Simultaneous determination of the zeroth-order coefficient and source term in a time-fractional diffusion equation
This paper investigates the simultaneous inversion problem of recovering the zeroth-order coefficient and source term in a time-fractional diffusion equation based on nonlocal integral observations. Compared with the linear simultaneous inversion problem, this problem is more difficult to solve. In...
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| Published in: | Chaos, solitons and fractals Vol. 202; p. 117511 |
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| Format: | Journal Article |
| Language: | English |
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01.01.2026
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| ISSN: | 0960-0779 |
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| Abstract | This paper investigates the simultaneous inversion problem of recovering the zeroth-order coefficient and source term in a time-fractional diffusion equation based on nonlocal integral observations. Compared with the linear simultaneous inversion problem, this problem is more difficult to solve. In terms of theoretical analysis, we define the weak solution for the forward problem and establish its existence and uniqueness, and the ill-posedness of the simultaneous inversion problem is analyzed. To address this ill-posed problem, we employ the Tikhonov regularization method to reformulate it as a variational problem. The existence of a minimizer for the variational functional is rigorously proved, and convergence rate estimates are derived under both the a priori and a posteriori regularization parameter choice strategies, subject to an appropriate source condition. For the numerical implementation, by introducing the sensitivity problems and the adjoint problem, we obtain the Fréchet gradients of the functional with respect to the zeroth-order coefficient and the source term. The variational problem is then solved using the conjugate gradient algorithm. Finally, several numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed method.
•Recovery of a coefficient and a source in a time-fractional diffusion model.•A conjugate gradient algorithm is designed for numerical implementation.•Numerical experiments verify the effectiveness of the proposed method. |
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| AbstractList | This paper investigates the simultaneous inversion problem of recovering the zeroth-order coefficient and source term in a time-fractional diffusion equation based on nonlocal integral observations. Compared with the linear simultaneous inversion problem, this problem is more difficult to solve. In terms of theoretical analysis, we define the weak solution for the forward problem and establish its existence and uniqueness, and the ill-posedness of the simultaneous inversion problem is analyzed. To address this ill-posed problem, we employ the Tikhonov regularization method to reformulate it as a variational problem. The existence of a minimizer for the variational functional is rigorously proved, and convergence rate estimates are derived under both the a priori and a posteriori regularization parameter choice strategies, subject to an appropriate source condition. For the numerical implementation, by introducing the sensitivity problems and the adjoint problem, we obtain the Fréchet gradients of the functional with respect to the zeroth-order coefficient and the source term. The variational problem is then solved using the conjugate gradient algorithm. Finally, several numerical examples are provided to demonstrate the feasibility and effectiveness of the proposed method.
•Recovery of a coefficient and a source in a time-fractional diffusion model.•A conjugate gradient algorithm is designed for numerical implementation.•Numerical experiments verify the effectiveness of the proposed method. |
| ArticleNumber | 117511 |
| Author | Xiong, Xiangtuan Qiao, Yu Li, Zhenping |
| Author_xml | – sequence: 1 givenname: Yu surname: Qiao fullname: Qiao, Yu email: 15514358907@163.com organization: School of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou 450002, China – sequence: 2 givenname: Xiangtuan surname: Xiong fullname: Xiong, Xiangtuan organization: Department of Mathematics, Northwest Normal University, Lanzhou 730070, China – sequence: 3 givenname: Zhenping surname: Li fullname: Li, Zhenping organization: Department of Mathematics and Physics, Luoyang Institute of Science and Technology, Luoyang 471023, China |
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| Keywords | 65M30 Ill-posed problem 35R25 Tikhonov regularization method Simultaneous inversion problem Error estimate |
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| Title | Simultaneous determination of the zeroth-order coefficient and source term in a time-fractional diffusion equation |
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