On the Riemann matrix boundary value problem in the analytical solution of gas theory boundary value problems

The paper discusses the issue of generalizing and expanding the method of the canonical matrix for solving boundary value problems for an equation like the linearized Boltzmann equation with a collision integral in the form of Bhatnagar-Gross-Kruk (BGK-equation) depending on the properties of the corresponding Riemann matrix boundary value problem. The eigenvalues of the characteristic equation constitute a discrete and continuous spectrum, and the eigenfunctions of the continuous spectrum belong to the class of generalized functions, the use of which leads to the need to solve the Riemann matrix edge problem. Described is an algorithm for constructing a canonical matrix of a problem, its properties are investigated. The necessary conditions for the applicability of the method for the general case are formulated. For the Smolukhovsky problem, a canonical matrix is constructed for the BGK equation in the case of a one-, two-, and polyatomic gas. The theorem on the completeness of the set of eigenfunctions in the space of functions Gelder on the positive real half-axis is proved.