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Numerical Analysis and Scientific Computingmathscidoc:2103.25008

Inverse Problems, 32, (1), 015009, 2016.1
We consider an inverse problem of recovering a time-dependent factor of an unknown source on some subboundary for a diffusion equation with time fractional derivative by nonlocal measurement data. Such fractional-order equations describe anomalous diffusion of some contaminants in heterogeneous media such as soil and model the contamination process from an unknown source located on a part of the boundary of the concerned domain. For this inverse problem, we firstly establish the well-posedness in some Sobolev space. Then we propose two regularizing schemes in order to reconstruct an unknown boundary source stably in terms of the noisy measurement data. The first regularizing scheme is based on an integral equation of the second kind which an unknown boundary source solves, and we prove a convergence rate of regularized solutions with a suitable choice strategy of the regularizing parameter. The second regularizing scheme relies directly on discretization by the radial basis function for the initial-boundary value problem for fractional diffusion equation, and we carry out numerical tests, which show the validity of our proposed regularizing scheme.
anomalous diffusion, fractional order derivative, ill-posedness, regularization, stability
```@inproceedings{jijun2016on,
title={On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement},
author={Jijun Liu, Masahiro Yamamoto, and Liang Yan},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210325180038848054768},
booktitle={Inverse Problems},
volume={32},
number={1},
pages={015009},
year={2016},
}
```
Jijun Liu, Masahiro Yamamoto, and Liang Yan. On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement. 2016. Vol. 32. In Inverse Problems. pp.015009. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20210325180038848054768.