On a nonlocal boundary-value problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy

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Abstract

The work is devoted to the proof of the uniqueness and existence of a solution of a nonlocal problem for a loaded parabolic-hyperbolic equation with three lines of change of type. Using the representation of the general solution, the uniqueness of the solution is proved, and the existence of the solution is proved by the method of integral equations. Necessary conditions for the parameters and specified functions are established for the unique solvability of Volterra integral equations of the second kind with a shift equivalent to the problem under study.

About the authors

Bozor I. Islomov

National University of Uzbekistan named after M. Ulugbek

Email: islomovbozor@yandex.com
ORCID iD: 0000-0002-4372-395X
Scopus Author ID: 37041356900
http://www.mathnet.ru/person59921

Dr. Phys. & Math. Sci., Professor; Chief Researcher; Dept. of Differential Equations and Mathematical Physics

4, Universitetskaya st., Tashkent, 100174, Uzbekistan

Jurat A. Xolbekov

Tashkent State Technical University named after I. Karimov

Author for correspondence.
Email: xolbekovja@mail.ru
ORCID iD: 0000-0002-1495-2761
http://www.mathnet.ru/person172679

Assistant; Dept. of Higher Mathematics

2, Universitetskaya st., Tashkent, 100174, Uzbekistan

References

  1. Nakhushev A. M. Loaded equations and their applications, Differ. Uravn., 1983, vol. 19, no. 1, pp. 86–94 (In Russian).
  2. Nakhushev A. M. Uravneniia matematicheskoi biologii [Equations of Mathematical Biology]. Moscow, Vyssh. Shk., 1995, 301 pp. (In Russian)
  3. Nakhushev A. M. Zadachi so smeshcheniem dlia uravnenii v chastnykh proizvodnykh [Problems with Shift for Partial Differential Equations]. Moscow, Nauka, 2006, 288 pp. (In Russian)
  4. Wiener J., Debnath L. A survey of partial differential equations with piecewise continuous arguments, Int. J. Math. Math. Sci., 1995, vol. 18, no. 2, pp. 209–228. https://doi.org/10.1155/S0161171295000275
  5. Islamov B. I., Kuryazov D. M. On a boundary value problem for loaded equation of the second order, Dokl. Akad. Nauk Resp. Uzb., 1996, no. 1–2, pp. 3–6 (In Russian).
  6. Dzhenaliev M. T. Loaded equations with periodic boundary conditions, Differ. Equ., 2001, vol. 37, no. 1, pp. 51–57. https://doi.org/10.1023/A:1019268231282
  7. Pul'kina L. S. A nonlocal problem for a loaded hyperbolic equation, Proc. Steklov Inst. Math., 2002, vol. 236, pp. 285–290.
  8. Kozhanov A. I. A nonlinear loaded parabolic equation and a related inverse problem, Math. Notes, 2004, vol. 76, no. 6, pp. 784–795. https://doi.org/10.1023/B:MATN.0000049678.16540.a5
  9. Alikhanov A. A. A priori estimates for parabolic equations with a movable load, In: Proceedings of the Third All-Russian Scientific Conference (29–31 May 2006). Part 3, Matem. Mod. Kraev. Zadachi. Samara, Samara State Technical Univ., 2006, pp. 22–25 (In Russian).
  10. Baltayeva U. I., Islomov B. I. Boundary value problems for the loaded third order equations of the hyperbolic and mixed types, Ufimsk. Mat. Zh., 2011, vol. 3, no. 3, pp. 15–25 (In Russian).
  11. Sabitov K. B., Melisheva E. P. The Dirichlet problem for a loaded mixed-type equation in a rectangular domain, Russian Math. (Iz. VUZ), 2013, vol. 57, no. 7, pp. 53–65. https://doi.org/10.3103/S1066369X13070062
  12. Sabitov K. B. Initial-boundary problem for parabolic-hyperbolic equation with loaded summands, Russian Math. (Iz. VUZ), 2015, vol. 59, no. 6, pp. 23–33. https://doi.org/10.3103/S1066369X15060055
  13. Islomov B., Baltaeva U. I. Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients, Electron. J. Diff. Equ., 2015, vol. 2015, no. 221, pp. 1–10. https://ejde.math.unt.edu/Volumes/2015/221/abstr.html
  14. Sadarangani K. B., Abdullaev O. Kh. About a problem for loaded parabolic-hyperbolic type equation with fractional derivatives, Int. J. Diff. Equ., 2016, vol. 6, 9815796. https://doi.org/10.1155/2016/9815796
  15. Dzhamalov S. Z., Ashurov R. R. On a nonlocal boundary-value problem for second kind second-order mixed type loaded equation in a rectangle, Uzbek Math. J., 2018, no. 3, pp. 63–72. https://doi.org/10.29229/uzmj.2018-3-6
  16. Berdyshev A. S., Rakhmatullaeva N. A. A problem with Bitsadze–Samarskiy type conditions for a parabolic-hyperbolic equation with three lines of degeneracy, Dokl. Akad. Nauk Resp. Uzb., 2010, no. 4, pp. 8–12 (In Russian).
  17. Islomov B., Kholbekov Zh. A. An analogue of the Tricomi problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy – I, Uzbek Math. J., 2015, no. 4, pp. 47–56 (In Russian).
  18. Islomov B., Kholbekov Zh. A. An analogue of the Tricomi problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy – II, Uzbek Math. J., 2016, no. 1, pp. 49–56 (In Russian).
  19. Mikhlin S. G. Linear Integral Equations, International Monographs on Advanced Mathematics and Physics. Delhi, Hindustan Publ., 1960, vii+223 pp.

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