najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2022-13-5-483-490najo-259Research ArticleСтатьиInverse problem for Fredholm integro-differential equation with final redefinition conditions at the end of the intervalYuldashevT. K.tursun.k.yuldashev@gmail.comZarifzodaS. K.sarvar8383@list.ruTashkent State University of EconomicsRussian FederationTajik National UniversityRussian Federation202206062025135483490Copyright © Yuldashev T.K., Zarifzoda S.K., 20252025Yuldashev T.K., Zarifzoda S.K.Yuldashev T.K., Zarifzoda S.K.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/259

The questions of solvability and construction of solutions of an inverse problem for second-order Fredholm integro-differential equation with degenerate kernel, final conditions at the end of the interval, two parameters, and two redefinition data are considered. The sets of regular parameter values are determined and the corresponding solutions are constructed. The specific features of the inverse problem are studied. Criteria for the unique solvability of the posed inverse problem are established.

integro-differential equationdegenerate kernelfinal conditions at the end of a segmentparametersredefinition datasolvabilityReferencesCavalcanti 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. Sciences, 2001, 24, P. 1043-1053.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. Sciences, 2001, 24, P. 1043-1053.Bykov Ya. V. On some problems in the theory of integro-differential equations. Izdatelstvo Kirg. Gos. Univ-ta, Frunze, 1957, 327 p. (in Russian).Bykov Ya. V. On some problems in the theory of integro-differential equations. Izdatelstvo Kirg. Gos. Univ-ta, Frunze, 1957, 327 p. (in Russian).Dzhumabaev D. S., Bakirova E. A. Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations. Differential Equations, 2010, 46(4), P. 553-567.Dzhumabaev D. S., Bakirova E. A. Criteria for the well-posedness of a linear two-point boundary value problem for systems of integro-differential equations. Differential Equations, 2010, 46(4), P. 553-567.Dzhumabaev D. S., Mynbayeva S. T. New general solution to a nonlinear Fredholm integro-differential equation. Eurasian Math. Journal, 2019, 10(4), P. 24-33.Dzhumabaev D. S., Mynbayeva S. T. New general solution to a nonlinear Fredholm integro-differential equation. Eurasian Math. Journal, 2019, 10(4), P. 24-33.Gianni R. Equation with nonlocal boundary condition. Mathematical Models and Methods in Applied Sciences, 1993. 3(6), P. 789-804.Gianni R. Equation with nonlocal boundary condition. Mathematical Models and Methods in Applied Sciences, 1993. 3(6), P. 789-804.Giacomin G., Lebowitz J. L. Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. of Statist. Phys., 1997, 87, P. 37-61.Giacomin G., Lebowitz J. L. Phase segregation dynamics in particle systems with long range interactions. I. Macroscopic limits. J. of Statist. Phys., 1997, 87, P. 37-61.Falaleev M. V.Integro-differential equations with a Fredholm operator at the highest derivative in Banach spaces and their applications. Izv. Irkutsk. Gos. Universiteta. Matematika, 2012, 5(2), P. 90-102 (in Russian).Falaleev M. V.Integro-differential equations with a Fredholm operator at the highest derivative in Banach spaces and their applications. Izv. Irkutsk. Gos. Universiteta. Matematika, 2012, 5(2), P. 90-102 (in Russian).Fedorov E. G., Popov I. Yu. Analysis of the limiting behavior of a biological neurons system with delay. J. Phys.: Conf. Ser, 2021, 2086, P. 012109.Fedorov E. G., Popov I. Yu. Analysis of the limiting behavior of a biological neurons system with delay. J. Phys.: Conf. Ser, 2021, 2086, P. 012109.Fedorov E. G., Popov I. Yu. Hopf bifurcations in a network of FitzHugh-Nagumo biological neurons.International Journal of Nonlinear Sciences and Numerical Simulation, 2021.Fedorov E. G., Popov I. Yu. Hopf bifurcations in a network of FitzHugh-Nagumo biological neurons.International Journal of Nonlinear Sciences and Numerical Simulation, 2021.Fedorov E. G. Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Math., 2021, 12(5), P. 553-562.Fedorov E. G. Properties of an oriented ring of neurons with the FitzHugh-Nagumo model. Nanosystems: Phys. Chem. Math., 2021, 12(5), P. 553-562.Sidorov N. A. Solution of the Cauchy problem for a class of integro-differential equations with analytic nonlinearities. Differ. Uravneniya, 1968, 4(7), P. 1309-1316 (in Russian).Sidorov N. A. Solution of the Cauchy problem for a class of integro-differential equations with analytic nonlinearities. Differ. Uravneniya, 1968, 4(7), P. 1309-1316 (in Russian).Ushakov E. I. Static stability of electrical circuits. Nauka, Novosibirsk, 1988. 273 p. (in Russian)Ushakov E. I. Static stability of electrical circuits. Nauka, Novosibirsk, 1988. 