kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-1-250-264kaz29-2519Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО РЕШЕНИИ НАЧАЛЬНО–КРАЕВОЙ ЗАДАЧИ ДЛЯ НАГРУЖЕННОГО ДИФФЕРЕНЦИАЛЬНОГО УРАВНЕНИЯ В ЧАСТНЫХ ПРОИЗВОДНЫХ ЧЕТВЕРТОГО ПОРЯДКАON THE SOLUTION OF AN INITIAL–BOUNDARY VALUE PROBLEM FOR A LOADED FOURTH–ORDER PARTIAL DIFFERENTIAL EQUATIONhttps://orcid.org/0009-0006-3617-4071СарманА. Д.SarmanA. D.

докторант, преподаватель

г. Актобе

PhD, Lecturer

Aktobe

sadrasul8@gmail.comhttps://orcid.org/0000-0001-8697-8920АсановаА. Т.AssanovaA. T.

д.ф.-м.н., к.б.н.

г. Алматы

Dr. Phys.-Math.Sc., Cand.Biol.Sc.

Almaty

assanova@math.kzhttps://orcid.org/0000-0002-3738-5923ТокмурзинЖ. С.TokmurzinZh. S.

PhD, ст. преподаватель

г. Актобе

PhD, Senior Lecturer

Aktobe

tokmurzinzh@gmail.comАктюбинский региональный университет им. К. ЖубановаКазахстанK. Zhubanov Aktobe Regional UniversityKazakhstanИнститут математики и математического моделированияКазахстанInstitute of Mathematics and Mathematical ModelingKazakhstan202629032026231250264Copyright © Сарман А.Д., Асанова А.Т., Токмурзин Ж.С., 20262026Сарман А.Д., Асанова А.Т., Токмурзин Ж.С.Sarman A.D., Assanova A.T., Tokmurzin Z.S.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/2519

Рассматривается начально-краевая задача для дифференциального уравнения с n-нагрузкой, зависящего от двух переменных и содержащего частные производные четвертого порядка. Путем введения новой неизвестной функции исходная задача сводится к семейству задач Коши для нагруженного дифференциального уравнения с частными производными первого порядка. Для неизвестной функции cнагрузкамистроится система функциональных уравнений по переменной x. Предложен алгоритм нахождения решения этой системы функциональных уравнений. Доказана теорема о существовании единственного решения семейства задач Коши для нагруженного дифференциального уравнения с частными производными первого порядка. Установлены условия существования и единственности решения начально-краевой задачи для дифференциального уравнения с нагрузкой, содержащего частные производные четвертого порядка и зависящего от двух переменных. Результат проиллюстрирован примером.

An initial–boundary value problem is considered for a differential equation with n loads, depending on two variables and involving fourth-order partial derivatives. By introducing a new unknown function, the original problem is reduced to a family of Cauchy problems for a loaded differential equation with first-order partial derivatives. For the unknown function with loads z(t_i,x), a system of functional equations with respect to the variable x is constructed. An algorithm for finding a solution to this system of functional equations is proposed. A theorem on the existence and uniqueness of a solution to the family of Cauchy problems for the loaded differential equation with first-order partial derivatives is proved. Conditions for the existence and uniqueness of a solution to the initial–boundary value problem for the loaded differential equation involving fourth-order partial derivatives and depending on two variables are established. The result is illustrated by an example.

