Identification of the time-dependent source term in a multi-term time-fractional diffusion equation

Abstract

The multi-term time-fractional diffusion equation is a useful tool in the modeling of complex systems. This paper aims to identifying a time-dependent source term in a multi-term time-fractional diffusion equation from the boundary Cauchy data. The regularity of the weak solution for the direct problem with homogeneous Neumann boundary condition is proved. We provide the uniqueness and a stability estimate for the inverse time-dependent source problem. On the other hand, the inverse time-dependent source term is formulated into a variational problem by the Tikhonov regularization, with the help of sensitivity problem and adjoint problem we use a conjugate gradient method to find the approximate time-dependent source term. Numerical experiments for five examples in one-dimensional and two-dimensional cases show that our proposed method is effective and stable.

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Correspondence to T. Wei.

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This paper was supported by the NSF of China (11371181, 11771192, 11601207) and Scientific Research Project of GanSu Political Science And Law Institute (2016XZDLW16).

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Li, Y.S., Sun, L.L., Zhang, Z.Q. et al. Identification of the time-dependent source term in a multi-term time-fractional diffusion equation. Numer Algor 82, 1279–1301 (2019). https://doi.org/10.1007/s11075-019-00654-5

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Keywords

  • Inverse source problem
  • Multi-term time-fractional diffusion equation
  • Conjugate gradient method