Geodesic
Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation
Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 2, pp. 327-338 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We consider singularly perturbed operator differential transport equations of a special form in the case where the transport operator acts on space–time variables; a linear operator acting on an additional variable describes the interaction that “scrambles” the solution with respect to that variable. We construct a formal asymptotic expansion of the solution of the Cauchy problem for a singularly perturbed operator differential transport equation with small nonlinearity and weak diffusion in the case of several spatial variables. Under some conditions assumed for these problems, the leading term of the asymptotics is described by a quasilinear parabolic equation. The remainder term is estimated with respect to the residual under certain conditions.
Keywords: small parameter, asymptotic expansion, operator differential transport equation.
Mots-clés : singular perturbation 
@article{TMF_2024_220_2_a6,
     author = {A. V. Nesterov},
     title = {Asymptotics of solutions of {the~Cauchy} problem for a~singularly perturbed operator differential transport equation},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {327--338},
     year = {2024},
     volume = {220},
     number = {2},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2024_220_2_a6/}
}
TY  - JOUR
AU  - A. V. Nesterov
TI  - Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2024
SP  - 327
EP  - 338
VL  - 220
IS  - 2
UR  - http://geodesic.mathdoc.fr/item/TMF_2024_220_2_a6/
LA  - ru
ID  - TMF_2024_220_2_a6
ER  - 
%0 Journal Article
%A A. V. Nesterov
%T Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2024
%P 327-338
%V 220
%N 2
%U http://geodesic.mathdoc.fr/item/TMF_2024_220_2_a6/
%G ru
%F TMF_2024_220_2_a6
A. V. Nesterov. Asymptotics of solutions of the Cauchy problem for a singularly perturbed operator differential transport equation. Teoretičeskaâ i matematičeskaâ fizika, Tome 220 (2024) no. 2, pp. 327-338. http://geodesic.mathdoc.fr/item/TMF_2024_220_2_a6/

[1] A. B. Vasileva, V. F. Butuzov, Singulyarno vozmuschennye uravneniya v kriticheskikh sluchayakh, Izd-vo Mosk. un-ta, M., 1978 | MR

[2] A. V. Zaborskii, A. V. Nesterov, “Asimptoticheskoe razlozhenie resheniya singulyarno vozmuschennogo differentsialno-operatornogo uravneniya v kriticheskom sluchae”, Matem. modelirovanie, 26:4 (2014), 65–79

[3] A. V. Zaborskii, A. V. Nesterov, “Asimptoticheskoe razlozhenie resheniya zadachi Koshi dlya singulyarno vozmuschennogo differentsialno-operatornogo nelineinogo uravneniya”, Vestnik NIYaU “MIFI”, 4:4 (2015), 333–338 | DOI

[4] A. V. Zaborskii, A. V. Nesterov, “Asimptoticheskoe razlozhenie resheniya singulyarno vozmuschennogo differentsialno-operatornogo nelineinogo uravneniya s peremennymi koeffitsientami”, Matem. modelirovanie, 28:1 (2016), 117–131 | MR

[5] A. V. Zaborskii, A. V. Nesterov, D. Yu. Nechaev, “Ob asimptotike resheniya zadachi Koshi dlya singulyarno vozmuschennogo differentsialno-operatornogo uravneniya perenosa s mnogimi prostranstvennymi peremennymi”, Zh. vychisl. matem. i matem. fiz., 61:12 (2021), 2050–2058 | DOI | DOI | MR

[6] A. V. Zaborskii, A. V. Nesterov, “Ob asimptotike resheniya zadachi Koshi dlya singulyarno vozmuschennogo differentsialno-operatornogo uravneniya perenosa s maloi diffuziei”, Zh. vychisl. matem. i matem. fiz., 63:2 (2023), 273–281 | DOI | DOI | MR

[7] A. B. Vasileva, “O vnutrennem perekhodnom sloe v reshenii sistemy uravnenii v chastnykh proizvodnykh pervogo poryadka”, Differents. uravneniya, 21:9 (1985), 1537–1544 | MR

[8] A. M. Ilin, Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[9] A. Omuraliev, E. Abylaeva, “Regularization of the singularly perturbed Cauchy problem for a hyperbolic system”, J. Math. Sci., 264:4 (2022), 415-422 | DOI

[10] V. A. Galkin, Analiz matematicheskikh modelei: sistemy zakonov sokhraneniya uravnenii Boltsmana i Smolukhovskogo, BINOM. Laboratoriya znanii, M., 2011

[11] B. L. Rozhdestvenskii, N. N. Yanenko, Sistemy kvazilineinykh uravnenii i ikh prilozheniya k gazovoi dinamike, Nauka, M., 1978 | MR | MR | Zbl | Zbl

[12] G. I. Marchuk, V. I. Lebedev, Chislennye metody v teorii perenosa neitronov, Atomizdat, M., 1981 | MR | MR | Zbl

[13] V. N. Latyshev, V. A. Tupchiev, “Asimptoticheskoe razlozhenie resheniya uravneniya perenosa neitronov v sluchae maloi srednei dliny svobodnogo probega”, Differents. uravneniya, 19:11 (1983), 1922–1927 | MR | Zbl

[14] V. A. Tupchiev, A. N. Chepurko, “Asimptotika resheniya spektralnoi zadachi perenosa neitronov v sloe”, Differents. uravneniya, 32:6 (1996), 849–850 | Zbl

[15] A. N. Chepurko, “Asimptotika resheniya singulyarno vozmuschennogo nestatsionarnogo uravneniya perenosa neitronov”, Zh. vychisl. matem. i matem. fiz., 38:2 (1998), 289–297 | MR | Zbl

[16] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | DOI | MR | MR | Zbl