Analytic aspects of solution of model kinetic equations

The use of Case's method and singular integral equations in the solution of model Boltzmann equations for rarefied-gas dynamics is demonstrated. Exact solutions of the Boltzmann equation with BGK (Bhatnagar–Gross–Krook) collision operator are obtained for the first time in the problem of weak evaporation and in the problem of temperature and density discontinuities; an exact solution of the Boltzmann equation with collision operator of mixed type in the Kramers problem is also obtained. For the scalar BGK Boltzmann equation, the simplest known exact solution of the Kramers problem is obtained. For this, an expansion of a polynomial of arbitrary degree in the eigenfunctions of the corresponding characteristic equation is demonstrated.