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

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.