We propose and analyze a time-stepping Crank-Nicolson(CN) alternating direction implicit(ADI) scheme combined with an arbitrary-order orthogonal spline collocation (OSC) methods in space for the numerical solution of the fractional integro-differential equation with a weakly singular kernel. We prove the stability of the numerical scheme and derive error estimates. The analysis presented allows variable time steps which, as will be shown, can efficiently be selected to match singularities in the solution induced by singularities in the kernel of the memory term. Finally, some numerical tests are given.

中文翻译：

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二维分数阶积分微分方程分级网格的快速ADI正交样条配置方法

我们提出并分析了时间步进的 Crank-Nicolson(CN) 交替方向隐式 (ADI) 方案结合任意阶正交样条搭配 (OSC) 方法在空间中用于分数阶积分微分方程的数值解单核。我们证明了数值方案的稳定性并推导出误差估计。所呈现的分析允许可变时间步长，如将显示的，可以有效地选择这些时间步长以匹配由记忆项的核中的奇点引起的解中的奇点。最后给出了一些数值试验。