On the Simultaneous Recovery of Boundary Heat Transfer Coefficient and Initial Heat Status

Consider the heat conduction process with heat flux exchanges on the boundary of a 2-dimensional domain. The aim is to identify both the boundary heat exchange coefficient and the initial heat distribution simultaneously from the final measurement data of the heat field. We prove the uniqueness for this nonlinear inverse problem for the strictly positive exchange coefficient and initial heat distribution in terms of the maximal principle and the eigenfunction expansions. Then a regularizing scheme combining the data mollification and quasi-reversibility method along time direction together is established to recover two unknowns, with the choice strategies for the regularizing parameters and the error estimates on the regularizing solutions. The reconstruction implementations are carried out in terms of the potential representation of the heat field, with numerical examples showing the validity of the proposed scheme.