In this paper, we develop two fast implicit difference schemes for solving a class of variable‐coefficient time–space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded L1 formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the graded mesh can capture the model problem with a weak singularity at initial time. The stability and convergence are rigorously proved via the M‐matrix analysis, which is from the spatial discretized matrix of IFL. Moreover, the proposed schemes use the fast sum‐of‐exponential approximation and Toeplitz matrix algorithms to reduce the computational cost for the nonlocal property of time and space fractional derivatives, respectively. The fast schemes greatly reduce the computational work of solving the discretized linear systems from $\mathcal{O}(M{N}^3+M^{2}N)$ by a direct solver to $\mathcal{O}(MN(\log N+N_{exp}))$ per preconditioned Krylov subspace iteration and a memory requirement from 𝒪(MN²) to 𝒪(NN_exp), where N and (N_exp ≪) M are the number of spatial and temporal grid nodes, respectively. The spectrum of preconditioned matrix is also given for ensuring the acceleration benefit of circulant preconditioners. Finally, numerical results are presented to show the utility of the proposed methods.

中文翻译：

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带有分数分数拉普拉斯算子的时空分数扩散方程的快速隐式差分格式

在本文中，我们开发了两种快速的隐式差分格式，用于解决一类具有整数分数拉普拉斯算子（IFL）的变系数时空分数扩散方程。拟议的方案利用Caputo分数阶导数的分级L 1公式和IFL的特殊有限差分离散化，其中分级网格可以在初始时间捕获具有弱奇异性的模型问题。通过M严格证明了稳定性和收敛性矩阵分析，来自IFL的空间离散矩阵。此外，提出的方案分别使用快速指数和和Toeplitz矩阵算法来分别降低时间和空间分数导数的非局部性质的计算成本。快速方案大大减少了求解离散线性系统的计算量。$\mathcal{Ø}（\mathrm{中号}{ñ}^3+{\mathrm{中号}}^2ñ）$ 由直接求解器 $\mathcal{Ø}（\mathrm{中号}ñ（\mathrm{日志}ñ+{ñ}_{ËXp}））$每预处理Krylov子空间迭代和从存储器需求𝒪（中号Ñ ²）至𝒪（Ñ Ñ _{Ë X p}），其中Ñ和（Ñ _{Ë X p}  «）中号是空间和时间网格节点的数量，分别。还给出了预处理矩阵的频谱，以确保循环预处理器的加速收益。最后，数值结果表明了所提方法的实用性。