Generally, solving linear systems from finite difference alternating direction implicit scheme of two-dimensional time-space fractional differential equations with Gaussian elimination requires \(\mathcal {O}\left ({NM}_{1}M_{2}\left ({M_{1}^{2}}+{M_{2}^{2}}+NM_{1}M_{2}\right )\right )\) complexity and \(\mathcal {O}\left ({N{M_{1}^{2}}{M_{2}^{2}}}\right )\) storage, where N is the number of temporal unknown and M₁, M₂ are the numbers of spatial unknown in x, y directions respectively. By exploring the structure of the coefficient matrix in fully coupled form, it possesses block lower-triangular Toeplitz structure and its blocks are block-dense Toeplitz matrices with dense-Toeplitz blocks. Based on this special structure and cooperating with time-marching or divide-and-conquer technique, two fast solvers with storage \(\mathcal {O}\left ({NM}_{1}M_{2}\right )\) are developed. The complexity for the fast solver via time-marching is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (N+\log \left (M_{1}M_{2}\right )\right )\right )\) and the one via divide-and-conquer technique is \(\mathcal {O}\left ({NM}_{1}M_{2}\left (\log ^{2} N+ \log \left (M_{1}M_{2}\right )\right )\right )\). It is worth to remark that the proposed solvers are not lossy. Some discussions on achieving convergence rate for smooth and non-smooth solutions are given. Numerical results show the high efficiency of the proposed fast solvers.

通常，从具有高斯消除的二维时空分数阶微分方程的有限差分交替方向隐式格式求解线性系统需要\（\ mathcal {O} \ left（{NM} _ {1} M_ {2} \ left（ {M_ {1} ^ {2}} + {M_ {2} ^ {2}} + NM_ {1} M_ {2} \ right} \ right）\）复杂度和\（\ mathcal {O} \ left（ {N {M_ {1} ^ {2}} {M_ {2} ^ {2}}} \ right）\）存储，其中N是时间未知数，M ₁，M ₂是空间未知数在x，y方向。通过探索完全耦合形式的系数矩阵的结构，它具有块下三角Toeplitz结构，并且其块是具有密集Toeplitz块的块密集Toeplitz矩阵。基于这种特殊的结构并与时间行进或分治技术配合使用，两个具有存储\（\ mathcal {O} \ left（{NM} _ {1} M_ {2} \ right} \）的快速求解器被开发。通过时间行进的快速求解器的复杂度为\（\ mathcal {O} \ left（{NM} _ {1} M_ {2} \ left（N + \ log \ left（M_ {1} M_ {2} \右）\右）\右）\），然后通过分治法获得的是\（\ mathcal {O} \ left（{NM} _ {1} M_ {2} \ left（\ log ^ {2 } N + \ log \ left（M_ {1} M_ {2} \ right）\ right）\ right）\）。值得指出的是，提出的求解器不是有损的。给出了关于实现平滑和非平滑解的收敛速度的一些讨论。数值结果表明，所提出的快速求解器具有很高的效率。