Preliminary group classification of the full Boltzmann equation with a source term

Abstract

The classical Boltzmann equation is an integro-differential equation which describes the time evolution of rarefied gas in terms of a molecular distribution function. For some kinetic problems where it is necessary to add in the Boltzmann equation a source term depending on the independent and dependent variables. This paper is devoted to applying preliminary group classification to the Boltzmann equation with a source function by using the Lie group L11 admitted by the classical Boltzmann equation.The developed strategy for deriving determining equation of an integro-differential equation with a source (in general form) using a known Lie group admitted by the corresponding equation without the source is applied to the Boltzmann equation with a source. Solving the determining equation for the source function for each subalgebra of the optimal system of subalgebras of the Lie algebra L11, a preliminary group classification of the Boltzmann equation with respect to the source function is obtained.Furthermore, representations of invariant solutions of the Boltzmann equation with a source are presented. The reduced equations are also shown for some representations of invariant solutions.