Theory of orthogonality of eigenfunctions of the characteristic equations as a method of solution boundary problems for model kinetic equations

TL;DR

This paper develops a theoretical framework based on the orthogonality of eigenfunctions of characteristic equations to solve boundary problems in kinetic theory, specifically for gas diffusion and slip problems.

Contribution

It introduces a new method using eigenfunction orthogonality derived from a Riemann boundary value problem for solving kinetic boundary value problems.

Findings

Analytical solutions for diffusion light component in binary gases.

Application of eigenfunction orthogonality to Kramers problem of isothermal slip.

Development of a theory connecting eigenfunction orthogonality with boundary problem solutions.

Abstract

We consider two classes of linear kinetic equations: with constant collision frequency and constant mean free path of gas molecules (i.e., frequency of molecular collisions, proportional to the modulus molecular velocity). Based homogeneous Riemann boundary value problem with a coefficient equal to the ratio of the boundary values dispersion function, develops the theory of the half-space orthogonality of generalized singular eigenfunctions corresponding characteristic equations, which leads separation of variables. And in this two boundary value problems of the kinetic theory (diffusion light component of a binary gas and Kramers problem about isothermal slip) shows the application of the theory orthogonality eigenfunctions for analytical solutions these tasks.