нагруженное дифференциальное уравнение в частных производных четвертого порядканачально–краевая задачасистема функциональных уравненийусловия разрешимостиhigh-order loaded partial differential equationinitial-boundary value problemsystem of functional equationssolvability conditionsReferencesBishop R.E.D. Longitudinal waves in beams//Aeronaut. – 1952. – Q. 3(2). – P. 280 – 293 p.Bishop, R.E.D. Longitudinal waves in beams. Aeronautical Quarterly, 3 (2), 280–293 (1952).Ptashnyck B.I. Ill-posed boundary value problems for partial differential equations. – Kiev: Naukova Dumka, 1984. – 265 p.Ptashnyck, B.I. Ill-posed boundary value problems for partial differential equations (Kiev: Naukova Dumka, 1984), 265 p.Нахушев А.М. Уравнения математической биологии. – М.: Высшая школа, 1995. – 305 c.Nakhushev, A.M. Equations of Mathematical Biology (Moscow: Vysshaya shkola, 1995), 305 p.Demidenko G.V., Uspenskii S.V. Partial differential equations and systems not solvable with respect to the highest-order derivative// Pure and Appl.Math. – Marcel Dekker, New York, 1998 – 256 p.Demidenko, G.V., and Uspenskii, S.V. Partial differential equations and systems not solvable with respect to the highest-order derivative (New York: Marcel Dekker, 1998), 256 p.Ptashnyk B.Yo., Il’kiv V.S., Kmit’ I.Ya., Polishchuk V.M. Nonlocal boundary value problems for partial differential equations. – Kyiv: Naukova Dumka, Ukraine, 2002. – 292 p.Ptashnyk, B.Yo., Il’kiv, V.S., Kmit’, I.Ya., and Polishchuk, V.M. Nonlocal boundary value problems for partial differential equations (Kyiv: Naukova Dumka, 2002), 292 p.Алгазин С.Д., Кийко И.А. Флаттер пластин и оболочек. – М.: Наука, 2006. – 245 с.Algazin, S.D., and Kiiko, I.A. Flutter of plates and shells (Moscow: Nauka, 2006), 245 p. (in Russian).Нахушев А.М. Задачи со смещением для уравнений в частных производных. – М.: Наука, 2006. – 287 с.Nakhushev, A.M. Boundary value problems with nonlocal conditions for partial differential equations (Moscow: Nauka, 2006), 287 p. (in Russian).Нахушев А.М. Нагруженные уравнения и их применение. – М.: Наука, 2012. – 232 с.Nakhushev, A.M. Loaded equations and their applications (Moscow: Nauka, 2012), 232 p. (in Russian).Мамедов И.Г. Задача оптимального управления в процессах, описываемых нелокальной задачей с нагружениями для гиперболического интегро дифференциального уравнения // Известия Национальной академии наук Азердбайжана, серия ФТМН. – 2004. – Т. 24. – №2. – С. 74–79.Mamedov, I.G. The optimal control in processes described by a nonlocal loaded problem for a hyperbolic integro-differential equation. Izvestiya of the National Academy of Sciences of Azerbaijan, Series Physico-Mathematical Sciences, 24 (2), 74–79 (2004). (in Russian).Midodashvili B. Generalized Goursat problem for a spatial fourth order hyperbolic equation with dominated low terms // Proc. of A. Razmadze Math. Insitute, – 2005. – Vol. 138. – P. 43–54.Midodashvili, B. Generalized Goursat problem for a spatial fourth order hyperbolic equation with dominated low terms. Proceedings of A. Razmadze Mathematical Institute, 138, 43–54 (2005).Kiguradze T. On solvability and well – posedness of boundary value problems for nonlinear hyperbolic equations of the fourth order // Georgian Mathematical Journal. – 2008. – Vol. 15. – No. 3. – P. 555–569.Kiguradze, T. On solvability and well-posedness of boundary value problems for nonlinear hyperbolic equations of the fourth order. Georgian Mathematical Journal, 15 (3), 555–569 (2008).Liu Y., Li H. H1 – Galerkin mixed finite element for pseudo – hyperbolic equations// Applied Mathematics and Computation. – 2009. – Vol. 212 (2). – P. 446–457.Liu, Y., and Li, H. H¹-Galerkin mixed finite element for pseudo-hyperbolic equations. Applied Mathematics and Computation, 212 (2), 446–457 (2009).Mamedov I.G. A fundamental solution to