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arXiv · 2006.04291

L1 scheme for solving an inverse problem subject to a fractional diffusion equation

Binjie Li·Xiaoping Xie·Yubin Yan
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Abstract

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.

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Binjie Li, Xiaoping Xie, Yubin Yan. 2020-06-08. L1 scheme for solving an inverse problem subject to a fractional diffusion equation. https://arxiv.org/abs/2006.04291

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