kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-3-191-200kaz29-1381Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО КРАЕВОЙ ЗАДАЧЕ ДЛЯ ГИПЕРБОЛИЧЕСКОГО УРАВНЕНИЯ ВЫСОКОГО ПОРЯДКА С ИМПУЛЬСНОЙ ДИСКРЕТНОЙ ПАМЯТЬЮON A BOUNDARY VALUE PROBLEM FOR HIGH-ORDER HYPERBOLIC EQUATION WITH IMPULSE DISCRETE MEMORYhttps://orcid.org/0000-0001-8697-8920АсановаА. Т.AssanovaA. T.

ф.-м.н., г.н.с. 

050010, г. Алматы

Dr. Phys.-Math. Sc., Principal Researcher 

050010, Almaty

assanova@math.kzhttps://orcid.org/0009-0002-9456-3025БименоваР. А.BimenovaR. A.

050010, г. Алматы

050010, Almaty

b.rabiga@mail.ruhttps://orcid.org/0000-0002-7195-4480МинглибаеваБ. Б.MinglibayevaB. B.

к.ф.-м.н., ст. преподаватель, с.н.с. 

050010, г. Алматы;050000, г. Алматы

Cand. Phys.-Math.Sc., Senior Lecturer, Senior Researcher 

050010, Almaty;050000, Almaty

bayan_math@mail.ruhttps://orcid.org/0000-0003-1921-0174СабалаховаА. П.SabalakhovaA. P.

ст. преподаватель 160012, г. Шымкент

Senior Lecturer 

160012, Shymkent

sabalahova@mail.ruИнститут математики и математического моделированияКазахстанInstitute of Mathematics and Mathematical ModelingKazakhstanИнститут математики и математического моделирования; Казахский национальный женский педагогический университетКазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women's Teacher Training UniversityKazakhstanЮжно-Казахстанский университет имени М. АуезоваКазахстанMukhtar Auezov South Kazakhstan UniversityKazakhstan202402102024213191200Copyright © Асанова А.Т., Бименова Р.А., Минглибаева Б.Б., Сабалахова А.П., 20242024Асанова А.Т., Бименова Р.А., Минглибаева Б.Б., Сабалахова А.П.Assanova A.T., Bimenova R.A., Minglibayeva B.B., Sabalakhova A.P.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestnik.kbtu.edu.kz/jour/article/view/1381

Исследуется краевая задача для гиперболического уравнения высокого порядка с импульсной дискретной памятью в прямоугольной области. С помощью введения новых функций рассматриваемая задача сводится к семейству краевых задач для дифференциального уравнения первого порядка с импульсной дискретной памятью, зависящей от неизвестных функций, и интегральным равенствам. К этой эквивалентной задаче применяется метод параметризации Д.С. Джумабаева. Разбиением области по временной переменной, введением функциональных параметров как значений дискретной памяти во внутренних областях исследуемое семейство краевых задач для дифференциального уравнения первого порядка с импульсной дискретной памятью, зависящее от неизвестных функций, и интегральные равенства переходят к эквивалентному семейству интегральных-многоточечных краевых задач с функциональными параметрами и неизвестными функциями. Полученные эквивалентные задачи содержат начальные задачи для дифференциальных уравнений первого порядка относительно новой функции. Решение начальных задач выражается через интегральные уравнения Вольтерра. Подставляя эти решения в краевые условия и импульсные условия, строится система линейных функциональных уравнений относительно функциональных параметров. Построен алгоритм нахождения решения эквивалентной задачи. Сформулированы достаточные условия однозначной разрешимости семейства интегральных-многоточечных краевых задач с функциональными параметрами и неизвестными функциями. В терминах исходных данных краевой задачи для гиперболического уравнения высокого порядка с импульсной дискретной памятью установлены достаточные условия ее однозначной разрешимости. 

