# Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

## Abstract

This paper is devoted to the study of a nonlocal boundary value problem for a loaded equation of parabolic-hyperbolic type in a special domain.Using representations of the general regular solution, are proven the unique solvability of the problem posed.

## First Page

250

## Last Page

261

## References

1. Bitsadze A.V. On the problem of equations of mixed type. Tr. MIAN USSR. Vol. 41, pp. 3-59 (1953).

2. Salakhitdinov M.S., Urinov A.K. Nonlocal boundary value problem in a doubly connected domain for an equation of mixed type with nonsmooth lines of degeneracy. Dokl. Academy of Sciences of the USSR. Vol. 299, No. 1, pp. 63-66 (1988).

3. Nakhushev A.M. Loaded equations and their applications. Differential equations. Vol. 19, No. 1, pp. 86-94 (1983).

4. Kaziev V.M. On the Darboux problem for one loaded second-order integro-differential equation. Differential equations. Vol. 14, N. 1, pp. 181-184 (1978).

5. Attaev A.Kh. Boundary value problems for the loaded wave equation. Differential equations: Abstract. report regional interuniversity seminar. May 20-25, Kuibishev, pp. 9-10 (1984).

6. Dzhenaliev N.T. On a boundary value problem for a linear loaded parabolic equation with nonlocal and boundary conditions. Differential equations. Vol. 27, No. 10, pp. 1925-1927 (1991).

7. Shkhanukov M.Kh. Difference method for solving one loaded equation of parabolic type. Differential equations. Vol. 13, No. 1, pp. 163-167 (1977).

8. Borodin A.B. About one estimate for elliptic equations and its application to loaded equations. Differential equations. Vol. 13, No.1, pp. 17-22 (1977).

9. Khubiev K.U. Gellerstedt problem for a loaded equation of mixed type with data on non-parallel characteristics. Dokl. Adyg. inter. academic. sciences. No. 1, pp. 1-4 (2008).

10. Islomov B., Kuryazov D.M. Boundary value problems for a mixed loaded equation of third order of parabolic-hyperbolic type. Uzbek mathematical journal. No. 2, pp. 29-35 (2000). [in Russian]

11. Ramazanov M.I. On a nonlocal problem for a loaded hyperbolic-elliptic type in a rectangular domain. Mathematical journal. Vol. 2, No. 4, pp. 75-81 (2002). [in Russian]

12. Sabitov K.B., Melisheva E.P. The dirichlet problem for a loaded mixed-type equation in a rectangular domain. Russian Mathematics (Izvestiya VUZ. Matematika). Vol. 57, Issue 7, pp. 53-65 (2013).

13. Melisheva E.P. Dirichlet problem for the loaded Lavrentievâ€“Bitsadze equation Vestn. Myself. state tech. univ. Phys.-math. Sciences Ser. Vol. 80, No. 6, pp. 39-47 (2010).

14. Islomov B., Kuryazov D.M. On a boundary value problem for a loaded second-order equation. Doclady RUz. No. 1-2, pp. 3-6 (1996). [in Russian]

15. Islomov B., Baltaeva U.I. Boundary value problems for loaded differential equations of hyperbolic and mixed types of third order. Ufa Mathematical Journal. Vol. 3, No. 3, pp. 15-25 (2011).

16. Islomov B., Baltaeva U.I. Boundary value problems for a third-order loaded parabolic-hyperbolic equation with variable coefficients. Electronic Journal of Differential Equations. No. 221, pp. 1-10 (2015).

17. Islomov B.I., Kholbekov Zh.A. On a nonlocal boundary-value problem for a loaded parabolic-hyperbolic equation with three lines of degeneracy. Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta, Seriya Fiziko-Matematicheskie Nauki. Vol. 25, No. 3, pp. 407-422 (2021).

18. Sabitov K.B. Initial-boundary problem for a parabolic-hyperbolic equation with loaded terms. Russian Mathematics (Izvestiya VUZ. Matematika). Vol. 59, Issue 6, pp. 23-33 (2015).

19. 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 Mathematical Journal. No. 3, pp. 63-72 (2018).

20. Sabitova Yu.K. Dirichlet problem for the Lavrentiev-Bitsadze equation with loaded terms. News of universities. Mathematics. No. 9, pp. 42-58 (2018).

21. Eleev V.A. On some boundary value problems for mixed loaded equations of the second and third order. Differential equations. Vol. 30, No. 2, pp. 230-237 (1994).

22. Abdullayev O.Kh. About a method of research of the non-local problem for the loaded mixed type equation in double-connected domain. Bulletin KRASEC. Phys. Math. Sci. Vol. 9, No. 2, pp. 3-12 (2014).

23. Abdullaev O.Kh. Boundary value problem for a loaded equation of elliptic-hyperbolic type in a doubly connected domain. Bulletin of KRAUNC. Phys. Math. Sciences. No. 1(8), pp. 33-48 (2014).

24. Islomov B.I., Abdullaev O.Kh. A boundary value problem of the type of the Bitsadze problem for a third-order equation of elliptic-hyperbolic type in a doubly connected domain. Reports of the Adyghe (Circassian) International Academy of Sciences. Vol. 7, No. 1, pp. 42-46 (2004). [in Russian]

25. Abdullaev O.Kh. Nonlocal problem for a loaded equation of mixed type with an integral operator. Vestn. Myself. state tech. un-ta. Ser. Phys. Math. Sciences. Vol. 20, No. 2, pp. 220-240 (2016).

26. Islamov B.I., Yunusov O.M. Nonlocal boundary value problem for a loaded equation of hyperbolic type in a special domain. DAN Republic of Uzbekistan. No. 4, pp. 8-12 (2016). [in Russian]

27. Yunusov O.M. Nonlocal boundary value problem for a loaded equation of hyperbolic type. Proceedings of the international conference Differential equations and related problems. Russia, pp. 152-155 (2018). [in Russian]

28. Juraev T.D. Boundary value problems for equations of mixed and mixed-composite type. Tashkent, Fan. 240 p. (1979). [in Russian].

29. Mikhlin S.G. Lectures on linear integral equations. Moscow, Fizmatgiz, 232 p. (1959).

30. Tikhonov A.N., Samarsky A.A. Equations of mathematical physics. Moscow, Science, 736 p. (1977).

## Recommended Citation

Islomov, Bozor and Yunusov, Oybek
(2023)
"Boundary value problem with the Bitsadze-Samarsky condition for a loaded equation of parabolic-hyperbolic type in a doubly connected region,"
*Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences*: Vol. 5:
Iss.
4, Article 4.

DOI: https://doi.org/10.56017/2181-1318.1263