# The numerical solution of the initial and boundary value problem for one-dimensional nonstationary nonlinear Boltzmann’s six-moment system equations with the Vladimirov-Marshak boundary conditions

### Abstract

The Boltzmann equation is a complex integral-differential equation and the basis of the kinetic theory of gases. It describes the behavior of a rarefied gas in space of time and velocity. It is used to the study of electron transport in solids and plasmas, neutron transport in nuclear reactors, in the tasks of remote sensing of the Earth from Space. The moment method is one of effective methods for solution of the Boltzmann equation. The system of Boltzmann’s moment equations is intermediate between kinetic and hydrodynamic levels of description of state of the rarefied gas and form class of nonlinear partial differential equations. If particle distribution function will be decomposed into an Fourier series on complete orthogonal system of functions, then Boltzmann’s equation will be equivalent to an infinite system of partial differential equations relative to the moments of the particle distribution function in the complete system of eigenfunctions of linearized operator. But solving infinite system of differential equations impossible. Therefore, an approximate solution of the initial and boundary value problem for the Boltzmann equation can be determined by the moment method. This article describes the one-dimensional nonlinear nonstationary Boltzmann’s moment system equations in the third approximation, which a hyperbolic system of partial differential equations and contains six equations for the moments of the particle distribution function. And formulates the statement of the initial and boundary value problem for the Boltzmann’s moment system equations in the third approximation and shows the results of the numerical solution with the Vladimirov-Marshak boundary conditions.### References

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[9] Sakabekov, Auzhan, and Auzhani, Yerkanat. “Boundary conditions for the onedimensional nonlinear nonstationary Boltzmann’s moment system equations.” Journal of Mathematical Physics (2014): 23–50.

[10] Mischler, Stephane. “Kinetic equations with Maxwell boundary conditions.” Annales Scientifiques de l’Ecole Normale Superieure (2010): 719–760.

[2] Grad, Harold. “Principle of the kinetic theory of gases.” Handbuch der Physik (1958): 205–294.

[3] Sakabekov, Auzhan. Initial-Boundary Value Problems for the Boltzmann’s Moment System Equations. Almaty: Gylym, 2002.

[4] Cercignani, Carlo. “Theory and Application of the Boltzmann Equation.” Journal of Applied Mechanics (1976): 502–521.

[5] Kogan, Mikhail. Dynamic of Rarefied Gas. Moscow: Nauka, 1967.

[6] Kumar, Kailash. “Polynomial expansions in kinetic theory of gases.” Annals of Physics (1966): 113–141.

[7] Neudachin, Vladimir G., and Smirnov, Yuriy F. Nucleon Association of Easy Kernel. Moscow: Nauka, 1969.

[8] Moshinsky, Marcos. The Harmonic Oscillator in Modern Physics: from Atoms to Quarks. New York: Gordon and Breach, 1960.

[9] Sakabekov, Auzhan, and Auzhani, Yerkanat. “Boundary conditions for the onedimensional nonlinear nonstationary Boltzmann’s moment system equations.” Journal of Mathematical Physics (2014): 23–50.

[10] Mischler, Stephane. “Kinetic equations with Maxwell boundary conditions.” Annales Scientifiques de l’Ecole Normale Superieure (2010): 719–760.

How to Cite

SAKABEKOV, A.; TLEUOVA, G..
The numerical solution of the initial and boundary value problem for one-dimensional nonstationary nonlinear Boltzmann’s six-moment system equations with the Vladimirov-Marshak boundary conditions.

**Journal of Mathematics, Mechanics and Computer Science**, [S.l.], v. 93, n. 1, p. 46-54, june 2018. ISSN 2617-4871. Available at: <https://bm.kaznu.kz/index.php/kaznu/article/view/433>. Date accessed: 20 jan. 2021.
Section

Mathematics

Keywords
Boltzmann’s six moment system equations, Vladimirov-Marshak boundary conditions, particle distribution function