Geodesic
Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 247-264 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice du chapitre de livre

We study the problem of finding a boundary condition in the heat equation for a hollow ball made of a composite material consisting of two homogeneous components. The Dirichlet condition is considered as boundary conditions inside the ball at $r=r_0$. In the inverse problem, the temperature inside the ball is assumed to be unknown on an infinite time interval. For finding it, the temperature of the heat flux at the media interface for $r=r_1$ is measured. We analyze the direct problem, which allows us to give a strict formulation of the inverse problem and determine the functional spaces in which it is natural to solve the inverse problem. Estimating the error of the approximate solution presents a major difficulty, which is dealt with in this paper by the method of projection regularization. Using this method, we find order-exact estimates.
Mots-clés : error estimation, Fourier transform
Keywords: modulus of continuity, ill-posed problem. 
@article{TIMM_2019_25_3_a20,
     author = {V. P. Tanana and A. I. Sidikova},
     title = {Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution},
     journal = {Trudy Instituta matematiki i mehaniki},
     pages = {247--264},
     year = {2019},
     volume = {25},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a20/}
}
TY  - JOUR
AU  - V. P. Tanana
AU  - A. I. Sidikova
TI  - Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution
JO  - Trudy Instituta matematiki i mehaniki
PY  - 2019
SP  - 247
EP  - 264
VL  - 25
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a20/
LA  - ru
ID  - TIMM_2019_25_3_a20
ER  - 
%0 Journal Article
%A V. P. Tanana
%A A. I. Sidikova
%T Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution
%J Trudy Instituta matematiki i mehaniki
%D 2019
%P 247-264
%V 25
%N 3
%U http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a20/
%G ru
%F TIMM_2019_25_3_a20
V. P. Tanana; A. I. Sidikova. Approximate solution of an inverse boundary value problem for a system of differential equations of parabolic type and estimation of the error of this solution. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 25 (2019) no. 3, pp. 247-264. http://geodesic.mathdoc.fr/item/TIMM_2019_25_3_a20/

[1] Vildeman V.E., Sokolkin Yu.V., Tashkinov A.A., Mekhanika neuprugogo deformirovaniya i razrusheniya kompozitsionnykh materialov, Nauka. Fizmatlit, M., 1997, 288 pp. | MR

[2] Alifanov O.M., Artyukhin E.A., Rumyantsev S.V., Ekstremalnye metody resheniya nekorrektnykh zadach, Nauka, M., 1988, 288 pp. | MR

[3] Tanana V.P., Danilin A.P., “Ob optimalnosti regulyarizuyuschikh algoritmov pri reshenii nekorrektnykh zadach”, Differents. uravneniya, 12:2 (1976), 1323–1326 | Zbl

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

[5] Kabanikhin S. I., Obratnye i nekorrektnye zadachi, Sib. nauch. izd-vo, Novosibirsk, 2009, 457 pp.

[6] Ivanov V. K., Vasin V.V., Tanana V.P., Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya, Nauka, M., 1978, 206 pp.

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

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

[9] Lavrentev M.M., Romanov V.G., Shishatskii S.P., Nekotorye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980, 285 pp. | MR

[10] Budak B.M., Samarskii A.A., Tikhonov A. N., Sbornik zadach po matematicheskoi fizike, Izd-vo tekhniko-teoret. literatury, M., 1956, 683 pp. | MR

[11] Lykov A.V., Teoriya teploprovodnosti, Izd-vo vysshaya shkola, M., 1967, 599 pp.

[12] Tanana V., Sidikova A., Optimal methods for ill-posed problems with applications to heat conduction: with applications to heat conduction, De Gruyter, Berlin, 2018, 130 pp. | MR

[13] Fikhtengolts G.M., Osnovy matematicheskogo analiza, v. 2, Fizmatlit, M., 2006, 864 pp.

[14] Privalov I.I., Vvedenie v teoriyu funktsii kompleksnogo peremennogo, Nauka, M., 1984, 432 pp. | MR