This study addresses an inverse source problem for a general stationary kinetic equation involving absorption and scattering terms. The primary objective is the simultaneous reconstruction of the phase space particle distribution and an unknown source in the model, utilizing boundary conditions defined by derivatives and additional interior measurements. In the theoretical part of the study, the uniqueness of the solution to the inverse problem is proved. For the numerical solution, a comprehensive computational framework is developed based on finite difference approximations for derivatives and the trapezoidal rule for the discretization of the integral terms. The reconstruction strategy employs bilinear interpolation techniques coupled with Tikhonov regularization. In addition, a Monte Carlo noise sensitivity analysis is carried out to assess the stability and robustness of the proposed method under perturbed data. Finally, the effectiveness of the method is illustrated through numerical examples.

In this work, we consider an inverse problem for a stationary kinetic equation. Our aim is to determine the source function from boundary measurements together with additional information provided at an interior point of the domain. Unlike existing works, the boundary information comprises not the direct problem solution itself, but the gradients the gradients of the solution and the source function are prescribed on the boundary. We develop a numerical algorithm based on a hybrid strategy that combines the finite-difference method with a bilinear interpolation polynomial approximation. The performance of the proposed approach is demonstrated through several numerical experiments, and the results are reported comparatively via graphs and tables. The numerical tests indicate that the reconstruction errors for the unknown functions remain sufficiently small.