kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-2-35-45kaz29-2876Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESИССЛЕДОВАНИЕ МНОГОПЕРИОДИЧЕСКОЙ ЗАДАЧИ ДЛЯ КОНЕЧНО-ЭРЕДИТАРНОЙ СИСТЕМЫ ИНТЕГРО-ДИФФЕРЕНЦИАЛЬНЫХ УРАВНЕНИЙINVESTIGATION OF A MULTIPERIODIC PROBLEM FOR A FINITEHEREDITARY SYSTEM OF INTEGRO-DIFFERENTIAL EQUATIONShttps://orcid.org/0000-0001-6280-4168AбдикаликоваГ. А.AbdikalikovaG. A.

К.ф.-м.н., профессор.

Актобе

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

abdikalikova.galliya@gmail.comhttps://orcid.org/0000-0003-2601-2678СартабановЖ. А.SartabanovZh. A.

Д.ф.-м.н., профессор.

Актобе

Dr. Phys.-Math. Sc., Professor.

Aktobe

sartabanov42@mail.ruhttps://orcid.org/0000-0002-4572-8252AйтеноваГ. M.AitenovaG. M.

PhD, ассоциированный профессор.

Уральск

PhD, Associate Professor (Docent).

Uralsk

gulsezimaitenova@gmail.comhttps://orcid.org/0000-0002-6007-3311ЖумагазиевА. Х.ZhumagaziyevA. Kh.

PhD.

Актобе

PhD.

Aktobe

azhumagaziyev@zhubanov.edu.kzАктюбинский региональный университет им. К. ЖубановаКазахстанK. Zhubanov Aktobe Regional UniversityKazakhstanЗападно-Казахстанский университет им. М. УтемисоваКазахстанM. Utemisov West Kazakhstan UniversityKazakhstanАктюбинский региональный университет им. К. ЖубановаКазахстанK. Zhubanov Aktobe Regional University; Kh. Dosmukhamedov Atyrau UniversityKazakhstan2026270620262323545Copyright © Aбдикаликова Г.А., Сартабанов Ж.А., Aйтенова Г.M., Жумагазиев А.Х., 20262026Aбдикаликова Г.А., Сартабанов Ж.А., Aйтенова Г.M., Жумагазиев А.Х.Abdikalikova G.A., Sartabanov Z.A., Aitenova G.M., Zhumagaziyev A.K.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/2876

К проблеме необходимости исследования интегро-дифференциальных уравнений приводят многочисленные задачи приложения ряда областей естествознания и техники. Исследование интегро-дифференциальных уравнений, описывающие предысторию процесса или явления, начались с работ В. Вольтерра с учетом влияния интегрального члена. Известны различные подходы учета «наследственности», «последействия». Отметим, что установление существования многопериодических и квазипериодических решений системы интегро-дифференциальных уравнений типа Вольтерра не всегда однозначно разрешимо. Рассматривается система интегро-дифференциальных уравнений в частных производных со специальным оператором дифференцирования. Используя метод периодических характеристик, построены интегральные представления многообразия решений таких систем с последействием. Установлены некоторые свойства итерированных ядер и резольвенты, а также получены оценки. Найдены условия существования матричной функции типа Грина многопериодической задачи и интегральное представление с соответствующей оценкой. Получены достаточные условия существования и единственности многопериодического решения, которые являются интегральным многообразием интегро-дифференциальной системы с конечной эредитарностью.

Numerous problems arising in various fields of natural sciences and engineering lead to the study of integrodifferential equations. The investigation of such equations, which account for the historical behavior of processes or phenomena, originates from the pioneering works of V. Volterra, where the role of the integral term was emphasized. Various approaches have since been developed to describe hereditary effects and aftereffects in these equations. It should be noted that the existence of multiperiodic and quasiperiodic solutions for systems of Volterra-type integrodifferential equations does not always guarantee their uniqueness. In this paper, a system of partial integro-differential equations with a special differentiation operator is considered. Using the method of periodic characteristics, integral representations of the solution manifold for such systems with aftereffects are constructed. The properties of iterated kernels and resolvents are studied, and corresponding estimates are obtained. Conditions for the existence of a Greentype matrix function for the multiperiodic problem are established, along with integral representations with suitable bounds. Finally, sufficient conditions for the existence and uniqueness of a multiperiodic solution, interpreted as an integral manifold of the integro-differential system with finite hereditary effects, are derived.

матрицантядроразрешающий оператордихотомичностьфункция типа Гринаmatricantkernelresolving operatordichotomyGreen type functionThis research is funded by the Science Committee of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP19676629)ReferencesVolterra, V. Theory of functionals, integral and integro-differential equations (Moscow: Dover Publications, 2005), 226.Volterra, V. Theory of functionals, integral and integro-differential equations (Moscow: Dover Publications, 2005), 226.De Bonis, M.C., Occorsio, D. Quadrature methods for integro-differential equations of Prandtl’s type in weighted spaces of continuous functions. Applied Mathematics and Computation, 393, 125721 (2021). https://doi.org/10.1016/j.amc.2020.125721De Bonis, M.C., Occorsio, D. Quadrature methods for integro-differential equations of Prandtl’s type in weighted spaces of continuous functions. 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The authors declare that there are no conflicts of interest present.