СМЕШАННАЯ ЗАДАЧА ДЛЯ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ ТИПА БЕННИ-ЛЮК ВЫСОКОГО ПОРЯДКА С ВЫРОЖДЕННЫМ ЯДРОМ
In a rectangular domain, a partial differential equation of the Benney-Luke type of even high order with
mixed conditions is considered. The issues of unique solvability of this problem are studied. The solution is
studied in the class of regular functions. The Fourier series method of separation of variables is used. When
proving the existence and uniqueness of the Fourier coefficient of an unknown function, the method of successive
approximations is used in combination with the method of contraction mapping.
1. Benney D.J., Luke J.C.: Interactions of permanent waves of finite amplitude. Journal Math. Physics,
43, 1964, pp.309-313.
2. Gordeziani D.G., Avilishbili G.A. Solving the nonlocal problems for one-dimensional medium oscillation,
Math. Model., 12 (1), 2000, pp.94-103 (in Russian).
3. Yuldashev T.K. Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with
degenerate kernel. Ukrainian Math. J. 68 (8), 2016, pp.1278-1296.
4. Yuldashev T.K. Solvability of a boundary value problem for a differential equation of the Boussinesq
type. Differential Equations, 54 (10), 2018, pp.1384-1393.
5. Yuldashev T.K. On a boundary-value problem for Boussinesq type nonlinear integro-differential
equation with reflecting argument. Lobachevskii Journal of Math. 41 (1), 2020, pp. 111-123.
6. Ashurov R.R., Mukhiddinova A.T. Inverse problem of determining the heat source density for the
subdiffusion equation. Differential Equations, 56 (12), 2020, pp.1550-1563.
7. Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M. A nonlocal problem for loaded partial di?erential
equations of fourth order. Bulletin of the Karaganda university-Mathematics 97 (1), 2020, pp.6-16.
8. Assanova A. An integral-boundary value problem for a partial differential equation of second order.
Turkish Journal of Math. 43 (4), 2019, pp.1967-1978.
9. Denisov, A.M., Efimov, A.A. Iterative method for the numerical solution of an inverse coefficient
problem for a system of partial differential equations. Differential Equations, 56 (7), 2000, pp.900-909.
10. Heydarzade N. A. On one nonlocal inverse boundary problem for the second-order elliptic equation.
Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 40 (4), 2020, pp.97-109.
11. Isgenderova G.N., Huseynova A. F. On solvability of an inverse boundary value problem for the pseudo
hyperbolic equation. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. 39 (4), 2019, pp.72-83.
12. Mamedov Kh.R. Uniqueness of the solution to the inverse problem of scattering theory for the Sturm–
Liouville operator with a spectral parameter in the boundary condition. Math. Notes 74 (1), 2003, pp.136-140.
13. Romanov V.G. Inverse phaseless problem for the electrodynamic equations in an anisotropic medium.
Doklady Math. 100 (2), 2019, pp.495-500.
14. Romanov V.G., Yamamoto M. Phaseless inverse problems with interference waves. Journal of Inverse
and Ill-Posed Problems. 26 (5), 2018, pp.681-688.
15. Yuldashev T.K. Inverse problem for a nonlinear Benney–Luke type integro-differential equations with
degenerate kernel. Russian Mathematics 60 (9), 2016, pp.53-60.
16. Yuldashev T.K. Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-
differential equations. Axioms 9 (2), ID 45, 2020, pp.21.
17. Yuldashev T.K., Rakhmonov F.D. On a boundary value problem for Benney–Luke type differential
equation with nonlinear function of redefinition and integral conditions. Trans. Natl. Acad. Sci. Azerb. Ser.
Phys.-Tech. Math. Sci. Mathematics, 41 (1), 2021, pp.172-183.
18. Yuldashev T.K., Rakhmonov F.D. On a Benney–Luke type Differential Equation with Nonlinear
Boundary Value Conditions. Lobachevski Journal of Math. 2021. 42 (15). 3761-3772.
19. Yuldashev T. K., Rakhmonov F. D. Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic
type di?erential equation. AIP Conference Proceedings. 2021. 2365 (060004). 1-20.
20. Рахмонов Ф.Д. Нелокальная краевая задача для дифференциального уравнения типа Бенни-
Люка высокого порядка с нелинейной функцией переопределения. Бюллетень Института математики ,
Vol. 4 (6), 2021, стр.100-112.
21. Рахмонов Ф.Д. Обратная задача для интегро-дифференциального уравнения высокого порядка
с вырожденным ядром. Бюллетень Института математики, Vol. 5 (1), 2022, стр.88-101.
22. Yuldashev T. K., Rakhmonov F. D., Ismoilov A.I. Integro-differential equation of Boussinesk with
integral conditions and with a small parameter for mixed derivatives. Itogi Nauki. VINITI, Moscow, 2022. 211.
P. 114-130. (in Russian).
23. Rakhmonov F. D. Identification of sources in a boundary value problem for Benney-Luke type
diferential equation with integral conditions. Bulletin of National University of Uzbekistan: Mathematics and
Natural Sciences. Tashkent. Uzbekistan. 2023. 6 (3). P. 141-155.
24. Рахмонов Ф.Д. Нелокальная краевая задача для дифференциального уравнения типа Бенни-
Люка высокого порядка. Журнал “Вестник НУУз”. Ташкент. Узбекистан. 2023. No 1/1. P. 69-78.
25. Onur Alp Ilhan, Danyal Soyba, Shakirbay G Kasimov, Farhod D Rakhmanov, Solvability of mixed
problems for heat equations with two nonlocal conditions. Journal Mathematica Slovaca. De Gruyter. 2022. 72
(6). P. 1573-1584.
26. Yuldashev T. K., Rakhmonov F. D.: Mixed problem for an integro-di?erential equation with a
multidimensional pseudoparabolic operator and nonlinear deviation. Itogi Nauki. VINITI, Moscow, 2021. 201.
P. 33-43. (in Russian).
27. Yuldashev T. K., Rakhmonov F. D.: Nonlocal Inverse Problem for a Pseudoheperbolic-Pseudoelliptic
Type Differential Equation. Uzbekistan-Malaysia international online conference “Computational Models and
Technologies (CMT2020)”, 24-25 August, 2020.
28. Рахмонов Ф.Д. Нелокальная краевая задача для дифференциального уравнения типа Бенни-
Люка высокого порядка с нелинейной функцией переопределения. Журнал “Вестник НУУз”. Ташкент.
Узбекистан. 2024. No 2/1. P. 152-168.
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