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

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 depen...

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Bibliographic Details
Published in:arXiv.org 2017-07
Main Authors: Karnbanjong, Adisak, Suriyawichitseranee, Amornrat, Grigoriev, Yurii N, Meleshko, Sergey V
Format: Article
Language:English
Subjects:
Boltzmann transport equation
Classification
Dependent variables
Differential equations
Distribution functions
Eulers equations
Independent variables
Invariants
Lie groups
Mathematical analysis
Rarefied gases
Representations
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Summary: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.
ISSN:2331-8422