273 p. (in Russian)Vainberg M. M.Integro-differential equations. Itogi Nauki, 1962, VINITI, Moscow, 1964, P. 5-37 (in Russian).Vainberg M. M.Integro-differential equations. Itogi Nauki, 1962, VINITI, Moscow, 1964, P. 5-37 (in Russian).Yuldashev T. K., Odinaev R. N., Zarifzoda S. K. On exact solutions of a class of singular partial integro-differential equations. Lobachevskii Journal of Mathematics, 2021, 42(3), P. 676-684.Yuldashev T. K., Odinaev R. N., Zarifzoda S. K. On exact solutions of a class of singular partial integro-differential equations. Lobachevskii Journal of Mathematics, 2021, 42(3), P. 676-684.Yuldashev T. K., Zarifzoda S. K. On a new class of singular integro-differential equations. Bulletin of the Karaganda University. Mathematics series, 2021, 101(1), P. 138-148.Yuldashev T. K., Zarifzoda S. K. On a new class of singular integro-differential equations. Bulletin of the Karaganda University. Mathematics series, 2021, 101(1), P. 138-148.Abildayeva A. T., Kaparova R. M., Assanova A. T. To a unique solvability of a problem with integral condition for integro-differential equation. Lobachevskii Journal of Mathematics, 2021, 42(12), P. 2697-2706.Abildayeva A. T., Kaparova R. M., Assanova A. T. To a unique solvability of a problem with integral condition for integro-differential equation. Lobachevskii Journal of Mathematics, 2021, 42(12), P. 2697-2706.Asanova A. T., Dzhumabaev D. S. Correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations. Doklady Mathematics, 2003, 68(1), P. 46-49.Asanova A. T., Dzhumabaev D. S. Correct solvability of a nonlocal boundary value problem for systems of hyperbolic equations. Doklady Mathematics, 2003, 68(1), P. 46-49.Assanova A. T., On the solvability of nonlocal problem for the system of Sobolev-type differential equations with integral condition. Georgian Mathematical Journal, 2021, 28(1), P. 49-57.Assanova A. T., On the solvability of nonlocal problem for the system of Sobolev-type differential equations with integral condition. Georgian Mathematical Journal, 2021, 28(1), P. 49-57.Asanova A. T., Dzhumabaev D. S. Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations. Journal of Mathematical Analysis and Applications, 2013, 402(1), P. 167-178.Asanova A. T., Dzhumabaev D. S. Well-posedness of nonlocal boundary value problems with integral condition for the system of hyperbolic equations. Journal of Mathematical Analysis and Applications, 2013, 402(1), P. 167-178.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, 2020, 97(1), P. 6-16.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, 2020, 97(1), P. 6-16.Assanova A. T., Tokmurzin Z. S. A nonlocal multipoint problem for a system of fourth-order partial differential equations. Eurasian Math. Journal, 2020, 11(3), P. 8-20.Assanova A. T., Tokmurzin Z. S. A nonlocal multipoint problem for a system of fourth-order partial differential equations. Eurasian Math. Journal, 2020, 11(3), P. 8-20.Gordeziani D. G., Avalishvili G. A. Gordeziani solutions of nonlocal problems for one-dimensional vibrations of a medium. Matematicheskoye Modelirovaniye, 2000, 12(1), P. 94-103 (in Russian).Gordeziani D. G., Avalishvili G. A. Gordeziani solutions of nonlocal problems for one-dimensional vibrations of a medium. Matematicheskoye Modelirovaniye, 2000, 12(1), P. 94-103 (in Russian).Dzhumabaev D. S. 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, 2018, 461(1), P. 1439-1462.Dzhumabaev D. S. 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, 2018, 461(1), P. 1439-1462.Ivanchov N. I. Boundary value problems for a parabolic equation with integral conditions. Differential Equations, 2004, 40(4), P. 591-609.Ivanchov N. I. Boundary value problems for a parabolic equation with integral conditions. Differential Equations, 2004, 40(4), P. 591-609.Pao C. V. Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. of Computational and Applied Mathematics, 2001, 136(1-2), P. 227-243.Pao C. V. Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. of Computational and Applied Mathematics, 2001, 136(1-2), P. 227-243.Ochilova N. K., Yuldashev T. K. On a nonlocal boundary value problem for a degenerate parabolic-hyperbolic equation with fractional derivative. Lobachevskii Journal of Mathematics, 2022, 43(1), P. 229-236.Ochilova N. K., Yuldashev T. K. On a nonlocal boundary value problem for a degenerate parabolic-hyperbolic equation with fractional derivative. Lobachevskii Journal of Mathematics, 2022, 43(1), P. 229-236.Tikhonov I. V. Theorems on uniqueness in linear nonlocal problems for abstract differential equations. Izvestiya: Mathematics, 2003, 67(2), P. 333-363.Tikhonov I. V. Theorems on uniqueness in linear nonlocal problems for abstract differential equations. Izvestiya: Mathematics, 2003, 67(2), P. 333-363.Yurko V. A. Inverse problems for first-order integro-differential operators. Math Notes, 2016, 100(6), P. 876-882.Yurko V. A. Inverse problems for first-order integro-differential operators. Math Notes, 2016, 100(6), P. 876-882.Yuldashev T. K. Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel. Lobachevskii Journal of Mathematics, 2017, 38(3), P. 547-553.Yuldashev T. K. Determination of the coefficient and boundary regime in boundary value problem for integro-differential equation with degenerate kernel. Lobachevskii Journal of Mathematics, 2017, 38(3), P. 547-553.Yuldashev T. K., Kadirkulov B. J. Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions. Lobachevskii Journal of Mathematics, 2021, 42(3), P. 649-662.Yuldashev T. K., Kadirkulov B. J. Inverse boundary value problem for a fractional differential equations of mixed type with integral redefinition conditions. Lobachevskii Journal of Mathematics, 2021, 42(3), P. 649-662.Yuldashev T. K., Saburov Kh. Kh., Abduvahobov T. A. Nonlocal problem for a nonlinear system of fractional order impulsive integro-differential equations with maxima. Chelyabinskii Phys.-Mathem. Journal, 2022, 7(1), P. 113-122.Yuldashev T. K., Saburov Kh. Kh., Abduvahobov T. A. Nonlocal problem for a nonlinear system of fractional order impulsive integro-differential equations with maxima. Chelyabinskii Phys.-Mathem. Journal, 2022, 7(1), P. 113-122.Yuldashev T. K., Rakhmonov F. D. Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters. AIP Conference Proceedings, 2021, 2365(060003), P. 1-20.Yuldashev T. K., Rakhmonov F. D. Nonlocal problem for a nonlinear fractional mixed type integro-differential equation with spectral parameters. AIP Conference Proceedings, 2021, 2365(060003), P. 1-20.Zaripov S. K. Construction of an analog of the Fredholm theorem for a class of model first order integro-differential equations with a singular point in the kernel. Vestnik Tomsk. gos. universiteta. Matematika i mekhanika, 46, 2017, P. 24-35.Zaripov S. K. Construction of an analog of the Fredholm theorem for a class of model first order integro-differential equations with a singular point in the kernel. Vestnik Tomsk. gos. universiteta. Matematika i mekhanika, 46, 2017, P. 24-35.Zaripov S. K. A construction of analog of Fredgolm theorems for one class of first order model integro-differential equation with logarithmic singularity in the kernel. Vestnik Samar. gos. tekh. univ. Fiz.-mat. nauki, 21(2), 2017, P. 236-248.Zaripov S. K. A construction of analog of Fredgolm theorems for one class of first order model integro-differential equation with logarithmic singularity in the kernel. Vestnik Samar. gos. tekh. univ. Fiz.-mat. nauki, 21(2), 2017, P. 236-248.Zaripov S. K. On a new method of solving of one class of model first-order integro-differential equations with singularity in the kernel. Matematicheskaya fizika i compyuternoe modelirovanie, 2017, 20(4), P. 68-75.Zaripov S. K. On a new method of solving of one class of model first-order integro-differential equations with singularity in the kernel. Matematicheskaya fizika i compyuternoe modelirovanie, 2017, 20(4), P. 68-75.Yuldashev T. K. On Fredholm partial integro-differential equation of the third order.Russian Mathematics, 2015, 59(9), P. 62-66.Yuldashev T. K. On Fredholm partial integro-differential equation of the third order.Russian Mathematics, 2015, 59(9), P. 62-66.Yuldashev T. K. On a boundary-value problem for a fourth-order partial integro-differential equation with degenerate kernel. Journal of Mathematical Sciences, 2020, 245(4), P. 508-523.Yuldashev T. K. On a boundary-value problem for a fourth-order partial integro-differential equation with degenerate kernel. Journal of Mathematical Sciences, 2020, 245(4), P. 508-523.Yuldashev T. K. 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, P. 793-807.Yuldashev T. K. 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, P. 793-807.Yuldashev T. K. Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel. Journal of Mathematical Sciences, 2020, 250(5), P. 847-858.Yuldashev T. K. Inverse boundary-value problem for an integro-differential Boussinesq-type equation with degenerate kernel. Journal of Mathematical Sciences, 2020, 250(5), P. 847-858.Yuldashev T. K., Apakov Yu. P., Zhuraev A. Kh. Boundary value problem for third order partial integro-differential equation with a degenerate kernel. Lobachevskii Journal of Mathematics, 2021, 42(6), P. 1317-1327.Yuldashev T. K., Apakov Yu. P., Zhuraev A. Kh. Boundary value problem for third order partial integro-differential equation with a degenerate kernel. Lobachevskii Journal of Mathematics, 2021, 42(6), P. 1317-1327.

The authors declare that there are no conflicts of interest present.