Inverse Problem for a Hyperbolic Integro-Differential Equation with two Redefinition Conditions at the End of the Interval and Involution

Year 2024, Volume: 14 Issue: 1, 3 - 22, 15.01.2024

Tursun Yuldashev Oybek Kilichev

https://doi.org/10.59849/2218-6816.2024.1.3

Abstract

In this paper, we consider an inhomogeneous hyperbolic type partial integrodifferential equation with degenerate kernel, two redefinition functions and involution.
Intermediate data are used to find these redefinition functions. Dirichlet boundary conditions with respect to spatial variable are used. The Fourier method of separation of
variables is applied. The countable system of functional-integral equations is obtained.
Theorem on a unique solvability of countable system of functional-integral equations is
proved. The method of successive approximations is used in combination with the method
of contraction mappings. The triple of solutions of the inverse problem is obtained in
the form of Fourier series. Absolute convergence of Fourier series is proved.

Keywords

inverse problem, two redefinition functions, final conditions at the endpoint of a segment, intermediate data, three parameters, involution, unique solvability

References

• [1] M. M. Cavalcanti, V. N. Domingos Cavalcanti, J. Ferreira, Existence and uniform decay for a nonlinear viscoelastic equation with strong damping, Math. Methods in the Appl. Sciences, 24, 2001, 1043–1053.
• [2] Ya. V. Bykov, On some problems in the theory of integro-differential equations, Izdatelstvo Kirg. Gos. Univ-ta, Frunze, 1957, 327 p. (in Russian).
• [3] D. S. Dzhumabaev, E. A. Bakirova, Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations, Differential Equations, 46(4), 2010, 553–567.
• [4] D. S. Dzhumabaev, S. T. Mynbayeva, New general solution to a nonlinear Fredholm integro-differential equation, Eurasian Math. Journal, 10(4), 2019, 24–33.
• [5] M. M. Vainberg, Integro-differential equations, Itogi Nauki. 1962. VINITI, Moscow, 1964, 5(37) (in Russian).
• [6] M. V. Falaleev, Integro-differential equations with a Fredholm operator at the highest derivative in Banach spaces and their applications, Izv. Irkutsk. Gos. Universiteta, Matematika, 5(2), 2012, 90–102 (in Russian).
• [7] N. A. Sidorov, Solution of the Cauchy problem for a class of integrodifferential equations with analytic nonlinearities, Differ. Uravneniya, 4(7), 1968, 1309–1316 (in Russian).
• [8] E. I. Ushakov, Static stability of electrical circuits, Nauka, Novosibirsk, 1988, 273 p. (in Russian).
• [9] T. K. Yuldashev, R. N. Odinaev, S. K. Zarifzoda, On exact solutions of a class of singular partial integro-differential equations, Lobachevskii Journal of Mathematics, 42(3), 2021, 676–684.
• [10] T. K. Yuldashev, S. K. Zarifzoda, On a new class of singular integrodifferential equations, Bulletin of the Karaganda University. Mathematics series, 101(1), 2021, 138–148.
• [11] A. T. Abildayeva, R. M. Kaparova, A. T. Assanova, To a unique solvability of a problem with integral condition for integro-differential equation, Lobachevskii Journal of Mathematics, 42(12), 2021, 2697–2706.
• [12] A. T. Asanova, D. S. Dzhumabaev, Correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations, Doklady Mathematics, 68(1), 2003, 46–49.
• [13] A. T. Assanova, On the solvability of nonlocal problem for the system of Sobolev-type differential equations with integral condition, Georgian Mathematical Journal, 28(1), 2021, 49–57.
• [14] A. T. Asanova, D. S. Dzhumabaev, Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations, Journal of Mathematical Analysis and Applications, 402(1), 2013, 167–178.
• [15] A. T. Assanova, A. E. Imanchiyev, Zh. M. Kadirbayeva, A nonlocal problem for loaded partial differential equations of fourth order, Bulletin of the Karaganda university-Mathematics, 97(1), 2020, 6–16.
