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Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations

Author

Listed:
  • Yushan Li

    (School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China
    Center for Applied Mathematics of Guangxi (GUET), Guilin 541004, China
    Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin 541004, China)

  • Yuxuan Yang

    (School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China
    Department of Electronics and Information Engineering, Bozhou University, Bozhou 236800, China)

  • Nanbo Chen

    (School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China
    Center for Applied Mathematics of Guangxi (GUET), Guilin 541004, China
    Guangxi Colleges and Universities Key Laboratory of Data Analysis and Computation, Guilin 541004, China)

Abstract

This paper investigates the inverse problem of identifying a time-dependent source term in multi-term time–space fractional diffusion Equations (TSFDE). First, we rigorously establish the existence and uniqueness of strong solutions for the associated direct problem under homogeneous Dirichlet boundary conditions. A novel implicit finite difference scheme incorporating matrix transfer technique is developed for solving the initial-boundary value problem numerically. Regarding the inverse problem, we prove the solution uniqueness and stability estimates based on interior measurement data. The source identification problem is reformulated as a variational problem using the Tikhonov regularization method, and an approximate solution to the inverse problem is obtained with the aid of the optimal perturbation algorithm. Extensive numerical simulations involving six test cases in both 1D and 2D configurations demonstrate the high effectiveness and satisfactory stability of the proposed methodology.

Suggested Citation

  • Yushan Li & Yuxuan Yang & Nanbo Chen, 2025. "Identification of a Time-Dependent Source Term in Multi-Term Time–Space Fractional Diffusion Equations," Mathematics, MDPI, vol. 13(13), pages 1-22, June.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:13:p:2123-:d:1690153
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    References listed on IDEAS

    as
    1. Li, Zhiyuan & Liu, Yikan & Yamamoto, Masahiro, 2015. "Initial-boundary value problems for multi-term time-fractional diffusion equations with positive constant coefficients," Applied Mathematics and Computation, Elsevier, vol. 257(C), pages 381-397.
    2. S. Li, Y. & Wei, T., 2018. "An inverse time-dependent source problem for a time–space fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 336(C), pages 257-271.
    3. Djennadi, Smina & Shawagfeh, Nabil & Abu Arqub, Omar, 2021. "A fractional Tikhonov regularization method for an inverse backward and source problems in the time-space fractional diffusion equations," Chaos, Solitons & Fractals, Elsevier, vol. 150(C).
    4. Zhang, Hui & Jiang, Xiaoyun & Yang, Xiu, 2018. "A time-space spectral method for the time-space fractional Fokker–Planck equation and its inverse problem," Applied Mathematics and Computation, Elsevier, vol. 320(C), pages 302-318.
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