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«THE BULLETIN OF IRKUTSK STATE UNIVERSITY». SERIES «MATHEMATICS»
«IZVESTIYA IRKUTSKOGO GOSUDARSTVENNOGO UNIVERSITETA». SERIYA «MATEMATIKA»
ISSN 1997-7670 (Print)
ISSN 2541-8785 (Online)

List of issues > Series «Mathematics». 2025. Vol 52

Inverse Problems of Recovering Parameters in the Lin–Reissner–Tsien Equation

Author(s)
Alexandr I. Kozhanov1,2, Lyubov A. Telesheva2

1Sobolev Institute of Mathematics SB RAS, Novosibirsk, Russian Federation

2Banzarov Buryat State University, Ulan-Ude, Russian Federation

Abstract

The article is devoted to the study of the solvability of new nonlinear inverse problems of finding an unknown constant together with solving the linearized Lin– Reissner–Tsien equation. For the problems under consideration, theorems of solvability are proved in classes of regular solutions, i.e., of solutions having all weak derivatives in the sense of S.L. Sobolev that occur in the corresponding equation.

The peculiarity of the problems under study is, firstly, that the unknown coefficient is a constant (which corresponds, for example, to a homogeneous medium). Secondly, a new overdetermination condition, not previously used by the predecessors, is introduced: an integral condition with respect to the time variable.

About the Authors

Aleksandr I. Kozhanov, Dr. Sci. (Phys.-Math.), Prof., Sobolev Institute of Mathematics SB RAS, 630090, Novosibirsk, Russian Federation, kozhanov@math.nsc.ru

Lyubov A. Telesheva, Cand. Sci. (Phys.Math.), Assoc. Prof., Banzarov Buryat State University, Ulan-Ude, 670000, Russian Federation

For citation
Kozhanov A. I.,Telesheva L. A. Inverse Problems of Recovering Parameters in the Lin–Reissner–Tsien Equation. The Bulletin of Irkutsk State University. Series Mathematics, 2025, vol. 52, pp. 71–87.