the Cauchy problem for a fourth order pseudoparabolic equation // Comput. Math. Phys. – 2009. – Vol. 49 (1). – P. 93–104.Mamedov, I.G. A fundamental solution to the Cauchy problem for a fourth order pseudoparabolic equation. Computational Mathematics and Mathematical Physics, 49 (1), 93–104 (2009).Guo H. Analysis of split weighted least–squares procedures for pseudohyperbolic equations // Applied Mathematics and Computation. – 2010. – Vol. 217 (8). – P. – 4109–4121.Guo, H. Analysis of split weighted least-squares procedures for pseudohyperbolic equations. Applied Mathematics and Computation, 217 (8), 4109–4121 (2010).Ferraioli D.C., Tenenblat K. Fourth order evolution equations which describe pseudospherical surfaces // J. Differtial Equations. – 2014. – Vol. 257. – P. 3165–3199. https://doi.org/10.1016/j.jde.2014.06.010.Ferraioli, D.C., and Tenenblat, K. Fourth order evolution equations which describe pseudospherical surfaces. Journal of Differential Equations, 257, 3165–3199 (2014). https://doi.org/10.1016/j.jde.2014.06.010.Fedotov I., Shatalov M., Marais J. Hyperbolic and pseudohyperbolic equations in the theory of vibration // Acta Mech. – 2016. – Vol. 227(12). – P. 3315–3324.Fedotov, I., Shatalov, M., and Marais, J. Hyperbolic and pseudohyperbolic equations in the theory of vibration. Acta Mechanica, 227 (12), 3315–3324 (2016).Yuldashev T.K. Inverse problem for a nonlinear Benny – Luke type integro – differential equations with degenerate kernel// Russian Mathematics. – 2016. – Vol. 60. – No. 9.– P. 53–60.Yuldashev, T.K. Inverse problem for a nonlinear Benny–Luke type integro-differential equation with degenerate kernel. Russian Mathematics, 60 (9), 53–60 (2016).Zhao Z., Li H. A continuous Galerkin method for pseudo – hyperbolic equations with variable coefficients // J. Math. Anal. and Appl. – 2019. – Vol. 473 (2). – P. 1053–1072.Zhao, Z., and Li, H. A continuous Galerkin method for pseudo-hyperbolic equations with variable coefficients. Journal of Mathematical Analysis and Applications, 473 (2), 1053–1072 (2019).Assanova A.T., Boichuk A.A., TokmurzinZh.S. On the initial – boundary value problem for system of the partial differential equations of fourth order // News of the NAS RK. Physico–Math. Ser. – 2019. – Vol. 323. – P. 14–21. https://doi.org/10.32014/2019.2518-1726.2.Assanova, A.T., Boichuk, A.A., and Tokmurzin, Zh.S. On the initial-boundary value problem for a system of fourth order partial differential equations. News of the National Academy of Sciences of the Republic of Kazakhstan, Physico-Mathematical Series, 323, 14–21 (2019). https://doi.org/10.32014/2019.2518-1726.2.Assanova A.T., Tokmurzin,Zh.S. On two – point initial boundary value problem for fourth order partial differential equations// Kazakh Mathematical Jounal. – 2019. – Vol. 19. – No. 3. – P. 66–78.Assanova, A.T., and Tokmurzin, Zh.S. On two-point initial-boundary value problem for fourth order partial differential equations. Kazakh Mathematical Journal, 19 (3), 66–78 (2019).Assanova A.T., Tokmurzin,Zh.S. An approach to the solution of the initial boundary – value problem for systems of fourth – order hyperbolic equations // Mathematical Notes. – 2020. – Vol. 108. – No. 1. – P. 3–14.Assanova, A.T., and Tokmurzin, Zh.S. An approach to the solution of the initial boundary-value problem for systems of fourth-order hyperbolic equations. Mathematical Notes, 108 (1), 3–14 (2020).Assanova A.T., TokmurzinZh.S. Boundary value problem for system of pseudo – hyperbolic equations of the fourth order with nonlocal condition // Russian Mathematics. – 2020. – Vol. 64. – No. 9. – P. 1–11.Assanova, A.T., and Tokmurzin, Zh.S. Boundary value problem for system of pseudo-hyperbolic equations of the fourth order with nonlocal condition. Russian Mathematics, 64 (9), 1–11 (2020).Assanova A.T., Tokmurzin Zh.S. A