• [16] A. T. Assanova, Z. S. Tokmurzin, A nonlocal multipoint problem for a system of fourth-order partial differential equations, Eurasian Math. Journal, 11(3), 2020, 8–20.
• [17] D. G. Gordeziani, G. A. Avalishvili, Gordeziani solutions of nonlocal problems for one-dimensional vibrations of a medium, Matematicheskoye Modelirovaniye, 12(1), 2000, 94–103 (in Russian).
• [18] D. S. Dzhumabaev, Well-posedness of nonlocal boundary value problem for a system of loaded hyperbolic equations and an algorithm for finding its solution, Journal of Mathematical Analysis and Applications, 461(1), 2018, 1439–1462.
• [19] N. Sh. Isgenderov, S. I. Allahverdiyeva, On solvability of an inverse boundary value problem for the Boussinesq-Love equation with periodic and integral condition, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 41(1), 2021, 118–132.
• [20] N. I. Ivanchov, Boundary value problems for a parabolic equation with integral conditions, Differential Equations, 40(4), 2004, 591–609.
• [21] G. S. Mammedzadeh, On a boundary value problem with spectral parameter quadratically contained in the boundary condition, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 42(1), 2022, 141–150.
• [22] N. K. Ochilova, T. K. Yuldashev, On a nonlocal boundary value problem for a degenerate parabolichyperbolic equation with fractional derivative, Lobachevskii Journal of Mathematics, 43(1), 2022, 229–236.
• [23] I. V. Tikhonov, Theorems on uniqueness in linear nonlocal problems for abstract differential equations, Izvestiya: Mathematics, 67(2), 2003, 333–363.
• [24] V. A. Yurko, Inverse problems for first-order integro-differential operators, Math. Notes, 100(6), 2016, 876–882.
• [25] T. K. Yuldashev, Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel, Lobachevskii Journal of Mathematics, 38(3), 2017, 547–553.
• [26] T. K. Yuldashev, F. D. Rakhmonov, Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters, AIP Conference Proceedings, 2365(060003), 2021, 1–20.
• [27] S. K. Zaripov, Construction of an analog of the Fredholm theorem for a class of model first order integrodifferential equations with a singular point in the kernel, Vestnik Tomsk. gos. universiteta. Matematika i mekhanika, 46, 2017, 24–35.
• [28] S. K. Zaripov, A construction of analog of Fredgolm theorems for one class of first order model integrodifferential equation with logarithmic singularity in the kernel, Vestnik Samar. gos. tekh. univ. Fiz.-mat. nauki, 21(2), 2017, 236–248.
• [29] S. K. Zaripov, On a new method of solving of one class of model firstorder integro-differential equations with singularity in the kernel, Matematicheskaya fizika i compyuternoe modelirovanie, 20(4), 2017, 68–75.
• [30] T. K. Yuldashev, On Fredholm partial integro-differential equation of the third order, Russian Mathematics, 59(9), 2015, 62–66.
• [31] T. K. Yuldashev, On a boundary-value problem for a fourth-order partial integro-differential equation with degenerate kernel, Journal of Mathematical Sciences, 245(4), 2020, 508–523.
• [32] T. K. Yuldashev, Determining of coefficients and the classical solvability of a nonlocal boundary-value problem for the Benney-Luke integro-differential equation with degenerate kernel, Journal of Mathematical Sciences, 254(6), 2021, 793–807.
• [33] T. K. Yuldashev, Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel, Journal of Mathematical Sciences, 250(5), 2020, 847–858.
• [34] T. K. Yuldashev, Yu. P. Apakov, A. Kh. Zhuraev, Boundary value problem for third order partial integro-differential equation with a degenerate kernel, Lobachevskii Journal of Mathematics, 42(6), 2021, 1317–1327.
• [35] T. K. Yuldashev, On a boundary value problem for a fifth order partial integro-differential equation, Azerbaijan Journal of Mathematics, 12(2), 2022, 154–172.
• [36] T. K. Yuldashev, On a nonlocal boundary value problem for a partial integrodifferential equations with degenerate kernel, Vladikavkaz Mathematical Journal, 24(2), 2022, 130–141.