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Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation

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Abstract

In this article, we are concerned with the analysis on the numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. This ill-posed problem is solved through a stabilized nonlinear minimization system by an appropriately selected Tikhonov regularization. The existence and the stability of the optimization system are demonstrated. The nonlinear optimization problem is approximated by a fully discrete scheme, whose convergence is established under a novel result verified in this study that the H1-norm of the solution to the discrete forward system is uniformly bounded. The iterative thresholding algorithm is proposed to solve the discrete minimization, and several numerical experiments are presented to show the efficiency and the accuracy of the algorithm.

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Acknowledgments

The authors thank Professor Bangti Jin (University College London) for his constructive discussions and appreciate the valuable comments by the anonymous referees.

Funding

The first author is financially supported by the National Natural Science Foundation of China (NSFC, Nos. 11871240, 11401241, 11571265) and NSFC-RGC (China-Hong Kong, No. 11661161017). The second author is financially supported by Japan Society for the Promotion of Science KAKENHI Grant Number JP15H05740. The last author is financially supported by the NSFC (Nos. 11871057 and 11501447).

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Authors and Affiliations

  1. School of Mathematics and Statistics & Hubei Key Laboratory of Mathematical Sciences, Central China Normal University, Wuhan, 430079, China

    Daijun Jiang

  2. Research Center of Mathematics for Social Creativity, Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo, 060-0812, Japan

    Yikan Liu

  3. Department of Mathematics and Center for Nonlinear Studies, Northwest University, Xi’an, 710075, Shaanxi, China

    Dongling Wang

Authors
  1. Daijun Jiang
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  2. Yikan Liu
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  3. Dongling Wang
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Correspondence to Yikan Liu.

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Communicated by: Martin Stynes

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Jiang, D., Liu, Y. & Wang, D. Numerical reconstruction of the spatial component in the source term of a time-fractional diffusion equation. Adv Comput Math 46, 43 (2020). https://doi.org/10.1007/s10444-020-09754-6

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  • DOI: https://doi.org/10.1007/s10444-020-09754-6

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