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Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences

Abstract

In three-dimensional domain a problem of identification of recourses for Benney-Luke type partial differential equation of the even order with integral form conditions, spectral parameter and small positive parameters in mixed derivatives is considered. The solution of this partial differential equation is studied in the class of regular functions. The Fourier series method is used. Using this Fourier method, is obtained a countable system of ordinary differential equations. So, the nonlocal boundary value problem is integrated as an ordinary differential equation. When we define the arbitrary integration constants there are possible five cases with respect to the spectral parameter. By the aid of given additional condition, we obtained the presentations with respect to redefinition functions. Using the Cauchy-Schwarz inequality and the Bessel inequality, we proved the absolute and uniform convergence of the obtained Fourier series.

First Page

141

Last Page

155

References

1. Ashurov R.R., Mukhiddinova A.T. Inverse problem of determining the heat source density for the subdiffusion equation. Diff. Equ. Vol. 56, No. 12, pp. 1550-1563 (2020).

2. Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M. A nonlocal problem for loaded partial differential equations of fourth order. Bulletin of the Karaganda university. Mathematics. Vol. 97, No. 1, pp. 6-16 (2020).

3. Assanova A.T. An integral-boundary value problem for a partial differential equation of second order. Turkish Journal of Math. Vol. 43, No. 4, pp. 1967-1978 (2019).

4. Benney D.J., Luke J.C. Interactions of permanent waves of finite amplitude. Journal Math. Phys. Vol. 43, pp. 309-313 (1964).

5. Cavalcanti M.M., Domingos Cavalcanti V.N., Ferreira J. Existence and uniform decay for a nonlinear viscoelastic equation with strong damping. Math. Methods in the Appl. Sci. Vol. 24, pp. 1043-1053 (2001).

6. Denisov A.M., Efimov A.A. Iterative method for the numerical solution of an inverse coefficient problem for a system of partial differential equations. Diff. Equ. Vol. 56, No. 7, pp. 900-909 (2020).

7. Gordeziani D.G., Avilishbili G.A. Solving the nonlocal problems for one-dimensional medium oscillation. Matematicheskoe modelirovanie. Vol. 12, No. 1, pp. 94-103 (2000). (in Russian)

8. 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. Vol. 40, No. 4, pp. 97-109 (2020).

9. Il’in V.A. Uniqueness of generalized solutions of mixed problems for the wave equation with nonlocal boundary conditions. Diff. Equ. Vol. 44, No. 5, pp. 692-700 (2008).

10. 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. Vol. 39, No. 4, pp. 72-83 (2019).

11. Kabanikhin S.I., Krivorotko O.I. An algorithm for source reconstruction in nonlinear shallow-water equations. Comput. Math. Math. Phys. Vol. 58, No. 8, pp. 1334-1343 (2018).

12. Kabanikhin S.I., Krivorot’ko O.I., Latyshenko V.A., Yermolenko D.V., Kashtanova V.N. Inverse problems of immunology and epidemiology. Eurasian Journal of Mathematical and Computer Applications. Vol. 5, No. 2, pp. 14-35 (2017).

13. Kabanikhin S.I., Shishlenin M.A. Recovery of the time-dependent diffusion coefficient by known non-local data. Num. Anal. Appl. Vol. 11, No. 1, pp. 38-44 (2018).

14. Kostin A.B. The inverse problem of recovering the source in a parabolic equation under a condition of nonlocal observation. Sbornik. Math. Vol. 204, No. 10, pp. 1391-1434 (2013).

15. 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. Vol. 74, No. 1, pp. 136-140 (2003).

16. Prilepko A.I., Kostin A.B., Solov’ev V.V. Inverse problems of finding the source and coefficients for elliptic and parabolic equations in Hulder and Sobolev spaces. Sib. zhurn. chist. i prikl. matem. Vol. 17, No. 3, pp. 67-85 (2017). https://doi.org/10.17377/PAM.2017.17.7

17. Romanov V.G. Inverse phaseless problem for the electrodynamic equations in an anisotropic medium. Doklady Math. Vol. 100, No. 2, pp. 495-500 (2019).

18. Romanov V.G., Yamamoto M. Phaseless inverse problems with interference waves. Journal of Inverse and Ill-Posed Problems. Vol. 26, No. 5, pp. 681-688 (2018).

19. Whitham G.B. Linear and nonlinear waves. New-York - London - Sydney - Toronto, A Willey-Interscience Publication (1974).

20. Yuldashev T.K. Nonlocal mixed-value problem for a Boussinesq-type integro-differential equation with degenerate kernel. Ukrainian Math. J. Vol. 68, No. 8, pp. 1278-1296 (2016).

21. Yuldashev T.K. Mixed problem for pseudoparabolic integro-differential equation with degenerate kernel. Diff. Equ. Vol. 53, No. 1, pp. 99-108 (2017).

22. Yuldashev T.K. Solvability of a boundary value problem for a differential equation of the Boussinesq type. Diff. Equ. Vol. 54, No. 10, pp. 1384-1393 (2018).

23. Yuldashev T.K. On a boundary-value problem for Boussinesq type nonlinear integro-differential equation with reflecting argument. Lobachevskii Journal of Math. Vol. 41, No. 1, pp. 111-123 (2020).

24. Yuldashev T.K. Nonlocal inverse problem for a pseudohyperbolic-pseudoelliptic type integro-differential equations. Axioms. Vol. 9, No. 2 (2020).

25. Yuldashev T.K., Kadirkulov B.J. Nonlocal problem for a mixed type fourth-order differential equation with Hilfer fractional operator. Ural Math. J. Vol. 6, No. 1, pp. 153-167 (2020).

26. Yusifova E.H. Inverse boundary value problem for one partial differential equation of third order. Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Math. Vol. 39, No. 4, pp. 175-189 (2019).

27. 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. Vol. 41, No. 1, pp. 172-183 (2021).

28. Yuldashev T.K., Rakhmonov F.D. On a Benney–Luke type Differential Equation with Nonlinear Boundary Value Conditions. Lobachevski Journal of Math. Vol. 41, No. 1, pp. 111-123 (2020).

29. Yuldashev T.K., Rakhmonov F.D. Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters. AIP Conference Proceedings. Vol. 2365, No. 060003, 20 p. (2021).

30. Yuldashev T.K., Rakhmonov F.D. Nonlocal inverse problem for a pseudoheperbolic-pseudoelliptic type differential equation. AIP Conference Proceedings. Vol. 2365, No. 060004, 21 p. (2021).

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