Identification of the boundary heat transfer coefficient from interior measurement of temperature field

Consider the heat conduction process for a homogeneous solid rod with one endpoint contacted with some liquid media. The aim is to identify the boundary heat transfer coefficient from the measured temperature field, which is essentially nonlinear and ill-posed. Based on the 1-dimensional heat equation model with Robin boundary condition, we prove the recognizability of the time-dependent Robin coefficient from the temperature field specified at only one interior point. To give a stable reconstruction for noisy inversion input data, we propose a regularizing scheme by minimizing a cost functional including penalty term with rigorous mathematical analysis. A steepest iterative scheme is established to solve this non-quadratic optimizing problem.