The investigation focuses on a boundary value problem for a high-order hyperbolic equation with impulse discrete memory in a rectangular domain. By introducing new functions, the problem is transformed into a set of boundary value problems for a first-order differential equation with impulse discrete memory, which depends on unknown functions and integral relations. D.S. Dzhumabaev's parametrization method is applied to this equivalent problem. The domain is divided according to the time variable, and functional parameters representing discrete memory values are introduced within the interior domains. As a result, the family of boundary value problems for the first-order differential equation with impulse discrete memory and unknown functions is converted into an equivalent family of integral-multipoint boundary value problems involving functional parameters and unknown functions. These equivalent problems include initial value problems for first-order differential equations related to the new functions. The solutions to the initial value problems are expressed using Volterra integral equations. By substituting these solutions into the boundary and impulse conditions, a system of linear functional equations concerning the functional parameters is derived. An algorithm is developed to solve the equivalent problem, and sufficient conditions for the unique solvability of the family of integral-multipoint boundary value problems with functional parameters and unknown functions are provided. Additionally, sufficient conditions for the unique solvability of the original boundary value problem for the high-order hyperbolic equation with impulse discrete memory are established based on the initial data.

импульсная дискретная памятьгиперболическое уравнение высокого порядкасемейство краевых задачфункционально-дифференциальные уравненияImpulse discrete memoryhigh-order hyperbolic equationfamily of boundary value problemsfunctional-differential equationsReferencesDzhumabayev D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. USSR Computational Mathematics and Mathematical Physics, 1989, vol. 29, no. 1, pp. 34–46.Dzhumabayev D.S. Criteria for the unique solvability of a linear boundary-value problem for an ordinary differential equation. USSR Computational Mathematics and Mathematical Physics, 1989, vol. 29, no. 1, pp. 34–46.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, vol. 402, no. 1, pp. 167–178. https://doi.org/10.1016/j.jmaa.2013.01.012.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, vol. 402, no. 1, pp. 167–178. https://doi.org/10.1016/j.jmaa.2013.01.012.Assanova A.T., Sabalakhova A.P. On the unique solvability of nonlocal problems with integral conditions for a hybrid system of partial differential equations. Eurasian Mathematical Journal, 2018, vol. 9, no. 3, pp. 14–24. https://doi.org/10.32523/2077-9879-2018-9-3-14-24.Assanova A.T., Sabalakhova A.P. On the unique solvability of nonlocal problems with integral conditions for a hybrid system of partial differential equations. Eurasian Mathematical Journal, 2018, vol. 9, no. 3, pp. 14–24. https://doi.org/10.32523/2077-9879-2018-9-3-14-24.Assanova A.T. An integral-boundary value problem for a partial differential equation of second order. Turkish Journal of Mathematics, 2019, vol. 43, no. 4, pp. 1967–1978. https://doi.org/10.3906/mat-1903-111.Assanova A.T. An integral-boundary value problem for a partial differential equation of second order. Turkish Journal of Mathematics, 2019, vol. 43, no. 4, pp. 1967–1978. https://doi.org/10.3906/mat-1903-111.Bakirova E.A. Dzhumabaev D.S. and Mynbayeva S.T. A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation. Mathematical Methods in the Applied Sciences, 2020, vol. 43, no. 7, pp. 1788–1802. https://doi.org/10.1002/mma.6003.Bakirova E.A. Dzhumabaev D.S. and Mynbayeva S.T. A method of solving a nonlinear boundary value problem with a