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Journal of Applied Mathematics
Volume 2016 (2016), Article ID 5834620, 8 pages
http://dx.doi.org/10.1155/2016/5834620
Research Article

Boltzmann’s Six-Moment One-Dimensional Nonlinear System Equations with the Maxwell-Auzhan Boundary Conditions

1Research Laboratory “Applied Modeling Oil and Gas Fields”, Kazakh-British Technical University, 59 Tole Bi Street, Almaty 050000, Kazakhstan
2Al-Farabi Kazakh National University, 71 Al-Farabi Avenue, Almaty 050040, Kazakhstan

Received 2 February 2016; Revised 1 June 2016; Accepted 5 June 2016

Academic Editor: Peter G. L. Leach

Copyright © 2016 A. Sakabekov and Y. Auzhani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We prove existence and uniqueness of the solution of the problem with initial and Maxwell-Auzhan boundary conditions for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations in space of functions continuous in time and summable in square by a spatial variable. In order to obtain a priori estimation of the initial and boundary value problem for nonstationary nonlinear one-dimensional Boltzmann’s six-moment system equations we get the integral equality and then use the spherical representation of vector. Then we obtain the initial value problem for Riccati equation. We have managed to obtain a particular solution of this equation in an explicit form.