Geodesic
Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions
Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 2, pp. 221-229 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider the one-dimensional integro-differential Boltzmann equation for Maxwell particles with inelastic collisions. We show that the equation has a five-dimensional algebra of point symmetries for all dissipation parameter values and obtain an optimal system of one-dimensional subalgebras and classes of invariant solutions.
Keywords: inelastic Boltzmann equation, Lie symmetry
Mots-clés : invariant solution, optimal system of subalgebras. 
@article{TMF_2016_186_2_a2,
     author = {O. V. Ilyin},
     title = {Symmetries and invariant solutions of the~one-dimensional {Boltzmann} equation for inelastic collisions},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {221--229},
     year = {2016},
     volume = {186},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_186_2_a2/}
}
TY  - JOUR
AU  - O. V. Ilyin
TI  - Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 221
EP  - 229
VL  - 186
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_186_2_a2/
LA  - ru
ID  - TMF_2016_186_2_a2
ER  - 
%0 Journal Article
%A O. V. Ilyin
%T Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 221-229
%V 186
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2016_186_2_a2/
%G ru
%F TMF_2016_186_2_a2
O. V. Ilyin. Symmetries and invariant solutions of the one-dimensional Boltzmann equation for inelastic collisions. Teoretičeskaâ i matematičeskaâ fizika, Tome 186 (2016) no. 2, pp. 221-229. http://geodesic.mathdoc.fr/item/TMF_2016_186_2_a2/

[1] L. V. Ovsiannikov, Group Analysis of Differential Equations, Academic Press, New York, 1982 | MR | Zbl

[2] P. Olver, Applications of Lie Groups to Differential Equations, Springer, New York, 1993 | MR | Zbl

[3] V. V. Zharinov, Lecture Notes on Geometrical Aspects of Partial Differential Equations, Series on Soviet and East European Mathematics, 9, World Sci., Singapore, 1992 | MR | Zbl

[4] N. Kh. Ibragimov, Gruppy preobrazovanii v matematicheskoi fizike, Nauka, M., 1983 | MR | Zbl

[5] Y. N. Grigoryev, N. H. Ibragimov, V. F. Kovalev, S. V. Meleshko, Symmetries of Integro-Differential Equations, Springer, Dordrecht, 2010 | MR

[6] Yu. N. Grigorev, S. V. Meleshko, Dokl. AN SSSR, 297:2 (1987), 323–327 | MR | Zbl

[7] A. V. Bobylev, Math. Models Meth. Appl. Sci., 3:4 (1993), 443–476 | DOI | MR | Zbl

[8] A. V. Bobylev, V. Dorodnitsyn, Discrete Contin. Dyn. Syst., 24:1 (2009), 35–57 | DOI | MR | Zbl

[9] Yu. N. Grigoryev, S. V. Meleshko, P. Sattayatham, J. Phys. A: Math. Gen., 32:28 (1999), L337–L342 | DOI | MR

[10] A. V. Bobylev, Dokl. AN SSSR, 225:6 (1975), 1296–1299 | MR | Zbl

[11] M. Krook, T. T. Wu, Phys. Rev. Lett., 36:19 (1976), 1107–1109 | DOI

[12] A. V. Bobylev, N. Kh. Ibragimov, Matem. modelirovanie, 1:3 (1989), 100–109 | MR | Zbl

[13] A. V. Bobylev, G. L. Caraffini, G. Spiga, J. Math. Phys., 37:6 (1996), 2787–2795 | DOI | MR | Zbl

[14] Yu. N. Grigorev, S. V. Meleshko, Sib. matem. zhurn., 38:3 (1997), 510–525 | DOI | MR | Zbl

[15] E. Ben-Naim, P. L. Krapivsky, “The inelastic Maxwell model”, Granular Gas Dynamics, 624, Springer, Berlin, 2003, 65–94 | DOI

[16] E. Ben-Naim, P. L. Krapivsky, Phys. Rev. E, 61:1 (2000), R5–R8, arXiv: cond-mat/9909176 | DOI