A computational approach to an inverse source problem for a kinetic equation with gradient-type boundary data and an interior point observation

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.