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Differential Equations

, Volume 54, Issue 9, pp 1180–1190 | Cite as

Integral-Functional Equation Arising in the Study of an Inverse Problem for a Quasilinear Hyperbolic Equation

  • A. M. Denisov
Integral Equations
  • 21 Downloads

Abstract

A problem with data on the characteristics is considered for a quasilinear hyperbolic equation. The inverse problem of determining two unknown coefficients of the equation from some additional information about the solution is posed. One of the unknown coefficients depends on the independent variable, and the other, on the solution of the equation. Uniqueness theorems are proved for the solution of the inverse problem. The proof is based on the derivation of the integro-functional equation and the analysis of the uniqueness of its solution.

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References

  1. 1.
    Courant, R. and Hilbert, D., Methods of Mathematical Physics, Germany, 1924.zbMATHGoogle Scholar
  2. 2.
    Tikhonov, A.N. and Samarskii, A.A., Uravneniya matematicheskoi fiziki (Equations of Mathematical Physics), Moscow: Mosk. Gos. Univ., 1999.Google Scholar
  3. 3.
    Lavrent’ev, M.A., Romanov, V.G., and Shishatskii, S.P., Nekorrektnye zadachi matematicheskoi fiziki i analiza (Ill-Posed Problems of Mathematical Physics and Analysis), Moscow: Nauka, 1980.Google Scholar
  4. 4.
    Romanov, V.G., Obratnye zadachi matematicheskoi fiziki (Inverse Problems of Mathematical Physics), Novosibirsk: Nauka, 1984.zbMATHGoogle Scholar
  5. 5.
    Kabanikhin, S.I. and Lorenzi, A., Identification Problems of Wave Phenomena, Utrecht, The Netherlands: VSP, 1999.Google Scholar
  6. 6.
    Prilepko, A.I., Orlovsky, D.G., and Vasin, I.V., Methods for Solving Inverse Problems in Mathematical Physics, New York: Marcel Dekker, 2000.zbMATHGoogle Scholar
  7. 7.
    Isakov, V., Inverse Problems for Partial Differential Equations, New York: Springer, 2006.zbMATHGoogle Scholar
  8. 8.
    Kabanikhin, S.I., Obratnye i nekorrektnye zadachi (Inverse and Ill-Posed Problems), Novosibirsk: Sibirsk. Nauchn. Izd., 2008.Google Scholar
  9. 9.
    Cannon, J.R. and Du Chateau, P., An inverse problem for an unknown source term in a wave equation, SIAM J. Appl. Math., 1983, vol. 43, no. 3, pp. 553–564.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cavaterra, C., An inverse problem for semilinear wave equation, Boll. Un. Mat. Ital. (B), 1988, vol. 2, no. 3, pp. 695–711.MathSciNetzbMATHGoogle Scholar
  11. 11.
    Graselli, M., Local existence and uniqueness for a quasilinear hyperbolic inverse problem, Appl. Anal., 1989, vol. 32, no. 1, pp. 15–30.MathSciNetCrossRefGoogle Scholar
  12. 12.
    Shcheglov, A.Yu., The inverse problem of determination of a nonlinear source in a hyperbolic equation, J. Inverse Ill-Posed Probl., 1998, vol. 6, no. 6, pp. 625–644.MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Denisov, A.M., Solvability of the inverse problem for a quasilinear hyperbolic equation, Differ. Equations, 2002, vol. 38, no. 9, pp. 1229–1238.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Denisov, A.M. and Shirokova, E.Yu., Inverse problem for a quasilinear hyperbolic equation with a nonlocal boundary condition containing a delay argument, Differ. Equations, 2013, vol. 49, no. 9, pp. 1053–1061.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2018

Authors and Affiliations

  1. 1.Lomonosov Moscow State UniversityMoscowRussia

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