On the reconstruction of unknown time-dependent boundary sources for time fractional diffusion process by distributing measurement

Jijun Liu Southeast University Masahiro Yamamoto The University of Tokyo Liang Yan Southeast University

Numerical Analysis and Scientific Computing mathscidoc: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 heterogeneous 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 measurement 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
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@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.
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