Hopf's lemma and uniqueness of simultaneously determining source profile and Robin coefficient in a fractional diffusionequation by interior data

Daijun Jiang; Zhiyuan Li

doi:10.3934/ipi.2024009

Article Contents

2024, Volume 18, Issue 5: 1121-1141. Doi: 10.3934/ipi.2024009

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Hopf's lemma and uniqueness of simultaneously determining source profile and Robin coefficient in a fractional diffusionequation by interior data

Daijun Jiang^{1, 2,} and
Zhiyuan Li^3, ,

1.
School of Mathematics and Statistics, Central China Normal University, China
2.
Key Laboratory of Nonlinear Analysis and Applications (Ministry of Education), Central China Normal University, China
3.
School of Mathematics and Statistics, Ningbo University, China

^*Corresponding author: Zhiyuan Li
^*Corresponding author: Zhiyuan Li

Received: October 2023

Revised: January 2024

Early access: February 2024

Published: October 2024

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Abstract

This paper is devoted to an inverse problem of simultaneously determining the spatially dependent source term and the Robin boundary coefficient in a time fractional diffusion equation, with the aid of extra measurement data at a subdomain near the accessible boundary. Firstly, the spatially varying source is uniquely determined in view of the unique continuation principle and Duhamel principle for the fractional diffusion equation. The Hopf lemma for a homogeneous time-fractional diffusion equation is proved and then used to prove the uniqueness of recovering the Robin boundary coefficient. Numerically, based on the theoretical uniqueness result, we apply the classical Tikhonov regularization method to transform the inverse problem into a minimization problem, which is solved by an iterative thresholding algorithm. Finally, several numerical examples are presented to show the accuracy and effectiveness of the proposed algorithm.

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Mathematics Subject Classification: Primary: 35R30, 35R11; Secondary: 47A52.

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Figure 1. Exact and reconstructed solutions for Example 5.6: $ K = 20 $, $ e_f = 0.0450 $, $ e_\gamma = 0.0182 $

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Figure 2. Exact and reconstructed solutions for Example 5.7: $ K = 25 $, $ e^K_f = 0.0389 $, $ e^K_\gamma = 0.0159 $

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