Solution of the inverse boundary problem of heat exchange for a hollow cylinder

A. I. Sidikova   ; A. S. Sushkov

Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 26 (2023) no. 3, pp. 331-344

Résumé

The article solves the problem of determining the temperature on the inner wall of a hollow cylinder. Using a time Fourier transform, the problem is reduced to an ordinary differential equation, with the help of which the Fourier transform of an exact solution of the inverse boundary value problem is found. A projection regularization method is considered, which makes it possible to obtain a stable solution to the problem and an accurate in order of magnitude estimate of the error of the approximate solution. Since high accuracy requirements are imposed on solutions of such problems in numerical calculations, an algorithm is developed to improve the accuracy and reliability of processing the results of thermal test data. To check the performance of the algorithm, test calculations are carried out.

Bibliographie

[1] Alifanov O.M., Artyukhin E.A., Rumyantsev S.V., Ekstremalnye metody resheniya nekorrektnykh zadach i ikh prilozheniya k obratnym zadacham teploobmena, Nauka, M., 1988 | MR

[2] Yagola A.G., Van Yanfei, Stepanova I.E., Titarenko V.N., Obratnye zadachi i metody ikh resheniya. Prilozheniya k geofizike, BINOM. Laboratoriya znanii, M., 2014

[3] Tikhonov A.N., Glasko V.B., “K voprosu o metodakh opredeleniya temperatury poverkhnosti tela”, Zhurn. vychisl. matem. i mat. fiziki, 1967, no. 4, 910–914 | Zbl

[4] Pektas B., Tamci E., “The heat flux identification problem for a nonlinear parabolic equation in 2D”, J. of Comput. and Appl. Mathematics, 312 (2017), 134–142 | DOI | MR | Zbl

[5] Kabanikhin S.I., Shislenin M.A., “Regularization of the decision prolongation problem for parabolic and elliptic elliptic equations from border part”, Eurasian J. of Mathematical and Computer Applications, 2:2 (2014), 81–91 | DOI | MR

[6] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekotorye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR

[7] Kabanikhin S.I., Obratnye i nekorrektnye zadachi, Sib. nauch. izd-vo, Novosibirsk, 2009

[8] Tanana V.P., “Ob optimalnosti po poryadku metoda proektsionnoi regulyarizatsii pri reshenii obratnykh zadach”, Sib. zhurn. industr. matematiki, 7:2 (2004), 117–132 | Zbl

[9] Tanana V.P., Sidikova A.I., “O garantirovannoi otsenke tochnosti priblizhennogo resheniya odnoi obratnoi zadachi teplovoi diagnostiki”, Tr. IMM UrO RAN, 16, no. 2, 2010, 238–252

[10] Danilin A.R., “Ob optimalnykh po poryadku otsenkakh konechnomernykh approksimatsii reshenii nekorrektnykh zadach”, Zhurn. vychisl. matem. i mat. fiziki, 25:8 (1985), 1123–1130 | MR | Zbl

[11] Vasin V.V., Osnovy teorii nekorrektnykh zadach, Izd-vo SO RAN, Novosibirsk, 2020

[12] Leonov A.S., “Effektivnye algoritmy vychisleniya globalnoi i lokalnoi aposteriornoi otsenki tochnosti reshenii lineinykh nekorrektnykh zadach”, Izvestiya vuzov. Matematika, 2020, no. 2, 29–38 | MR | Zbl

[13] Tanana V.P., Ershova A.A., “O reshenii obratnoi granichnoi zadachi dlya kompozitnykh materialov”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 474–488 | MR | Zbl

[14] Johansson V.T., “Determining the temperature from incomplete boundary data”, Math. Nachr., 280:16 (2007), 1765–1779 | DOI | MR | Zbl

[15] Lavrentev M.M., O nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki, Nauka, Novosibirsk, 1962 | MR

[16] Ivanov V.K., “O ravnomernoi regulyarizatsii neustoichivykh zadach”, Sib. mat. zhurnal, 7:3 (1966), 546–558 | Zbl

[17] Tikhonov A.N., “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Doklady AN SSSR, 151:3 (1963), 501–504 | Zbl

[18] Kolmogorov A.N., Fomin S.V., Elementy teorii funktsii i funktsionalnogo analiza, Nauka, M., 1989 | MR

[19] Vatson G.N., Teoriya besselevykh funktsii, v. 1, Izd-vo inostrannoi literatury, M., 1949

[20] Grei E., Metyuz G.B., Funktsii Besselya i ikh prilozheniya k fizike i mekhanike, Izd-vo inostrannoi literatury, M., 1953