kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2024-21-1-64-74kaz29-1023Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESОБ ОДНОЗНАЧНОЙ РАЗРЕШИМОСТИ КРАЕВОЙ ЗАДАЧИ ДЛЯ ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ С ПАРАМЕТРОМON THE UNIQUE SOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR DIFFERENTIAL EQUATIONS WITH PARAMETERhttps://orcid.org/0000-0002-3820-5373БакироваЭ. С.BakirovaE. A.

к.ф-м.н.

г. Алматы

Can. Phys.-Math.Sc.

Almaty

   

bakirova1974@mail.ruhttps://orcid.org/0000-0002-0680-4099ИскаковаН. Б.IskakovaN. B.

к.ф-м.н.

г. Алматы

Can. Phys.-Math.Sc.

Almaty

   

narkesh@mail.ruhttps://orcid.org/0000-0002-3341-4539ТемешеваС. М.ТemeshevaS. M.

д. ф.-м. н.

г. Алматы

Dr. Phys.-Math.Sc.

Almaty

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

к.ф-м.н.

г. Алматы

Can. Phys.-Math.Sc.

Almaty

zhkadirbayeva@gmail.comИнститут математики и математического моделирования; Казахский национальный женский педагогический университетКазахстанInstitute of Mathematics and Mathematical Modeling; Kazakh National Women's Teacher Training UniversityKazakhstanИнститут математики и математического моделированияКазахстанInstitute of Mathematics and Mathematical ModelingKazakhstanИнститут математики и математического моделирования; Казахский национальный университет им. аль-ФарабиКазахстанInstitute of Mathematics and Mathematical Modeling; Al-Farabi Kazakh National UniversityKazakhstanИнститут математики и математического моделирования; Международный университет информационных технологийКазахстанInstitute of Mathematics and Mathematical Modeling; International Information Technology UniversityKazakhstan2024250320242116474Copyright © Бакирова Э.С., Искакова Н.Б., Темешева С.М., Кадирбаева Ж.М., 20242024Бакирова Э.С., Искакова Н.Б., Темешева С.М., Кадирбаева Ж.М.Bakirova E.A., Iskakova N.B., Тemesheva S.M., Каdirbayeva Z.М.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/1023

На конечном интервале методом параметризации исследуется линейная краевая задача для дифференциального уравнения с параметром. Путем разбиения интервала, введения дополнительных параметров в точках разбиения и новых функций исследуемая краевая задача с параметром сводится к эквивалентной многоточечной краевой задаче с параметрами. Полученная эквивалентная краевая задача содержит задачи Коши для обыкновенных дифференциальных уравнений относительно новых функций. С помощью подстановки представления решения задачи Коши в краевые условия и условия непрерывности решения составляется система линейных алгебраических уравнений относительно введенных параметров. Построен алгоритм нахождения решения краевой задачи с параметрами. Приведена формулировка теоремы о достаточных условиях однозначной разрешимости  краевой задачи с параметрами. В терминах данных исходной краевой задачи получены достаточные условия ее однозначной разрешимости. Приводится пример, показывающий выполнение условий теорем.

A linear boundary value problem for a differential equation with a parameter is investigated on a finite interval by the parameterization method. The studied boundary value problem with parameter is reduced to an equivalent multipoint boundary value problem with parameters by splitting the interval, introducing additional parameters at the points of splitting and new functions. The obtained equivalent boundary value problem contains Cauchy problems for ordinary differential equations with respect to new functions. By substituting the solution representation of the Cauchy problem into the boundary conditions and continuity conditions of the solution, a system of linear algebraic equations with respect to the introduced parameters is compiled. An algorithm for finding a solution to the boundary value problem with parameters is constructed. The formulation of the theorem on sufficient conditions of unique solvability of the boundary value problem with parameters is given. Sufficient conditions of its unique solvability are obtained in terms of the data of the original boundary value problem. An example showing the fulfillment of the conditions of the theorem is given.

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