Abstract In this paper, we consider a fast algorithm to calculate a two-dimensional nonlinear time distributed-order and space fractional diffusion equation, which is called the time two-mesh (TT-M) finite element (FE) method. In time, the TT-M algorithm combined with both the implicit second-order σ backward difference formula and Crank–Nicolson scheme for computing the numerical solution at time t 1 is used to speed up the calculation. At the same time, the spatial direction is approximated by the FE method. The detailed analyses of stability and error are also given, and the second-order time convergence accuracy can be arrived at. Finally, some numerical examples are shown to illustrate the effectiveness of our numerical method.

中文翻译：

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二维非线性时间分布阶次空间分数扩散方程的TT-M有限元方法

摘要 在本文中，我们考虑了一种快速算法来计算二维非线性时间分布阶次和空间分数扩散方程，称为时间二维网格（TT-M）有限元（FE）方法。随着时间的推移，TT-M 算法结合隐式二阶 σ 后向差分公式和 Crank-Nicolson 方案来计算时间 t 1 的数值解，以加快计算速度。同时，空间方向通过有限元方法近似。还给出了稳定性和误差的详细分析，可以得到二阶时间收敛精度。最后，给出了一些数值例子来说明我们数值方法的有效性。