Recovering time-dependent coefficients in a parabolic equation from two nonlocal measurements

Shufang Qiu; Hang Deng; Zewen Wang; Di Liu; Shufang Qiu; Hang Deng; Zewen Wang; Di Liu

doi:10.3934/math.2026035

AIMS Mathematics

2026, Volume 11, Issue 1: 810-824. doi: 10.3934/math.2026035

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Research article

Recovering time-dependent coefficients in a parabolic equation from two nonlocal measurements

1.
School of Artificial Intelligence, Guangzhou Maritime University, Guangzhou 510725, China
2.
Army Infantry Academy of PLA, Nanchang 330103, China
3.
School of Arts and Sciences, Guangzhou Maritime University, Guangzhou 510725, China

Received: 16 October 2025 Revised: 25 November 2025 Accepted: 01 December 2025 Published: 12 January 2026
MSC : 65M32, 35R30

This work investigates the inverse recovery of two time-dependent coefficients in a one-dimensional parabolic initial–boundary value problem using two nonlocal measurements. Under suitable regularity assumptions and a natural identifiability condition, we establish the existence and uniqueness of the reconstructed coefficients. We further clarify the intrinsic ill-posedness of the problem, noting that the reconstruction step inherently amplifies high-frequency noise, thereby necessitating regularization. Building on these insights, we develop a Crank–Nicolson finite-difference inversion method, which at each half time step reduces to solving a small linear system for the two coefficients. To stabilize the measured data, we employ mollification. Numerical experiments with synthetic data demonstrate accurate reconstructions under small noise levels and reveal that the recovery of the first coefficient is more sensitive to noise than that of the second. The results highlight the critical roles of identifiability and data smoothing in achieving stable performance.
- parabolic equation,
- inverse problem,
- finite difference,
- nonlocal measurement,
- mollification method
Citation: Shufang Qiu, Hang Deng, Zewen Wang, Di Liu. Recovering time-dependent coefficients in a parabolic equation from two nonlocal measurements[J]. AIMS Mathematics, 2026, 11(1): 810-824. doi: 10.3934/math.2026035

Related Papers:

Abstract

This work investigates the inverse recovery of two time-dependent coefficients in a one-dimensional parabolic initial–boundary value problem using two nonlocal measurements. Under suitable regularity assumptions and a natural identifiability condition, we establish the existence and uniqueness of the reconstructed coefficients. We further clarify the intrinsic ill-posedness of the problem, noting that the reconstruction step inherently amplifies high-frequency noise, thereby necessitating regularization. Building on these insights, we develop a Crank–Nicolson finite-difference inversion method, which at each half time step reduces to solving a small linear system for the two coefficients. To stabilize the measured data, we employ mollification. Numerical experiments with synthetic data demonstrate accurate reconstructions under small noise levels and reveal that the recovery of the first coefficient is more sensitive to noise than that of the second. The results highlight the critical roles of identifiability and data smoothing in achieving stable performance.

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