Geodesic
Analytical solution of the vector model kinetic equations with constant kernel and their applications
Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 2, pp. 283-303 
A. V. Latyshev. Analytical solution of the vector model kinetic equations with constant kernel and their applications. Teoretičeskaâ i matematičeskaâ fizika, Tome 97 (1993) no. 2, pp. 283-303. http://geodesic.mathdoc.fr/item/TMF_1993_97_2_a9/
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     author = {A. V. Latyshev},
     title = {Analytical solution of the vector model kinetic equations with constant kernel and their applications},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {283--303},
     year = {1993},
     volume = {97},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_1993_97_2_a9/}
}
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Voir la notice de l'article provenant de la source Math-Net.Ru

Exact solutions are obtained for the first time for the half-space boundary-value problem for the vector model kinetic equations $$\begin {gathered} \mu \frac {\partial }{\partial x}\Psi (x,\mu )+\Sigma \Psi (x,\mu )=C\int _{-\infty }^{\infty }\exp \left (-{\mu '}^2\right )\Psi (x,\mu ')\,d\mu ',\\ \lim _{x\to 0+}\Psi (x,\mu )=\Psi _0(\mu ),\qquad \mu >0,\\ \lim _{x\to +\infty }\Psi (x,\mu )=A,\qquad \mu <0, \end {gathered} $$ is obtained. Here $x>0$, $\mu \in (-\infty ,0)\cup (0,+\infty )$, $\Sigma =\operatorname {diag}\{\sigma _1,\sigma _2\}$, $C=\left [c_{ij}\right ]$ – $2\times 2$-matrix, $\Psi (x,\mu )$ is vector-column with elements $\psi _1$ and $\psi _2$. 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.

[1] Case K. M., Ann. Phys., 9:1 (1960), 1–23 | DOI | MR | Zbl

[2] Keiz K., Tsvaifel P., Lineinaya teoriya perenosa, Mir, M., 1972

[3] Zelazny R., Kuszell A., Ann. Phys., 16:1 (1961), 81–95 | DOI | MR | Zbl

[4] Siewert C. E., Zweifel P. F., Ann. Phys., 36:1 (1966), 61–85 | DOI | MR

[5] Siewert C. E., Zweifel P. F., J. Math. Phys., 7:11 (1966), 2092–2102 | DOI

[6] Siewert C. E., Shieh P. S., J. Nucl. Energy, 21:3 (1967), 383–392 | DOI

[7] Metcalf D. R., Zweifel P. F., Nucl. Sci. Eng., 33:3 (1968), 307–317

[8] Yoshimura T., Katsuragi S., Nucl. Sci. Eng., 33:3 (1968), 297–302

[9] Pahor S., Shultis J. K., J. Math. Phys., 10:12 (1969), 2220–2226 | DOI | MR | Zbl

[10] Siewert C. E., Ishiguro Y., J. Nucl. Energy, 26:5 (1972), 251–269 | DOI

[11] Siewert C. E., Burniston E. E., Kriese J. T., J. Nucl. Energy, 26:9 (1972), 469–482 | DOI

[12] Vladimirov B. C., Uravneniya matematicheskoi fiziki, Nauka, M., 1988 | MR

[13] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR | Zbl

[14] McCormack F. J., Phys. Fluids, 16:12 (1973), 2095–2105 | DOI

[15] Zhdanov V. M., Smirnov R. V., Zh. prikl. matem.i tekhn. fiziki, 1978, no. 5, 103–115

[16] Chepmen S., Kauling T., Matematicheskaya teoriya neodnorodnykh gazov, IL, M., 1960 | MR

[17] Fertsiger Dzh., Kaper G., Matematicheskaya teoriya protsessov perenosa v gazakh, Mir, M., 1976

[18] Latyshev A. V., Gaidukov M. N., “Analiticheskoe reshenie zadachi diffuzionnogo skolzheniya”, Kh-ya Vsesoyuznaya konf. po dinamike razrezhennykh gazov. Tezisy dokladov, Nauka, M., 1989