建三维 (3D) 多项时空分数 Bloch-Torrey 方程中的适用性。我们没有像以前大多数研究中那样使用 Riesz 分数阶导数，而是考虑空间中的分数拉普拉斯算子和时间中的多项 Caputo 分数阶导数。结合矩阵传递技术和加权移位的 Grünwald-Letnikov 公式，首先导出了一种以顶点为中心的有限体积方法来离散化这个 3D 分数动态系统。由于计算矩阵表示的分数幂$米(-\Delta )$拉普拉斯算子的结果是一个高度密集的矩阵，我们使用在 Krylov 子空间框架中开发的矩阵函数逼近，并利用$米(-\Delta )$以减少计算开销。然后采用Lanczos方法逼近矩阵函数的向量积。应用预条件器来加速收敛过程，表现出优异的性能并大大减少了内存。特别是，我们引入了块矩阵函数来处理耦合系统。进一步推导了两项时空分数扩散方程的解析解，并用多项式Mittag-Leffler函数表示。通过一些数值例子验证了所提出数值技术的可行性和有效性。与二维分数模型或单项时间分数动力学系统相比，这种多项时空分数模型为表征生物组织中的异常扩散提供了更大的灵活性。