Kinetic equations (the Boltzmann, the neutron transport equation etc.) are known to converge to fluid equations (the Euler, the heat equation etc.) in certain regimes, but when kinetic and fluid regime co-exist,  how to couple the two systems remains an open problem. The key is to understand the half-space problem that resembles the boundary layer at the interface. In this talk, I will present a unified proof for the well-posedness of a class of half-space equations with general incoming data, propose an efficient spectral solver, and utilize it to couple fluid with kinetics. Moreover, I will present complete error analysis for the proposed spectral solver. Numerical results will be shown to demonstrate the accuracy of the algorithm.