Geodesic
Analytic solutions to heat transfer problems on a~basis of determination of a~front of heat disturbance
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2016), pp. 27-41.

Voir la notice de l'article provenant de la source Math-Net.Ru

With the use of additional boundary conditions in integral method of heat balance, we obtain analytic solution to nonstationary problem of heat conductivity for infinite plate. Relying on determination of a front of heat disturbance, we perform a division of heat conductivity process into two stages in time. The first stage comes to the end after the front of disturbance arrives the center of the plate. At the second stage the heat exchange occurs at the whole thickness of the plate, and we introduce an additional sought-for function which characterizes the temperature change in its center. Practically the assigned exactness of solutions at both stages is provided by introduction on boundaries of a domain and on the front of heat perturbation the additional boundary conditions. Their fulfillment is equivalent to the sought-for solution in differential equation therein. We show that with the increasing of number of approximations the accuracy of fulfulment of the equation increases. Note that the usage of an integral of heat balance allows the application of the given method to solve differential equations that do not admit a separation of variables (nonlinear, with variable physical properties etc.).
Keywords: nonstationary heat conductivity, integral method of heat balance, analytic solution, eigenvalues, additional boundary conditions.
Mots-clés : front of heat perturbation 
@article{IVM_2016_11_a2,
     author = {I. V. Kudinov and V. A. Kudinov and E. V. Kotova},
     title = {Analytic solutions to heat transfer problems on a~basis of determination of a~front of heat disturbance},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {27--41},
     publisher = {mathdoc},
     number = {11},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2016_11_a2/}
}
TY  - JOUR
AU  - I. V. Kudinov
AU  - V. A. Kudinov
AU  - E. V. Kotova
TI  - Analytic solutions to heat transfer problems on a~basis of determination of a~front of heat disturbance
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2016
SP  - 27
EP  - 41
IS  - 11
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2016_11_a2/
LA  - ru
ID  - IVM_2016_11_a2
ER  - 
%0 Journal Article
%A I. V. Kudinov
%A V. A. Kudinov
%A E. V. Kotova
%T Analytic solutions to heat transfer problems on a~basis of determination of a~front of heat disturbance
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2016
%P 27-41
%N 11
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2016_11_a2/
%G ru
%F IVM_2016_11_a2
I. V. Kudinov; V. A. Kudinov; E. V. Kotova. Analytic solutions to heat transfer problems on a~basis of determination of a~front of heat disturbance. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 11 (2016), pp. 27-41. http://geodesic.mathdoc.fr/item/IVM_2016_11_a2/

[1] Kudinov I. V., Kudinov V. A., Analiticheskie resheniya parabolicheskikh i giperbolicheskikh uravnenii teplomassoperenosa, Infra-M, M., 2013

[2] Lykov A. V., “Metody resheniya nelineinykh uravnenii nestatsionarnoi teploprovodnosti”, Energetika i transport, 1970, no. 5, 109–150 | Zbl

[3] Gudmen T., “Primenenie integralnykh metodov v nelineinykh zadachakh nestatsionarnogo teploobmena”, Problemy teploobmena, Sb. nauchn. tr., Atomizdat, M., 1967, 41–96

[4] Bio M., Variatsionnye printsipy v teorii teploobmena, Energiya, M., 1975

[5] Veinik A. I., Priblizhennyi raschet protsessov teploprovodnosti, Gosenergoizdat, M.–L., 1959

[6] Shvets M. E., “O priblizhennom reshenii nekotorykh zadach gidrodinamiki pogranichnogo sloya”, PMM, 13:3 (1949), 257–266

[7] Timoshpolskii V. I., Postolnik Yu. S., Andrianov D. N., Teoreticheskie osnovy teplofiziki i termomekhaniki v metallurgii, Belorussk. navuka, Minsk, 2005

[8] Glazunov Yu. T., Variatsionnye metody, NITs “Regulyarnaya i khaoticheskaya dinamika”, In-t komp. issledov., Moskva–Izhevsk, 2006

[9] Lykov A. V., Teoriya teploprovodnosti, Vysshaya shkola, M., 1967

[10] Kartashov E. M., Analiticheskie metody v teorii teploprovodnosti tverdykh tel, Vysshaya shkola, M., 2001

[11] Kudinov V. A., Stefanyuk E. V., “Analiticheskii metod resheniya zadach teploprovodnosti na osnove vvedeniya fronta temperaturnogo vozmuscheniya i dopolnitelnykh granichnykh uslovii”, Inzhenerno-fizicheskii zhurnal, 82:3 (2009), 540–558 | MR

[12] Stefanyuk E. V., Kudinov V. A., “Poluchenie priblizhennykh analiticheskikh reshenii pri rassoglasovanii nachalnykh i granichnykh uslovii v zadachakh teorii teploprovodnosti”, Izv. vuzov. Matem., 2010, no. 4, 63–71 | MR | Zbl

[13] Belyaev N. M., Ryadno A. A., Metody nestatsionarnoi teploprovodnosti, Vysshaya shkola, M., 1978

[14] Kantorovich L. V., “Ob odnom metode priblizhennogo resheniya differentsialnykh uravnenii v chastnykh proizvodnykh”, DAN SSSR, 2:2 (1934), 532–534

[15] Fedorov F. M., Granichnyi metod resheniya prikladnykh zadach matematicheskoi fiziki, Nauka, Novosibirsk, 2000 | MR

[16] Tsoi P. V., Sistemnye metody rascheta kraevykh zadach teplomassoperenosa, Izd-vo MEI, M., 2005

[17] Kudinov V. A., Kudinov I. V., Skvortsova M. P., “Obobschennye funktsii i dopolnitelnye granichnye usloviya v zadachakh teploprovodnosti dlya mnogosloinykh tel”, Zhurn. vychisl. matem. i matem. fiz., 55:4 (2015), 669–680 | DOI | MR | Zbl