Abstract A high-order compact alternating direction implicit scheme is considered to solve the two-dimensional time-fractional integro-differential equation with weak singularity near the initial time in this paper. The L1 formula and trapezoidal PI rule on nonuniform meshes, which greatly improve the temporal accuracy compared to the method on uniform grids, are adopted to approximate the Caputo derivative and the Riemann-Liouville integral, respectively. With the help of a modified discrete fractional Gronwall inequality and some crucial skills, the stability and convergence of the proposed scheme are analyzed. Numerical results confirm the sharpness of the error analysis.

中文翻译：

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具有奇异核的二维时间分数积分微分方程的紧凑 L1-ADI 方案的尖锐误差估计

摘要 本文考虑了求解初始时刻附近弱奇异性二维时间分数阶积分微分方程的一种高阶紧交替方向隐式格式。与均匀网格上的方法相比，非均匀网格上的 L1 公式和梯形 PI 规则大大提高了时间精度，分别用于逼近 Caputo 导数和 Riemann-Liouville 积分。借助改进的离散分数格朗沃尔不等式和一些关键技能，分析了所提出方案的稳定性和收敛性。数值结果证实了误差分析的尖锐性。