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Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation

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Abstract

The present paper is devoted to identifying the space-dependent source term and initial value simultaneously for a time-fractional diffusion equation. This inverse problem is reformulated on the basis of Fourier method as operator equations of the first kind. The construction of the solution to the inverse problem is derived by using the Landweber iteration method for the solution of the corresponding conjugate operator equation. In this paper, we also present error estimates between the exact solution and the regularized solution by the a priori and the a posteriori parameter choice rules. Numerical verification on the efficiency and accuracy of the proposed algorithm is performed by solving three numerical examples in one-dimensional and two-dimensional cases.

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Acknowledgements

The authors would like to express their most sincere thanks to the editor and referees for their very helpful comments and suggestions, which greatly improved the quality of this paper. The work described in this article was supported by the NNSF of China (11326234), NSF of Gansu Province (145RJZA099), Scientific research project of Higher School in Gansu Province (2014A-012), and Project of NWNU-LKQN2020-08.

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Jin Wen, Zhuan-Xia Liu & Chong-Wang Yue

  2. School of Traffic and Transportation, Lanzhou Jiaotong University, Lanzhou, 730070, People’s Republic of China

    Shi-Juan Wang

  3. School of Information Engineering, Lanzhou University of Finance and Economics, Lanzhou, 730020, People’s Republic of China

    Shi-Juan Wang

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  1. Jin Wen
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Wen, J., Liu, ZX., Yue, CW. et al. Landweber iteration method for simultaneous inversion of the source term and initial data in a time-fractional diffusion equation. J. Appl. Math. Comput. 68, 3219–3250 (2022). https://doi.org/10.1007/s12190-021-01656-0

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