This paper considers the temporal discretization of an inverse problem subject to a time fractional diffusion equation. Firstly, the convergence of the L1 scheme is established with an arbitrary sectorial operator of spectral angle $< \pi/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< \theta\}$ for some $ \pi/2 < \theta < \pi $. The relationship between the time fractional order $\alpha \in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ \alpha $ approaches $ 1 $. Then an inverse problem of a fractional diffusion equation is analyzed, and the convergence analysis of a temporal discretization of this inverse problem is given. Finally, numerical results are provided to confirm the theoretical results.
翻译：本文考虑了在时间分化等式条件下对反问题的时间分解。 首先, L1 方案与光谱角的任意部门操作员 $ <\pi/2 $, 也就是此操作员的解析数集包含$z\ nin\ mathbb C\ setminus ⁇ 0 ⁇ :\\ ⁇ operatorname{Arg}z\\\\\theta ⁇ $, $\pi/2 <\theta < \pi$。 时间分数顺序 $\alpha\ in (0, 1) 和错误估计中的常数之间的关系得到了精确的描述, 显示L1 方案以 $\ alpha 接近$ $ 。 然后分析一个分数扩散方程式的反向问题, 并给出关于这一反问题的时间分解的趋同分析。 最后, 提供了数字结果以证实理论结果 。
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