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arXiv · 1707.00895

Application of the admitted Lie group of the \\ classical Boltzmann equation to classification of the \\ Boltzmann equation with a source term

Adisak Karnbanjong·Amornrat Suriyawichitseranee·Yurii N. Grigoriev·Sergey V. Meleshko
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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 $L_{11}$ 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 $L_{11}$, 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.

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Adisak Karnbanjong, Amornrat Suriyawichitseranee, Yurii N. Grigoriev, Sergey V. Meleshko. 2017-07-04. Application of the admitted Lie group of the \\ classical Boltzmann equation to classification of the \\ Boltzmann equation with a source term. https://arxiv.org/abs/1707.00895

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