nonlocal multipoint problem for a system of fourth – order partial differential equations// EurasianMathematical Journal. – 2020. – Vol. 11(3). – P. 8–20. https://doi.org/10.32523/2077-9879-2020-11-08-20.Assanova, A.T., and Tokmurzin, Zh.S. A nonlocal multipoint problem for a system of fourth-order partial differential equations. Eurasian Mathematical Journal, 11 (3), 8–20 (2020). https://doi.org/10.32523/2077-9879-2020-11-08-20.Assanova A.T., Tokmurzin,Zh.S. Method of functional parametrization equations // Bulletin of the Karaganda university – Series Mathematics. – 2020. – Vol. 100. – No. 4. – P. 5–16.Assanova, A.T., and Tokmurzin, Zh.S. Method of functional parametrization of equations. Bulletin of the Karaganda University – Series Mathematics, 100 (4), 5–16 (2020).Assanova A.T., Imanchiyev A.E., Kadirbayeva Z.M. A nonlocal problem for ioaded partial differential equations of fourth order // Bulletin of the Karaganda university – Series Mathematics. – 2020. – Vol. 97. – No. 1. – P. 6–16. https://doi.org/10.31489/2020M1/6-16.Assanova, A.T., Imanchiyev, A.E., and Kadirbayeva, Zh.M. A nonlocal problem for loaded partial differential equations of fourth order. Bulletin of the Karaganda University – Series Mathematics, 97 (1), 6–16 (2020). https://doi.org/10.31489/2020M1/6-16.Абдикаликова Г.А., Асанова А.Т., Шекербекова Ш.Т. О нелокальной задаче для нагруженных гиперболических уравнений четвертого порядка // Известия вузов. Математика. – 2022. – № 8. – С. 3–23.Abdikalikova, G.A., Assanova, A.T., and Shekerbekova, Sh.T. A nonlocal problem for fourth-order loaded hyperbolic equations. Russian Mathematical Journal, 8, 3–23 (2022). (in Russian).Assanova A.T., Kadirbaev Zh.M., Bakirova E.A. On the unique solvability of a nonlocal boundary – value problem for systems of loaded hyperbolic equations with impulsive actions // Ukrainian Mathematical Journal. – 2018. – Vol. 69(8). – P. 1175 –1195. https://doi.org/101007/s11253-017-1424-5.Assanova, A.T., Kadirbayeva, Zh.M., and Bakirova, E.A. On the unique solvability of a nonlocal boundary-value problem for systems of loaded hyperbolic equations with impulsive actions. Ukrainian Mathematical Journal, 69 (8), 1175–1195 (2018). https://doi.org/10.1007/s11253-017-1424-5.Assanova A.T., Kadirbayeva Zh.M. Periodic problem for an impulsive system of the loaded hyperbolic equations // Electronic Journal of Differential Equations. – 2018. – Vol. 72. – P. 1– 8.Assanova, A.T., and Kadirbayeva, Zh.M. Periodic problem for an impulsive system of loaded hyperbolic equations. Electronic Journal of Differential Equations, 72, 1–8 (2018).Assanova A.T., Imanchiyev A.E., Kadirbayeva Zh.M. Numerical solution of systems of loaded ordinary differential equations with multipoint conditions // Computational Mathematics and Mathematical Physics. – 2018. – Vol. 58(4) – P. 508–516. https://doi.org/10.1134/s096554251804005X.Assanova, A.T., Imanchiyev, A.E., and Kadirbayeva, Zh.M. Numerical solution of systems of loaded ordinary differential equations with multipoint conditions. Computational Mathematics and Mathematical Physics, 58 (4), 508–516 (2018). https://doi.org/10.1134/S096554251804005X.Assanova A.T., Kadirbayeva Zh.M. On the numerical algorithms of parametrization method for solving a two – point boundary – value problem for impulsive systems of loaded differential equations // Computational and Applied Mathematics. – 2018. – Vol. 37(4). – P. 4966–4976. https://doi.org/10.1007/s40314-018-0611-9.Assanova, A.T., and Kadirbayeva, Zh.M. On the numerical algorithms of the parametrization method for solving a two-point boundary-value problem for impulsive systems of loaded differential equations. Computational and Applied Mathematics, 37 (4), 4966–4976 (2018). https://doi.org/10.1007/s40314-018-0611-9.

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