Inverse source problem for a diffusion equation involving the fractional spectral Laplacian
In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial inter...
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| Veröffentlicht in: | Mathematical methods in the applied sciences Jg. 44; H. 1; S. 917 - 936 |
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15.01.2021
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| Abstract | In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach. |
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| AbstractList | In this paper, we consider an inverse problem related to a fractional diffusion equation. The model problem is governed by a nonlinear partial differential equation involving the fractional spectral Laplacian. This study is focused on the reconstruction of an unknown source term from a partial internal measured data. The considered ill‐posed inverse problem is formulated as a minimization one. The existence, uniqueness, and stability of the solution are discussed. Some theoretical results are established. The numerical reconstruction of the unknown source term is investigated using an iterative process. The proposed method involves a denoising procedure at each iteration step and provides a sequence of source term approximations converging in norm to the actual solution of the minimization problem. Some numerical results are presented to show the efficiency and the accuracy of the proposed approach. |
| Author | Hassine, Maatoug BenSalah, Mohamed |
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| Cites_doi | 10.1007/s11118-014-9443-4 10.1080/03605301003735680 10.1007/978-3-642-65024-6 10.1016/j.bulsci.2011.12.004 10.1007/s10543-014-0484-2 10.1109/TIP.2003.814255 10.1115/1.4000563 10.1002/cpa.20042 10.1016/j.na.2018.10.016 10.1002/mana.201500041 10.1016/S0165-1684(03)00150-6 10.1016/j.aim.2010.01.025 10.2307/2372313 10.1007/s00526-014-0815-9 10.1103/PhysRevE.94.052147 10.1016/j.cnsns.2015.01.005 10.1103/PhysRevLett.115.180403 10.3934/dcdss.2014.7.857 10.1515/fca-2017-0002 10.1002/2015WR018515 10.1142/p614 10.1039/C4CP03465A 10.1016/j.anihpc.2015.01.004 |
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| SubjectTerms | fractional diffusion equation fractional spectral Laplacian Inverse problems inverse source problem Iterative methods minimization problem Noise reduction Nonlinear differential equations nonlocal operator numerical reconstruction algorithm Optimization Partial differential equations Reconstruction source term |
| Title | Inverse source problem for a diffusion equation involving the fractional spectral Laplacian |
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