Inverse Neumann Boundary Flux Identification in Two-Dimensional Heat Transfer via an Explicit Integral Transform Approach

This work addresses the inverse identification of a Neumann-type boundary heat flux, simultaneously dependent on the spatial coordinate and time, in a two-dimensional transient conduction problem in a rectangular domain. The direct problem is solved by means of the Classical Integral Transform Technique, which provides an explicit modal representation for the temperature field in terms of normalized eigenfunctions associated with the Neumann eigenvalue problem. This formulation allows , in the transformed space, the separation of the contributions of the initial condition and the boundary flux, leading to an analytical expression for the transformed temperatures as a function of the time-dependent modal coefficients of the unknown flux. In the inverse problem, the boundary flux is approximated by a truncated expansion in the eigenfunctions in the transverse direction, and the temporal coefficients of this expansion are explicitly recovered from transformed temperature data, through a centered finite-difference discretization of the transformed modal ordinary differential equations. A functional sensitivity analysis shows that, for each transverse mode, the choice of the first longitudinal mode maximizes the sensitivity of the transformed temperatures with respect to the flux coefficients, providing a simple and robust criterion for selecting the most informative mode. Additionally, the modal truncation level in the transverse direction is automatically determined from an adapted version of Morozov’s discrepancy principle, formulated in terms of the mean squared error between reconstructed temperatures and noisy data. The proposed methodology is numerically assessed with synthetic data, considering three flux profiles and different noise levels. The results show that the explicit formulation is able to reconstruct Neumann fluxes with good accuracy, while remaining simple to implement and computationally inexpensive.