parameter for a loaded differential equation. Mathematical Methods in the Applied Sciences, 2020, vol. 43, no. 7, pp. 1788–1802. https://doi.org/10.1002/mma.6003.Assanova A.T. On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition. Georgian Mathematical Journal, 2021, vol. 28, no. 1, pp. 49–57. https://doi.org/10.1515/gmj-2019-2011.Assanova A.T. On the solvability of a nonlocal problem for the system of Sobolev-type differential equations with integral condition. Georgian Mathematical Journal, 2021, vol. 28, no. 1, pp. 49–57. https://doi.org/10.1515/gmj-2019-2011.Bakirova E.A., Assanova A.T. and Kadirbayeva Zh.M. A problem with parameter for the integrodifferential equations. Mathematical Modelling and Analysis, 2021, vol. 26, no. 1, pp. 34–54. https://doi.org/10.3846/mma. 2021.11977.Bakirova E.A., Assanova A.T. and Kadirbayeva Zh.M. A problem with parameter for the integrodifferential equations. Mathematical Modelling and Analysis, 2021, vol. 26, no. 1, pp. 34–54. https://doi.org/10.3846/mma. 2021.11977.Minglibayeva B.B., Assanova A.T. An existence of an isolated solution to nonlinear two-point boundary value problem with parameter. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 3, pp. 587–597. https://doi.org/10.1134/S199508022103015X.Minglibayeva B.B., Assanova A.T. An existence of an isolated solution to nonlinear two-point boundary value problem with parameter. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 3, pp. 587–597. https://doi.org/10.1134/S199508022103015X.Assanova A.T., Sabalakhova A.P., and Toleukhanova Z.M. On the unique solvability of a family of boundary value problems for integro-differential equations of mixed type. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 6, pp. 1228–1238. https://doi.org/10.1134/S1995080221060044.Assanova A.T., Sabalakhova A.P., and Toleukhanova Z.M. On the unique solvability of a family of boundary value problems for integro-differential equations of mixed type. Lobachevskii Journal of Mathematics, 2021, vol. 42, no. 6, pp. 1228–1238. https://doi.org/10.1134/S1995080221060044.Assanova A.T. and Imanchiyev A.E. The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solution. Quaestiones Mathematicae, 2022, vol. 45, no. 10, pp. 1641–1653. https://doi.org/ 10.2989/16073606.2021.1967503.Assanova A.T. and Imanchiyev A.E. The problem with non-separated multipoint-integral conditions for high-order differential equations and a new general solution. Quaestiones Mathematicae, 2022, vol. 45, no. 10, pp. 1641–1653. https://doi.org/ 10.2989/16073606.2021.1967503.Assanova A.T. and Uteshova R. Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type. Chaos, Solitons & Fractals, 2022, vol. 165, no. 12, art. no.112816. https://doi.org/10.1016/ j.chaos.2022.112816.Assanova A.T. and Uteshova R. Solution of a nonlocal problem for hyperbolic equations with piecewise constant argument of generalized type. Chaos, Solitons & Fractals, 2022, vol. 165, no. 12, art. no.112816. https://doi.org/10.1016/ j.chaos.2022.112816.Assanova A.T. A generalized integral problem for a system of hyperbolic equations and its applications. Hacettepe Journal of Mathematics and Statistics, 2023, vol. 52, no. 6, pp. 1513–1532. https://doi.org/10.15672/hujms.1094454.Assanova A.T. A generalized integral problem for a system of hyperbolic equations and its applications. Hacettepe Journal of Mathematics and Statistics, 2023, vol. 52, no. 6, pp. 1513–1532. https://doi.org/10.15672/hujms.1094454.Assanova A.T. and Imanchiyev A.E. A nonlocal problem with multipoint conditions for partial differential equations of higher order, Filomat, 2024, vol. 38, no. 1, pp. 295–304. https://doi.org/10.2298/FIL2401295A.Assanova A.T. and Imanchiyev A.E. A nonlocal problem with multipoint conditions for partial differential equations of higher order, Filomat, 2024, vol. 38, no. 1, pp. 295–304. https://doi.org/10.2298/FIL2401295A.Assanova A.T. and