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Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation

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  • Yan, Xiong-bin
  • Zhang, Zheng-qiang
  • Wei, Ting

Abstract

The main purpose of this paper is to identify simultaneously a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation from two points observed data. First of all, using the fixed point theorem, we prove the existence and uniqueness of the solution for the direct problem. Secondly, the stability of the inverse problem is proved and the uniqueness is a direct result of the stability estimate. In addition, we illustrate the ill-posedness of the inverse problem and use a non-stationary iterative Tikhonov regularization method to recover numerically the time dependent potential coefficient and source term. At the same time, we give the existence of the minimizer for the minimization functional. In order to solve the minimization problem, we apply an alternating minimization method to find the minimizer and prove the solving sub-problems are stable on noisy data as well as prove the data fidelity item decreases monotonously with the iterative running. Finally, some numerical examples are provided to shed light on the validity and robustness of the numerical algorithm.

Suggested Citation

  • Yan, Xiong-bin & Zhang, Zheng-qiang & Wei, Ting, 2022. "Simultaneous inversion of a time-dependent potential coefficient and a time source term in a time fractional diffusion-wave equation," Chaos, Solitons & Fractals, Elsevier, vol. 157(C).
    • Handle: RePEc:eee:chsofr:v:157:y:2022:i:c:s0960077922001114
    DOI: 10.1016/j.chaos.2022.111901
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    References listed on IDEAS

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    1. Hao, Zhaopeng & Fan, Kai & Cao, Wanrong & Sun, Zhizhong, 2016. "A finite difference scheme for semilinear space-fractional diffusion equations with time delay," Applied Mathematics and Computation, Elsevier, vol. 275(C), pages 238-254.
    2. Giona, Massimiliano & Cerbelli, Stefano & Roman, H.Eduardo, 1992. "Fractional diffusion equation and relaxation in complex viscoelastic materials," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 191(1), pages 449-453.
    3. M. Hanke & C. W. Groetsch, 1998. "Nonstationary Iterated Tikhonov Regularization," Journal of Optimization Theory and Applications, Springer, vol. 98(1), pages 37-53, July.
    4. An, Na & Huang, Chaobao & Yu, Xijun, 2019. "Error analysis of direct discontinuous Galerkin method for two-dimensional fractional diffusion-wave equation," Applied Mathematics and Computation, Elsevier, vol. 349(C), pages 148-157.
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