Abstract
This paper deals with an inverse problem on simultaneously determining a time-dependent potential term and a time source function from two-point measured data in a multi-term time-fractional diffusion equation. First we study the existence, uniqueness and some regularities of the solution for the direct problem by using the fixed point theorem. Then a nice conditional stability estimate of inversion coefficients problem is obtained based on the regularity of the solution to the direct problem and a fine property of the Caputo fractional derivative. In addition, the ill-posedness of the inverse problem is illustrated and we transfer the inverse problem into a variational problem. Moreover, the existence and convergence of the minimizer for the variational problem are given. Finally, we use a modified Levenberg–Marquardt method to reconstruct numerically the approximate functions of two unknown time-dependent coefficients effectively. Numerical experiments for three examples in one- and two-dimensional cases are provided to show the validity and robustness of the proposed method.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 12201502
Funding source: Science Fund for Distinguished Young Scholars of Gansu Province
Award Identifier / Grant number: 20JR10RA099
Funding source: Special Fund Project of Guiding Scientific and Technological Innovation Development of Gansu Province
Award Identifier / Grant number: 2020B-088
Funding statement: This work is supported by the NSF of China (grant no. 12201502), the Youth Science and Technology Fund of Gansu Province (grant no. 20JR10RA099) and the Innovation Capacity Improvement Project for Colleges and Universities of Gansu Province (grant no. 2020B-088).
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