Tleulessova A.B. Nonlocal problem for a system of partial differential equations of higher order with pulsed actions. Ukranian Mathematical Journal, 2020, vol. 71, no. 12, pp. 1821–1842. https://doi.org/10.1007/s11253-020-01750-9.Assanova A.T. and Tleulessova A.B. Nonlocal problem for a system of partial differential equations of higher order with pulsed actions. Ukranian Mathematical Journal, 2020, vol. 71, no. 12, pp. 1821–1842. https://doi.org/10.1007/s11253-020-01750-9.Assanova A.T., Abildayeva A.D., Tleulessova A.B. Nonlocal problems for the fourth order impulsive partial differential equations. Springer Proceedings in Mathematics and Statistics, 2020, vol. 333, pp. 81–94. https://doi.org/10.1007/978-3-030-56323-3_7.Assanova A.T., Abildayeva A.D., Tleulessova A.B. Nonlocal problems for the fourth order impulsive partial differential equations. Springer Proceedings in Mathematics and Statistics, 2020, vol. 333, pp. 81–94. https://doi.org/10.1007/978-3-030-56323-3_7.Akhmet M.U. Principles of discontinuous dynamical systems, Springer, New-York, 2010, 176 p.Akhmet M.U. Principles of discontinuous dynamical systems, Springer, New-York, 2010, 176 p.Akhmet M.U. Nonlinear hybrid continuous/discrete-time models, Atlantis Press, Paris, 2011, 216 p.Akhmet M.U. Nonlinear hybrid continuous/discrete-time models, Atlantis Press, Paris, 2011, 216 p.Akhmet M.U. and Yilmaz E. Neural Networks with Discontinuous/Impact Activations, Springer, New-York, 2013, 168 p.Akhmet M.U. and Yilmaz E. Neural Networks with Discontinuous/Impact Activations, Springer, New-York, 2013, 168 p.Akhmet M.U. Almost Periodicity, Chaos, and Asymptotic Equivalence, Springer, Switzerland, 2020, 368 p.Akhmet M.U. Almost Periodicity, Chaos, and Asymptotic Equivalence, Springer, Switzerland, 2020, 368 p.Nieto J.J. and Rodriguez-Lopez R. Green's function for second order periodic BVPs with piecewise constant argument. Journal of Mathematical Analysis and Applications, 2005, vol. 304, no. 1, pp. 33–57. https://doi.org/10.1016/j.jmaa.2004.09.023.Nieto J.J. and Rodriguez-Lopez R. Green's function for second order periodic BVPs with piecewise constant argument. Journal of Mathematical Analysis and Applications, 2005, vol. 304, no. 1, pp. 33–57. https://doi.org/10.1016/j.jmaa.2004.09.023.Dominguez-Perez M.A., Rodriguez-Lopez R. Multipoint BVPs of Neumann type for functional differential equations. Nonlinear Analysis: Real World Applications, 2012, vol. 13, no. 4, pp. 1662–1675. https://doi.org/10.1016/j.nonrwa.2011.11. 023.Dominguez-Perez M.A., Rodriguez-Lopez R. Multipoint BVPs of Neumann type for functional differential equations. Nonlinear Analysis: Real World Applications, 2012, vol. 13, no. 4, pp. 1662–1675. https://doi.org/10.1016/j.nonrwa.2011.11. 023.Abildayeva A., Assanova A., Imanchiyev A. A multi-point problem for a system of differential equations with piecewise-constant argument of generalized type as a neural network model. Eurasian Mathematical Journal, 2022, vol. 13, no. 2, pp. 8–17. https://doi.org/10.32523/2077-9879-2022-13-2-08-17.Abildayeva A., Assanova A., Imanchiyev A. A multi-point problem for a system of differential equations with piecewise-constant argument of generalized type as a neural network model. Eurasian Mathematical Journal, 2022, vol. 13, no. 2, pp. 8–17. https://doi.org/10.32523/2077-9879-2022-13-2-08-17.Imanchiyev A.E., Assanova A.T., Molybaikyzy A. Properties of a nonlocal problem for hyperbolic equations with impulse discrete memory. Lobachevskii Journal of Mathematics, 2023, vol. 44, no. 10, pp. 4299–4309. https://doi.org/10.1134/S1995080223100177.Imanchiyev A.E., Assanova A.T., Molybaikyzy A. Properties of a nonlocal problem for hyperbolic equations with impulse discrete memory. Lobachevskii Journal of Mathematics, 2023, vol. 44, no. 10, pp. 4299–4309. https://doi.org/10.1134/S1995080223100177.

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