https://doi.org/10.26516/1997-7670.2025.52.71

Keywords
hyperbolic equations, inverse problems, unknown parameters, integral overdetermination condition, regular solutions
UDC
517.95
MSC
35L20, 35R30
DOI
https://doi.org/10.26516/1997-7670.2025.52.71
References
  1. Glazatov S.N. On solvability of a spatial periodic problem for the Lin–Reissner– Tsien equation of transonic gas dynamics. Mathematical Notes, 2010, vol. 87, iss. 1, pp. 130–134. https://doi.org/10.1134/S0001434610010177
  2. Kabanikhin S.I. Inverse and Ill-Posed Problems.Novosibirsk, Sib. Knizh. Izdat. Publ., 2009. (in Russian)
  3. Kozhanov A.I. Inverse Problems of Finding the Absorption Parameter in the Diffusion Equation. Mathematical Notes, 2019, vol. 106, iss. 3, pp. 378–389. https://doi.org/10.1134/S0001434619090074 (in Russian)
  4. Kozhanov A.I. The Heat Transfer Equation with an Unknown Heat Capacity Coefficient. Journal of Applied and Industrial Mathematics, 2020, vol. 14, iss. 1, pp. 104–114. https://doi.org/10.1134/S1990478920010111 (in Russian)
  5. Kozhanov A.I. On the solvability of the inverse problems of parameter recovery in elliptic equations. Mathematical notes of NEFU, 2020, vol. 27, iss. 4, pp. 14–29. https://doi.org/10.25587/SVFU.2020.57.53.002 (in Russian)
  6. Kozhanov A.I., Safiullova R.R. Determination of parameters in telegraph equation. Ufa Mathematical Journal, 2017, vol. 9, iss. 1, pp. 62–74. https://doi.org/10.13108/2017-9-1-62 (in Russian)
  7. Kozhanov A.I., Telesheva L.A. Inverse problems of restoring parameters in parabolic and hyperbolic equations. Mathematical Notes of NEFU, 2022, vol. 29, No. 3 , pp. 57–69. https://doi.org/10.25587/SVFU.2022.85.24.005 (in Russian)
  8. Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N. Linear and Quasilinear Equations of Parabolic Type. Moscow, Nauka Publ., 1967. (in Russian)
  9. Lyubanova A.Sh. Identificacion of a coefficient in the leading term of a pseudoparabolic equation of filtration. Sib. Math. J., 2013 , vol. 54, no. 6, pp. 1046–1058. https://doi.org/10.1134
  10. Mamontov E.V. Ob uravneniyakh malykh vozmushcheniy v nestatsionarnom okolozvukovom potoke gaza.[On the equations of small perturbations in a nonstationary transonic flow of gas]. Nestacionarnye problemy mekhaniki [Nonstationary problems of mechanics]. Novosibirsk, Hidrodynamics Institute SB USSR Academy of Sciences, 1978, no. 37, pp. 1390–143. (in Russian)
  11. Nakhushev A.M. Nagruzhennye uravneniya i ikh primenenie. Moscow, Nauka Publ., 2012, 232 p. (in Russian)
  12. Romanov V.G.Stability in Inverse Problems. Moscow, Nauchny Mir Publ., 2005. (in Russian)
  13. Sobolev S.L.Some Applications of Functional Analysis in Mathematical Physics. Moscow, Nauka Publ., 1973. (in Russian)
  14. Trnogin V.A.Finctional Analysis. Moscow, Fizmatlit Publ., 2002. (in Russian)
  15. Anikonov Yu.E. Inverse problems for kinetic and other evolution equations. Utrecht, VSP, 2001. https://doi.org/10.1515/9783110940909
  16. Belov Yu.Ya. Inverse problems for Partial Differential equations. Utrecht, VSP, 2002. https://doi.org/10.1515/9783110944631
  17. Isakov V. Inverse Problems for Partial Differential Equations. Springer Science, 2006. http://dx.doi.org/10.1007/0-387-32183-7
  18. Ivanchov M. Inverse problems for equations of parabolic type. Mathematical studies. Monograph series. Vol. 10. https://doi.org/10.1201/9781482292985-6
  19. Kozhanov A.I. Composite type equations and inverse problems. Utrecht, VSP, 1999. https://doi.org/10.1515/9783110943276
  20. Kozhanov A.I. Hyperbolic Equations with Unknown Coefficients. Symmetry, 2020, vol. 12, iss. 9, pp. 1539. https://doi.org/10.3390/sym12091539.
  21. Kozhanov A.I., Abylkayrov U.U., Ashurova G.R. Inverse problems of parameter recovery in differential equations with multiple characteristics. Journal of Mathematics, Mechanics and Computer Science, 2022. vol. 113, no. 1, pp. 3-16. https://doi.org/10.26577/JMMCS.2022.v113.i1.01
  22. Lin C.C., Reissner E., Tsien H.S. On two-dimensional non-steady motion of a slender body in a compressible fluid. Journal of Mathematical Physics, 1948, vol. 27, no. 3, pp. 220–231. http://dx.doi.org/10.1016/B978-0-12-398277-3.50028-2
  23. Lorenzi A. An Introduction to Mathematikal Problems via Functional Analysis. Utrecht, VSP, 2001. https://doi.org/10.1515/9783110940923
  24. Lorenzi A., Mola G. Identification of real constant in linear evolution equation in a Hilbert spaces. Inverse Problems Imaging, 2011, vol. 5, no. 3, pp. 695-714. https://doi.org/10.3934/ipi.2011.5.695
  25. Lorenzi A., Recovering two constants in a linear parabolic equaton. Inverse problem in applied sciences, 2007, vol. 73, pp. 1-15. https://doi.org/10.1088/1742-6596/73/1/012014Lorenzi A., Mola G. Recovering the reaction and the diffusion coefficients in a linear parabolic equation. Inverse Problems, 2012, vol. 28, no. 7. http://dx.doi.org/10.1088/0266-5611/28/7/075006
  26. Lyubanova A.Sh. Identification of a constant coefficient in an elliptic equation. Applicable Analysis, 2008, vol. 87, no. 10-11, pp. 1121-1128. https://doi.org/10.1080/00036810802189654
  27. Lyubanova A.Sh., Velisevich A.V. An inverse problem for a quasilinear elliptic equation. Journal of Mathematical Sciences, 2023, vol. 270. no. 4. https://doi.org/10.1007/s10958-023-06370-9
  28. Lyubanova A.Sh. Identificacion of a coefficient in the leading term of a pseudoparabolic equation of filtration. Siberian Mathematical Journal, 2013, vol. 54, no. 6, pp. 1046–1058. https://doi.org/10.1134
  29. Mola G. Identification of the Diffussion Coefficient in Linear Evolution Equations in Hilbert Spaces. J. Abstr. Diff. Equat. Appl., 2011, vol. 2, pp. 18–28. http://dx.doi.org/10.3934/ipi.2011.5.695
  30. Mola G., Okazawa N., Yokota T. Reconstruction of two constant coefficients in linear anisotropic diffusion model. Inverse Problems, 2016 , vol. 32, no. 11. http://dx.doi.org/10.1088/0266-5611/32/11/115016
  31. Prilepko A.I., Orlovsky D.C., Vasin I.A. Methods for solving inverse problems in mathimatical physics. New York, Dekker, 1999. https://doi.org/10.1201/9781482292985
  32. Triebel H. Interpolation Theory, Function Spaces, Differential Operators. Amsterdam, Noth-Holland Publ., 1978. https://doi.org/10.1002/zamm.19790591227
  33. Velisevich A.V. On an Inverse Problem for the Stationary Equation with a Boundary Condition of the Third Kind. Journal of Siberian Federal University. Mathematics & Physics, 2021, vol. 14, no.5, pp. 659–666. https://doi.org/10.17516/1997-1397-2021-14-5-659-666

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