New method of solving of boundary problems in kinetic theory

TL;DR

This paper introduces a new method for solving boundary problems in kinetic theory, exemplified by the Kramers problem, using integral equations and Neumann series to achieve solutions with arbitrary accuracy.

Contribution

The paper presents a novel approach that transforms boundary conditions into sources within the kinetic equation, enabling highly accurate solutions for boundary problems.

Findings

Method allows solutions with any desired accuracy.

Uses integral equations of Fredholm type.

Employs Neumann series for solution expansion.

Abstract

The classical Kramers problem with specular -- diffuse boundary conditions of the kinetic theory is considered. On an example of Kramers problem the new method of the decision of the boundary problems of the kinetic theory is stated. The method allows to receive the decision with any degree of accuracy. At the basis of a method lays the idea of representation of a boundary condition on distribution function in the form of a source in the kinetic equation. By means of integrals Furier the kinetic equation with a source is reduced to the integral equation of Fredholm type of the second kind. The decision is received in the form of Neumann's series.