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Solving the backward problem for space-fractional diffusion equation by a fractional Tikhonov regularization method

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  • Zheng, Guang-Hui
  • Zhang, Quan-Guo

Abstract

In this paper, we consider the backward problem for diffusion equation with space-fractional Laplacian. In order to overcome the ill-posedness of the backward problem, we propose a fractional Tikhonov regularization method to solve it. Based on the conditional stability estimate and an a posteriori regularization parameter choice rule, the convergence rate estimate is presented under a-priori bound assumption for the exact solution. Finally, several numerical examples are given to show that the proposed numerical methods are effective.

Suggested Citation

  • Zheng, Guang-Hui & Zhang, Quan-Guo, 2018. "Solving the backward problem for space-fractional diffusion equation by a fractional Tikhonov regularization method," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 148(C), pages 37-47.
    • Handle: RePEc:eee:matcom:v:148:y:2018:i:c:p:37-47
    DOI: 10.1016/j.matcom.2017.12.005
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    1. Gorenflo, Rudolf & Mainardi, Francesco & Moretti, Daniele & Pagnini, Gianni & Paradisi, Paolo, 2002. "Fractional diffusion: probability distributions and random walk models," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 305(1), pages 106-112.
    2. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    3. Tian, WenYi & Li, Can & Deng, Weihua & Wu, Yujiang, 2012. "Regularization methods for unknown source in space fractional diffusion equation," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 85(C), pages 45-56.
    4. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
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    Cited by:

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    2. Songshu Liu, 2022. "Recovering a Space-Dependent Source Term in the Fractional Diffusion Equation with the Riemann–Liouville Derivative," Mathematics, MDPI, vol. 10(17), pages 1-13, September.

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