kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-3-210-220kaz29-2117Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESРАЗРЕШИМОСТЬ МНОГОТОЧЕЧНОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ НАГРУЖЕННЫХ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ПАРАМЕТРОМSOLVABILITY OF A MULTIPOINT BOUNDARY VALUE PROBLEM FOR LOADED DIFFERENTIAL EQUATIONS WITH A PARAMETERhttps://orcid.org/0000-0002-3341-4539ТемешеваС. М.ТemeshevaS. M.

д.ф.-м.н., ведущий научный сотрудник, и.о. профессора

г. Алматы

Dr.Phys.-Math.Sc., Leading Researcher, Acting Professor

Almaty

temeshevasvetlana@gmail.comhttps://orcid.org/0000-0001-8861-4100КадирбаеваЖ. М.КadirbayevaZh. M.

к.ф.-м.н., ведущий научный сотрудник, профессор

г. Алматы

Cand.Phys.-Math.Sc., Leading Researcher, Professor

Almaty

zhkadirbayeva@gmail.comhttps://orcid.org/0009-0008-6654-7531АшировС. Б.AshirovS. B.

магистрант

г. Алматы

Master's student 

Almaty

ashirov_27@mail.ruИнститут математики и математического моделирования; Казахский национальный университет им. аль-Фараби,КазахстанInstitute of Mathematics and Mathematical Modeling; Al-Farabi Kazakh National UniversityKazakhstanИнститут математики и математического моделирования; Казахский национальный женский педагогический университетКазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women's Teacher Training UniversityKazakhstanИнститут математики и математического моделирования; Казахский национальный университет им. аль-ФарабиКазахстанInstitute of Mathematics and Mathematical Modeling; Al-Farabi Kazakh National UniversityKazakhstan202527092025223210220Copyright © Темешева С.М., Кадирбаева Ж.М., Аширов С.Б., 20252025Темешева С.М., Кадирбаева Ж.М., Аширов С.Б.Тemesheva S.M., Кadirbayeva Z.M., Ashirov S.B.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/2117

Данная статья посвящена вопросам разрешимости многоточечной краевой задачи для системы нагруженных дифференциальных уравнений с параметром. Исследование проводится методом параметризации Джумабаева. Это позволяет свести решение исходной краевой задачи к решению системы алгебраических уравнений и задач Коши для обыкновенных дифференциальных уравнений. В алгоритмы метода параметризации внесены изменения, которые позволяют ослабить влияние краевых условий на сходимость алгоритма. Следуя специфике метода параметризации, были получены достаточные условия существования и единственности решения многоточечной краевой задачи для системы нагруженных дифференциальных уравнений с параметром, которые также обеспечивают сходимость измененных алгоритмов метода. Результаты исследования представляют собой конструктивный инструмент анализа поведения различных моделей, которые описываются краевыми задачами для нагруженных дифференциальных уравнений с параметрами. Приведенный пример наглядно показывает реализуемость и эффективность предложенного метода.

This article is devoted to the solvability questions of a multipoint boundary value problem for a system of loaded differential equations with a parameter. The investigation is carried out using the Dzhumabaev parameterization method. This allows us to reduce the solving of the original boundary value problem to the solving of a system of algebraic equations and Cauchy problems for ordinary differential equations. Modifications have been introduced into the parametrization method algorithms to reduce the influence of boundary conditions on the convergence of the algorithm. Following the specifics of the parametrization method, sufficient conditions for the existence and uniqueness of a solution to the multipoint boundary value problem for systems of loaded differential equations with a parameter have been established. These conditions also ensure the convergence of the modified algorithms. The research results provide a constructive tool for analyzing the behavior of various models described by boundary value problems for loaded differential equations with parameters. The presented example clearly demonstrates the feasibility and effectiveness of the proposed method.

многоточечная краевая задачанагружениепараметрметод параметризацииразрешимостьmultipoint boundary value problemloadingparameterparameterization methodsolvabilityБұл зерттеуді Қазақстан Республикасының Ғылым және жоғары білім министрлігінің Ғылым комитеті (грант № AP23488811) қаржыландырады.ReferencesGeng F., Cui M. Multi-point boundary value problem for optimal bridge design // Int. J. Comput. Math. – 2010. – Vol. 87. – P. 1051–1056. https://doi.org/10.1080/00207160903023573.Geng, F. and Cui, M. Int. J. Comput. Math., 87, 1051–1056 (2010). https://doi.org/10.1080/00207160903023573.Hajji M.A. Multi-point special boundary-value problems and applications to fluid flow through porous media // Proc. of the international multi-conference of engineers and computer scientists (18–20 March). – Hong Kong, 2009.Hajji, M.A. Proc. of the international multi-conference of engineers and computer scientists (Hong Kong, 18–20 March 2009).Agarwal R.P. 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The authors declare that there are no conflicts of interest present.