We study a one-dimensional heat equation with dynamic integral boundary conditions together with an inverse problem for recovering an unknown boundary source from interior temperature measurements. For the direct problem, the existence and uniqueness of a smooth solution are discussed, and a weighted finite difference scheme with weight parameter σ is constructed and analyzed with respect to approximation, stability, and convergence. For the inverse problem, we propose a two-step reconstruction algorithm based on weighted and improved weighted finite difference discretizations. The numerical results show that the scheme with weight σ=0.5 provides higher accuracy for noise-free or weakly perturbed data, while the scheme with σ=1 exhibits better stability as the noise level increases or the measurement point moves farther from the unknown boundary. To improve stability in the presence of noisy data, a regularized reconstruction approach is introduced. Numerical experiments for different measurement locations and noise levels up to 10% illustrate the convergence and stability behavior of the proposed methods and confirm the efficiency of the regularized algorithm.
Koleva et al. (Tue,) studied this question.