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

Yushan Li, Yuxuan Yang and Nanbo Chen ()
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Yushan Li: School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China
Yuxuan Yang: School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China
Nanbo Chen: School of Mathematics and Computing Sciences, Guilin University of Electronic Technology, Guilin 541004, China

Mathematics, 2025, vol. 13, issue 13, 1-22

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.

Keywords: Caputo fractional derivative; fractional Laplacian; inverse source problem; multi-term time–space fractional diffusion equation; optimal perturbation algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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