# The analytical solution of the problem on plasma oscillations in half-space with specular boundary conditions

## TL;DR

This paper provides an analytical solution to plasma oscillations in a half-space with specular boundary conditions, using kinetic equations and eigenfunction expansions to describe electron distribution and electric fields.

## Contribution

It introduces an exact analytical method for solving plasma oscillation problems with arbitrary electron degeneracy and specular boundary conditions.

## Key findings

- Derived explicit distribution functions for electrons.
- Obtained electric field solutions in plasma.
- Validated the eigenfunction expansion approach.

## Abstract

The boundary problem about behavior (oscillations) of the electronic plasmas with arbitrary degree of degeneration of electronic gas in half-space with specular boundary conditions is analytically solved. The kinetic equation of Vlasov--Boltzmann with integral of collisions of type BGK (Bhatnagar, Gross, Krook) and Maxwell equation for electric field are applied. Distribution function for electrons and electric field in plasma in the form of expansion under eigen solutions of the initial system of equations are received. Coefficients of these expansions are found by means of the boundary conditions.