论文标题
L1方案解决了由分数扩散方程的逆问题
L1 scheme for solving an inverse problem subject to a fractional diffusion equation
论文作者
论文摘要
本文考虑了由时间分数扩散方程到的逆问题的时间离散化。首先，L1方案的收敛性是通过频谱角$ <π/2 $的任意部门操作员建立的，即该操作员的分辨集合包含$ \ {z \ in \ in \ Mathbb c \ setMinus \ setMinus \ \ setminus \ \ \ \} 0 \ \}：\ | \ | \ | \ | \ | \ | \ | \ | \ | \ | | <} π$。 （0，1）$中的时间分数订单$α\与误差估计中的常数之间的关系是精确表征的，表明L1方案是可靠的，因为$α$接近$ 1 $。然后，分析了分数扩散方程的反问题，并给出了该反问题的时间离散化的收敛分析。最后，提供数值结果以确认理论结果。
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 $< π/2 $, that is the resolvent set of this operator contains $ \{z\in\mathbb C\setminus\{0\}:\ |\operatorname{Arg} z|< θ\}$ for some $ π/2 < θ< π$. The relationship between the time fractional order $α\in (0, 1)$ and the constants in the error estimates is precisely characterized, revealing that the L1 scheme is robust as $ α$ 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.
