kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-1-231-239kaz29-2517Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESРАЗРЕШИМОСТЬ КРАЕВОЙ ЗАДАЧИ ДЛЯ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙ ДРОБНОГО ПОРЯДКА С ИНВОЛЮЦИЕЙSOLVABILITY OF A BOUNDARY VALUE PROBLEM FOR FRACTIONAL-ORDER INTEGRO-DIFFERENTIAL EQUATIONS WITH INVOLUTIONhttps://orcid.org/0000-0002-4311-5807УсмановК. И.UsmanovK. K.

к.ф.-м.н, доцент

г. Туркестан

Cand. Phys.-Math. Sc., Associate Professor

Turkestan

kairat.usmanov@ayu.edu.kzhttps://orcid.org/0000-0002-2093-1879НазароваК. Ж.NazarovaK. Zh.

к.ф.-м.н, доцент

г. Туркестан

Cand. Phys.-Math. Sc., Associate Professor

Turkestan

kulzina.nazarova@ayu.edu.kzhttps://orcid.org/0009-0005-7507-4535ЕркишеваЖ. С.YerkishevaZh. S.

PhD

г. Туркестан

PhD

Turkestan

zhazira.erkisheva@ayu.edu.kzМеждународный казахско-турецкий университет им. А. ЯсавиКазахстанInternational Kazakh-Turkish University named after H.A. YasawiKazakhstan202629032026231231239Copyright © Усманов К.И., Назарова К.Ж., Еркишева Ж.С., 20262026Усманов К.И., Назарова К.Ж., Еркишева Ж.С.Usmanov K.K., Nazarova K.Z., Yerkisheva 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/2517

В данной работе рассматривается краевая задача для интегро-дифференциального уравнения дробного порядка 0 <α <1 с инволютивным преобразованием. Для исследования разрешимости поставленной краевой задачи используется метод параметризации, предложенный профессором Д. Джумабаевым. Для этого вводится параметр μ = x (0) и выполняется замена переменных 0 <α <1. Введенная замена переменных разбивает рассматриваемую задачу формально на две части, т.е. задачу Коши для интегро-дифференциального уравнения дробного порядка 0 <α <1 с инволютивным преобразованием и линейное уравнение относительно введенного параметра. Определяя решение задачи Коши для интегро-дифференциального уравнения дробного порядка 0 <α <1 с инволютивным преобразованием и подставляя его в краевое условие, получим линейное уравнение относительно введенного параметра. Считая, что коэффициент данного уравнения не равен нулю, находим единственное решение исследуемой краевой задачи. Установлена взаимосвязь между разрешимостью исследуемой задачи и коэффициентом полученного уравнения.

This paper considers a boundary value problem for a fractional-order integro-differential equation 0<α<1 with an involutivity transformation. To investigate the solvability of the boundary value problem, we use the parameterization method proposed by Professor D. Dzhumabaev. This is done by introducing a parameter μ = x(0) and changing variables. x (t ) = u (t ) +μ This change of variables formally divides the problem into two parts: a Cauchy problem for a fractional-order integro-differential equation with an involutive transformation 0 < 𝛼𝛼 < 1 and a linear equation with respect to the introduced parameter. By determining the solution to the Cauchy problem for a fractional-order integro-differential equation with an involutive transformation 0 < 𝛼𝛼 < 1 and substituting it into the boundary condition, we obtain a linear equation with respect to the introduced parameter. Assuming that the coefficient of this equation is nonzero, we find a unique solution to the boundary value problem. A relationship is established between the solvability of the problem and the coefficient of the resulting equation.

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