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Theoretical and Mathematical Physics

, Volume 97, Issue 2, pp 1299–1311 | Cite as

Analytic solution of vector model kinetic equations with constant kernel and their applications

  • A. V. Latyshev
Article

Abstract

Exact solutions are obtained for the first time for the half-space boundary-value problem for the vector model kinetic equations
$$\begin{gathered} \mu \frac{\partial }{{\partial x}}\Psi (x,\mu ) + \sum \Psi (x,\mu ) = C\int_{ - \infty }^\infty {\exp ( - } \mu '^2 )\Psi (x,\mu ')d\mu ', \hfill \\ \mathop {\lim }\limits_{x \to 0 + } \Psi (x,\mu ) = \Psi _0 (\mu ), \mu > 0, \mathop {\lim }\limits_{x \to + 0} \Psi (x,\mu ) = {\rm A}, \mu< 0, \hfill \\ \end{gathered}$$
where
$$x > 0, \mu \in ( - \infty , 0) \cup (0, + \infty ), \sum = diag\{ \sigma _1 ,\sigma _2 \} ,$$
C=[cij] is a square second-order matrix, and ψ(x, μ) is a vector column with the elements ψ1(x,μ) and ψ2(x,μ). As an application, an exact solution is obtained for the first time to the problem of the diffusion slip of a binary gas for a model Boltzmann equation with collision operator in the form proposed by MacCormack.

Keywords

Exact Solution Kinetic Equation Boltzmann Equation Vector Column Vector Model 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    K. M. Case,Ann. Phys. (N.Y.),9, 1 (1960).Google Scholar
  2. 2.
    K. M. Case and P. F. Zweifel,Linear Transport Theory, Addison-Wesley (1967).Google Scholar
  3. 3.
    R. Zelazny and A. Kuszell,Ann. Phys. (N.Y.),16, 81 (1961).Google Scholar
  4. 4.
    C. E. Siewert and P. F. Zweifel,Ann. Phys. (N.Y.),36, 61 (1966).Google Scholar
  5. 5.
    C. E. Siewert and P. F. Zweifel,J. Math. Phys.,7, 2092 (1966).Google Scholar
  6. 6.
    C. E. Siewert and P. S. Shieh,J. Nucl. Energy,21, 383 (1967).Google Scholar
  7. 7.
    D. R. Metcalf and P. F. Zweifel,Nucl. Sci. Eng.,33, 307 (1968).Google Scholar
  8. 8.
    T. Yoshimura and S. Katsuragi,Nucl. Sci. Eng.,33, 297 (1968).Google Scholar
  9. 9.
    S. Pahor and J. K. Shultis,J. Math. Phys.,10, 2220 (1969).Google Scholar
  10. 10.
    C. E. Siewert and Y. Ishiguro,J. Nucl. Energy,26, 251 (1972).Google Scholar
  11. 11.
    C. E. Siewert, E. E. Burniston, and J. T. Kriese,J. Nucl. Energy,26, 469 (1972).Google Scholar
  12. 12.
    V. S. Vladimirov,The Equations of Mathematical Physics [in Russian], Nauka, Moscow (1988).Google Scholar
  13. 13.
    N. I. Muskhelishvili,Singular Integral Equations, Groningen (1953).Google Scholar
  14. 14.
    F. J. MacCormack,Phys. Fluids,16, 2095 (1973).Google Scholar
  15. 15.
    V. M. Zhdanov and R. V. Smirnov,Zh. Prikl. Mat. Tekh. Fiz., No.5, 103 (1978).Google Scholar
  16. 16.
    S. Chapman and T. G. Cowling,Mathematical Theory of Nonuniform Gases, Cambridge (1952).Google Scholar
  17. 17.
    J. H. Ferziger and H. G. Kaper,Mathematical Theory of Transport Processes in Gases, Amsterdam (1972).Google Scholar
  18. 18.
    A. V. Latyshev and M. N. Gaidukov, “Analytic solution of diffusion slip problem,” in:Abstracts of Papers at 10th All-Union Conference on Rarefied Gas Dynamics [in Russian], Nauka, Moscow (1989).Google Scholar

Copyright information

© Plenum Publishing Corporation 1994

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  • A. V. Latyshev

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