Geodesic
Analytic solution of boundary-value problems for nonstationary model kinetic equations
Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 1, pp. 127-138 
A. V. Latyshev; A. A. Yushkanov. Analytic solution of boundary-value problems for nonstationary model kinetic equations. Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 1, pp. 127-138. http://geodesic.mathdoc.fr/item/TMF_1992_92_1_a11/
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     title = {Analytic solution of boundary-value problems for nonstationary model kinetic equations},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
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     url = {http://geodesic.mathdoc.fr/item/TMF_1992_92_1_a11/}
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A theory for constructing the solutions ofboundary-valueproblems for nonstationary model kinetic equations is constructed. This theory was incorrectly presented in the recent well-known monographs of Cercignani [1, 2]. After application of a Laplace transformation to the studied equation, separation of the variables is used, this leading to a characteristic equation. Eigenfunctions are found in the space of generalized functions, and the eigenvalue spectrum is investigated. An existence and uniqueness theorem for the expansion of the Laplace transform of the solution with respect to the eigenfunctions is proved. The proof is constructive and gives explicit expressions for the expansion coefficients. An application to the Rayleigh problem is obtained, and the corresponding result of Cercignani is corrected.

[1] Cherchinyani K., Matematicheskie metody v kineticheskoi teorii, Mir, M., 1973

[2] Cherchinyani K., Teoriya i prilozheniya uravneniya Boltsmana, Mir, M., 1978 | MR

[3] Greenberg W., van der Mee C. V. M., Protopopescu V., Boundary value problems in abstract kinetic theory, Birkhäuser Verlag, Basel, 1987 | MR

[4] Beals R., Protopopescu V., J. Math. Anal. Appl., 121:2 (1987), 370–405 | DOI | MR | Zbl

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

[6] Cercignani C., Sernagiotto F., Ann. Phys., 30:1 (1964), 154–164 | DOI | MR

[7] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR

[8] Latyshev A. V., TMF, 85:3 (1990), 428–442 | MR | Zbl