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Russia Chelyabinsk
Section Mathematics
Title Numerical solution of the inverse boundary value heat transfer problem for an inhomogeneous rod
Author(-s) Sidikova A.I.a, Sushkov A.S.b
Affiliations South Ural State Universitya, Chelyabinsk State Universityb
Abstract The article is devoted to solving an inverse boundary value problem for a rod consisting of composite materials. In the inverse problem, it is required, using information about the temperature of the heat flow in the media section, to determine the temperature at one of the ends of the rod. The paper presents a method of projection regularization, which made it possible to approximately estimate the error of the obtained solution to the inverse problem. To check the computational efficiency of this method, test calculations were carried out.
Keywords error estimation, modulus of conditional correctness, Fourier series transformation, ill-posed problem
UDC 517.983.54
MSC 80A23, 35K05, 35R30
DOI 10.35634/vm210207
Received 15 September 2020
Language Russian
Citation Sidikova A.I., Sushkov A.S. Numerical solution of the inverse boundary value heat transfer problem for an inhomogeneous rod, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, vol. 31, issue 2, pp. 253-264.
  1. Alifanov O.M., Artyukhin E.A., Rumyantsev S.V. Ekstremal'nye metody resheniya nekorrektnykh zadach i ikh prilozheniya k obratnym zadacham teploobmena (Extreme methods for solving ill-posed problems and their application to inverse problems of heat transfer), Moscow: Nauka, 1988.
  2. Romanov V.G. An inverse phaseless problem for electrodynamic equations in an anisotropic medium, Doklady Akademii Nauk, 2019, vol. 488, no. 4, pp. 367-371 (in Russian).
  3. Hasanov A., Itou H. A priori estimates for the general dynamic Euler-Bernoulli beam equation: Supported and cantilever beams, Applied Mathematics Letters, 2019, vol. 87, pp. 141-146.
  4. Prilepko A.I., Kamynin V.L., Kostin A.B. Inverse source problem for parabolic equation with the condition of integral observation in time, Journal of Inverse and Ill-Posed Problems, 2018, vol. 26, no. 4, pp. 523-539.
  5. Danilin A.R. Asymptotics of the solution of a bisingular optimal boundary control problem in a bounded domain, Computational Mathematics and Mathematical Physics, 2018, vol. 58, no. 11, pp. 1737-1747.
  6. Vasin V.V., Belyaev V.V. Approximation of solution components for ill-posed problems by the Tikhonov method with total variation, Doklady Mathematics, 2018, vol. 97, no. 3, pp. 266-270.
  7. Pektas B., Tamci E. The heat flux identification problem for a nonlinear parabolic equation in 2D, Journal of Computational and Applied Mathematics, 2017, vol. 312, pp. 134-142.
  8. Hasanov A. An inverse source problem with single Dirichlet type measured output data for a linear parabolic equation, Applied Mathematics Letters, 2011, vol. 24, no. 7, pp. 1269-1273.
  9. Korotkii A.I., Kovtunov D.A. Optimal boundary control of a system describing thermal convection, Proceedings of the Steklov Institute of Mathematics, 2011, vol. 272, suppl. 1, pp. 74-100.
  10. Korolev Yu.M., Kubo H., Yagola A.G. Parameter identification problem for a parabolic equation - application to the Black-Scholes option pricing model, Journal of Inverse and Ill-posed Problems, 2012, vol. 20, no. 3, pp. 327-337.
  11. Kabanikhin S.I. Obratnye i nekorrektnye zadachi (Inverse and Ill-posed problems), Novosibirsk: Siberian Branch of the Russian Academy of Sciences, 2018,
  12. Lavrent'ev M.M., Romanov V.G., Shishatskii S.P. Nekotorye zadachi matematicheskoi fiziki i analiza (Some problems of mathematical physics and analysis), Moskow: Nauka, 1980.
  13. Denisov V.Ya. Vvedenie v teoriyu obratnykh zadach (Introduction to the theory of inverse problems), Moscow: Mocsow State University, 1994.
  14. Denisov A.M. Uniqueness and nonuniqueness of the solution to the problem of determining the source in the heat equation, Computational Mathematics and Mathematical Physics, 2016, vol. 56, no. 10, pp. 1737-1742.
  15. Tanana V.R. A comparison of error estimates at a point and on a set when solving ill-posed problems, Journal of Inverse and Ill-posed Problems, 2018, vol. 26, no. 4, pp. 541-550.
  16. Tikhonov A.N., Glasko V.B. Methods of determining the surface temperature of a body, USSR Computational Mathematics and Mathematical Physics, 1967, vol. 7, no. 4, pp. 267-273.
  17. Tanana V.P. On reducing an inverse boundary-value problem to the synthesis of two ill-posed problems and their solution, Numerical Analysis and Applications, 2020, vol. 13, issue 2, pp. 180-192.
  18. Yagola A.G., Titarenko V.N., Stepanova I.E., Van Yanfei. Obratnye zadachi i metody ikh resheniya (Inverse problems and methods for their solution), Moscow: BINOM, 2014.
  19. Lavrent'ev M.M. O nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki (On some ill-posed problems of mathematical physics), Novosibirsk: Nauka, 1962.
  20. Landis E.M. Some questions in the qualitative theory of elliptic and parabolic equations, Six Papers on Partial Differential Equations, American Mathematical Society, 1962, pp. 173-238.
  21. 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.
  22. Ivanov V.K., Vasin V.V., Tanana V.P. Theory of linear ill-posed problems and its applications, De Gruyter, 2002.
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