Anomalous diffusion in the presence or absence of an external force field is often modelled in terms of the fractional evolution equations, which can involve the hyper-singular source term. For this case, conventional time stepping methods may exhibit a severe order reduction. Although a second-order numerical algorithm is provided for the subdiffusion model with a simple hyper-singular source term $t^{\mu}$, $-2<\mu<-1$ in [arXiv:2207.08447], the convergence analysis remain to be proved. To fill in these gaps, we present a simple and robust smoothing method for the hyper-singular source term, where the Hadamard finite-part integral is introduced. This method is based on the smoothing/ID$m$-BDF$k$ method proposed by the authors [Shi and Chen, SIAM J. Numer. Anal., to appear] for subdiffusion equation with a weakly singular source term. We prove that the $k$th-order convergence rate can be restored for the diffusion-wave case $\gamma \in (1,2)$ and sketch the proof for the subdiffusion case $\gamma \in (0,1)$, even if the source term is hyper-singular and the initial data is not compatible. Numerical experiments are provided to confirm the theoretical results.

翻译：在存在或不存在外力场情况下的反常扩散通常用分数阶演化方程来建模，这类方程可能包含超奇异源项。针对这种情况，传统时间步进方法可能会出现严重的阶数降低。尽管在[arXiv:2207.08447]中针对带有简单超奇异源项$t^{\mu}$（$-2<\mu<-1$）的次扩散模型提出了二阶数值算法，但收敛性分析仍有待证明。为填补这些空白，我们提出了一种简洁且稳健的超奇异源项光滑化方法，其中引入了Hadamard有限部分积分。该方法基于作者[Shi and Chen, SIAM J. Numer. Anal., to appear]针对具有弱奇异源项的次扩散方程提出的光滑化/ID$m$-BDF$k$方法。我们证明了对于扩散-波情形$\gamma \in (1,2)$可恢复$k$阶收敛速率，并概述了次扩散情形$\gamma \in (0,1)$的证明思路，即使源项是超奇异的且初始数据不兼容。数值实验证实了理论结果。