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An inverse time-dependent source problem for a time–space fractional diffusion equation

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  • S. Li, Y.
  • Wei, T.

Abstract

This paper is devoted to identify a time-dependent source term in a time–space fractional diffusion equation by using the usual initial and boundary data and an additional measurement data at an inner point. The existence and uniqueness of a weak solution for the corresponding direct problem with homogeneous Dirichlet boundary condition are proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. Based on the separation of variables, we transform the inverse source problem into a first kind Volterra integral equation with the source term as the unknown function and then show the ill-posedness of the problem. Further, we use a boundary element method combined with a generalized Tikhonov regularization to solve the Volterra integral equation of the fist kind. The generalized cross validation rule for the choice of regularization parameter is applied to obtain a stable numerical approximation to the time-dependent source term. Numerical experiments for six examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:apmaco:v:336:y:2018:i:c:p:257-271
    DOI: 10.1016/j.amc.2018.05.016
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    Cited by:

    1. Zhenping Li & Xiangtuan Xiong & Qiang Cheng, 2022. "Identifying the Unknown Source in Linear Parabolic Equation by a Convoluting Equation Method," Mathematics, MDPI, vol. 10(13), pages 1-17, June.
    2. Lopushansky, Andriy & Lopushansky, Oleh & Sharyn, Sergii, 2021. "Nonlinear inverse problem of control diffusivity parameter determination for a space-time fractional diffusion equation," Applied Mathematics and Computation, Elsevier, vol. 390(C).
    3. Li, Yixin & Hu, Xianliang, 2022. "Artificial neural network approximations of Cauchy inverse problem for linear PDEs," Applied Mathematics and Computation, Elsevier, vol. 414(C).

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