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## Archive of Issues

Russia Chelyabinsk; Yekaterinburg
Year
2018
Volume
28
Issue
4
Pages
474-488
 Section Mathematics Title On the solution of an inverse boundary value problem for composite materials Author(-s) Tanana V.P.ab, Ershova A.A.c Affiliations Chelyabinsk State Universitya, South Ural State Universityb, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesc Abstract In the present paper, an inverse boundary value problem of thermal conduction is formulated, posed and solved, provided that the thermal diffusivity is piecewise constant. This task holds a prominent place in technology, since thermally loaded units of technical constructions are covered with a heat insulating layer, the thermal characteristics of which are very different from the thermal characteristics of the structure itself. Such tasks are used in the planning of bench tests of aircraft. Modern composite materials solve this problem, giving developers a number of advantages. In rocket engines, the inner wall of the internal combustion chamber is covered with a heat-shielding layer, which is made of composite materials. Due to the properties of these materials, the heat-shielding layer significantly reduces the temperature of the internal combustion wall. When solving an inverse boundary problem, it is necessary to take into account the difference in the thermal conductivity coefficients of the component parts of composite materials, which make the wall of the chamber. The problem was investigated using a Fourier series in eigenfunctions for an equation with a discontinuous coefficient. It is proved that for the solution of the inverse problem the Fourier transform with respect to $t$ is applicable. To solve the inverse problem, the Fourier transform was used, which made it possible to reduce the inverse problem to the operator equation, which was solved by the discrepancy method. Keywords projection regularization method, inverse problem of thermal conduction, piecewise constant thermal diffusivity UDC 517.983.54 MSC 31A25, 49N45, 65R32 DOI 10.20537/vm180404 Received 1 October 2018 Language Russian Citation Tanana V.P., Ershova A.A. On the solution of an inverse boundary value problem for composite materials, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 4, pp. 474-488. References Budak B.M., Samarskii A.A., Tikhonov A.N. Sbornik zadach po matematicheskoi fizike (Collection of problems in mathematical physics), Moscow: Fizmatlit, 2004, 688 p. Kolmogorov A.N., Fomin S.V. Elementy teorii funktsii i funktsional'nogo analiza (Elements of the theory of functions and functional analysis), Moscow: Fizmatlit, 2004, 572 p. Phillips D.L. A technique for the numerical solution of certain integral equations of the first kind, Journal of the ACM, 1962, vol. 9, no. 1, pp. 84-97. DOI: 10.1145/321105.321114 Ivanov V.K. The approximate solution of operator equations of the first kind, USSR Computational Mathematics and Mathematical Physics, 1966, vol. 6, issue 6, pp. 197-205. DOI: 10.1016/0041-5553(66)90171-6 Ivanov V.K., Vasin V.V., Tanana V.P. Teoriya lineinykh nekorrektnykh zadach i ee prilozheniya (Theory of linear ill-posed problems and its applications), Moscow: Nauka, 1978, 208 p. Tanana V.P. On the optimality of methods for solving nonlinear unstable problems, Dokl. Akad. Nauk SSSR, 1975, vol. 220, no. 5, pp. 1035-1037 (in Russian). Ivanov V.K., Korolyuk T.I. Error estimates for solutions of incorrectly posed linear problems, USSR Computational Mathematics and Mathematical Physics, 1969, vol. 9, issue 1, pp. 35-49. DOI: 10.1016/0041-5553(69)90005-6 Tanana V.P., Kolesnikova N.Yu. On the estimation of the error of the approximate solution of an inverse problem of thermal diagnostics, Izv. Ural. Gos. Univ. Ser. Mat. Mekh. Inform., 2008, no. 58, pp. 155-162 (in Russian). http://hdl.handle.net/10995/24615 Tanana V.P., Sidikova A.I. On the guaranteed accuracy estimate of an approximate solution of one inverse problem of thermal diagnostics, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 2, pp. 238-252 (in Russian). http://mi.mathnet.ru/eng/timm565 Tanana V.P., Bredikhina A.B., Kamaltdinova T.S. On an error estimate for the approximate solution of an inverse problem in the class of piecewise smooth functions, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2012, vol. 18, no. 1, pp. 281-288 (in Russian). http://mi.mathnet.ru/eng/timm797 Full text