Identification of a time-dependent source term for a time fractional diffusion problem

Abstract

In this paper, we study an inverse problem of identifying a time-dependent term of an unknown source for a time fractional diffusion equation using nonlocal measurement data. Firstly, we establish the conditional stability for this inverse problem. Then two regularization methods are proposed to for reconstructing the time-dependent source term from noisy measurements. The first method is an integral equation method which formulates the inverse source problem into an integral equation of the second kind; and a prior convergence rate of regularized solutions is derived with a suitable choice strategy of regularization parameters. The second method is a standard Tikhonov regularization method and formulates the inverse source problem as a minimizing problem of the Tikhonov functional. Based on the superposition principle and the technique of finite-element interpolation, a numerical scheme is proposed to implement the second regularization method. One- and two-dimensional examples are carried out to verify efficiency and stability of the second regularization method.

Keywords:

• Ill-posed problem
• inverse source problem
• regularization
• time fractional diffusion equation

AMS Subject Classifications:

• 65M30
• 65M32
• 35R30

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant number 11561003], [grant number 11661004]; Natural Science Foundation of Jiangxi Province of China [grant number 20151BAB201018]; Ground Project of Science and Technology of Jiangxi Universities [grant number KJLD14051]; Science and technology research project of Education Department of Jiangxi Province [grant number GJJ150568]; Key Laboratory for Radioactive Geology and Exploration Technology, Fundamental Science for National Defense [grant number RGET1513].