Advertisement

Differential Equations

, Volume 55, Issue 1, pp 34–45 | Cite as

On a Linear Inverse Problem for a Multidimensional Mixed-Type Equation

  • S. Z. DzhamalovEmail author
  • R. R. Ashurov
Partial Differential Equations
  • 6 Downloads

Abstract

We study the well-posedness of a linear inverse problem for a multidimensional mixed-type equation including the classical equations of elliptic, hyperbolic, and parabolic types as special cases. For this problem, using the “ε-regularization,” a priori estimate, and successive approximationmethods, we prove the existence and uniqueness theorems for the solution in some function class.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Lavrent’ev, M.M., Romanov, V.G., and Vasil’ev, V.G., Mnogomernye obratnye zadachi dlya differentsial’nykh uravnenii (Multidimensional Inverse Problems for Differential Equations), Novosibirsk: Nauka, 1969.Google Scholar
  2. 2.
    Anikonov, Yu.E., Nekotorye methody issledovaniya mnogomernykh obratnykh zadach dlya differentsial’nykh uravnenii (Several Methods for Studying Multidimensional Inverse Problems for Differential Equations), Novosibirsk: Nauka, 1978.Google Scholar
  3. 3.
    Kabanikhin, S.I., Obratnye i nekorrektnye zadachi (Inverse and Ill-Posed Problems), Novosibirsk: Sibirsk. Nauchn. Izd., 2008.Google Scholar
  4. 4.
    Bubnov, B.A., To the Problem of Solvability of Multidimensional Inverse Problems for Parabolic and Hyperbolic Equations, Preprint Computer Center Siberian Branch Akad. Nauk SSSR, no. 713, Novosibirsk, 1987.Google Scholar
  5. 5.
    Bubnov, B.A., To the Problem of Solvability of Multidimensional Inverse Problems for Parabolic and Hyperbolic Equations, Preprint Computer Center Siberian Branch Akad. Nauk SSSR, no. 714, Novosibirsk, 1987.Google Scholar
  6. 6.
    Kozhanov, A.I., Nonlinear loaded equations and inverse problems, Comput. Math. Math. Phys., 2004, vol. 44, no. 4, pp. 658–678.MathSciNetzbMATHGoogle Scholar
  7. 7.
    Sabitov, K.B. and Martem’yanova, N.V., A nonlocal inverse problem for a mixed-type equation, Russ. Math., 2011, vol. 55, no. 2, pp. 61–74.CrossRefzbMATHGoogle Scholar
  8. 8.
    Sabitov, K.B. and Sidorov, S.N., Inverse problem for degenerate parabolic-hyperbolic equation with nonlocal boundary condition, Russ. Math., 2015, vol. 59, no. 1, pp. 39–50.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sabitov, K.B. and Martem’yanova, N.V., The inverse problem for the Lavrent’ev–Bitsadze equation connected with the search of elements in the right-hand side, Russ. Math., 2017, vol. 61, no. 2, pp. 36–48.CrossRefzbMATHGoogle Scholar
  10. 10.
    Dzhamalov, S.Z., On solvability of the linear inverse problem with periodic conditions for a mixed-type equation, in Primenenie metodov funktsional’nogo analiza k neklassicheskim uravneniyam matematicheskoi fiziki (Application of Methods of Functional Analysis to Nonclassical Equations of Mathematical Physics), Novosibirsk: Inst. Mat., 1988, pp. 132–139.Google Scholar
  11. 11.
    Dzhamalov, S.Z., The linear inverse problem for the Tricomi equation in three-dimensional space, Vestn. KRAUNTs, 2016, no. 2 (14), pp. 12–17.MathSciNetGoogle Scholar
  12. 12.
    Dzhamalov, S.Z., Linear inverse problem for second-order mixed-type equation of the second kind with nonlocal boundary conditions in the three-dimensional case, Vestn. KRAUNTs, 2017, no. 1 (17), pp. 7–13.MathSciNetzbMATHGoogle Scholar
  13. 13.
    Dzhenaliev, M.T., K teorii kraevykh zadach dlya nagruzhennykh differentsial’nykh uravnenii (To the Theory of Boundary Value Problems for Loaded Differential Equations), Almaty: Inst. Teor. Prikl. Mat., 1995.Google Scholar
  14. 14.
    Nakhushev, A.M., Loaded equations and their applications, Differ. Equations, 1983, vol. 19, no. 1, pp. 74–81.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Djamalov, S.Z., On the correctness of a nonlocal problem for the second order mixed type equations of the second kind in a rectangle, IIUM J., 2016, vol. 17, no. 2, pp. 95–104.CrossRefGoogle Scholar
  16. 16.
    Vragov, V.N., Kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki (Boundary Value Problems for Nonclassical Equations of Mathematical Physics), Novosibirsk: Novosibirsk. Gos. Univ., 1983, p. 84.Google Scholar
  17. 17.
    Glazatov, S.N., Nonlocal boundary-value problems for some equations of mixed type in the rectangle, Sib. Mat. Zh., 1985, vol. 26, no. 6, pp. 162–164.MathSciNetzbMATHGoogle Scholar
  18. 18.
    Karatopraklieva, M.G., A nonlocal boundary-value problem for an equation of mixed type, Differ. Equations, 1991, vol. 27, no. 1, pp. 54–63.MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dzhamalov, S.Z., On the well-posedness of nonlocatl boundary-value problems for a multidimensional mixed-type equation of the second kind, in Kraevye zadachi dlya neklassicheskikh uravnenii matematicheskoi fiziki (Boundary Value Problems for Nonclassical Equations of Mathematical Physics), Novosibirsk: Inst. Mat., 1989, pp. 63–70.Google Scholar
  20. 20.
    Dzhamalov, S.Z., On a nonlocal boundary value problem for a second-order mixed type equation of the second kind, Uzb. Mat. Zh., 2014, vol. 1, no. 1, pp. 5–14.MathSciNetGoogle Scholar
  21. 21.
    Ladyzhenskaya, O.A., Kraevye zadachi matematicheskoi fiziki (Boundary Value Problems of Mathematical Physics), Moscow: Nauka, 1973.Google Scholar
  22. 22.
    Trenogin, V.A., Funktsional’nyi analiz (Functional Analysis), Moscow: Nauka, 1980.zbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Romanovskiy Institute of MathematicsUzbekistan Academy of SciencesTashkentUzbekistan

Personalised recommendations