kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2025-22-2-155-164kaz29-1996Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО НЕКОТОРЫХ СПЕКТРАЛЬНЫХ ЗАДАЧАХ ДЛЯ НЕЛОКАЛЬНОГО АНАЛОГА БИГАРМОНИЧЕСКОГО ОПЕРАТОРАON SOME SPECTRAL PROBLEMS FOR THE NONLOCAL ANALOGUE OF THE BIHARMONIC OPERATORhttps://orcid.org/0000-0002-1377-4633КошановаМ. Д.KoshanovaM. D.

 к.т.н., доцент 

 г. Туркестан 

 Cand. Tech. Sc., Associate Professor 

 Turkistan 

maira.koshanova@ayu.edu.kzМуратбековаМ. А.MuratbekovaM. A.

 PhD, доцент и.о. 

 г. Туркестан 

 PhD, a.d 

 Turkistan 

moldir.muratbekova@ayu.edu.kzhttps://orcid.org/0000-0001-7735-6484ТурметовБ. Х.TurmetovB. H.

 д.ф.-м.н., профессор 

г. Туркестан

г. Ташкент

 d.f.-m.s., professor 

 Turkistan 

 Tashkent 

batirkhan.turmetov@ayu.edu.kzМеждународный казахско-турецкий университет А. ЯсавиКазахстанKhoja Akhmet Yassawi International Kazakh-Turkish UniversityKazakhstanМеждународный казахско-турецкий университет А. Ясави; Университет АльфраганусКазахстанKhoja Akhmet Yassawi International Kazakh-Turkish University; Alfraganus UniversityKazakhstan202506072025222155164Copyright © Кошанова М.Д., Муратбекова М.А., Турметов Б.Х., 20252025Кошанова М.Д., Муратбекова М.А., Турметов Б.Х.Koshanova M.D., Muratbekova M.A., Turmetov B.H.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/1996

Известно, что система собственных функций классического бигармонического оператора с краевым условием Дирихле является полной и ортонормированной в пространстве . Соответствующие этим собственным функциям собственные значения являются положительными, и их можно пронумеровать в порядке возрастания. В некоторых случаях аналогичными свойствами обладают собственные функции и собственные значения краевых задач для возмущенного бигармонического оператора. В данной работе при помощи ортогональных матриц вводится нелокальный аналог возмущенного бигармонического оператора. Для данного оператора исследуются спектральные вопросы двух краевых задач. В первой задаче рассматриваются краевые условия Дирихле, во второй – условия типа Дирихле. При исследовании первой задачи мы используем полноту системы собственных функций задачи Дирихле для возмущенного бигармонического оператора. Используя свойства данных систем, а также свойства отображений с ортогональными матрицами, находим собственные функции и собственные значения основной задачи. Во второй задаче используем собственные функции и собственные значения задачи Дирихле для оператора Лапласа. При использовании явного вида, а также свойств этих систем строятся собственные функции и собственные значения второй задачи. Доказаны теоремы о полноте систем собственных функций, рассматриваемых задач в пространстве L2.

It is known that the system of eigenfunctions of the classical biharmonic operator with the Dirichlet boundary condition is complete and orthonormal in space. The eigenvalues corresponding to these eigenfunctions are positive and can be numbered in ascending order. In some cases, eigenfunctions and eigenvalues of boundary value problems for a perturbed biharmonic operator have similar properties. In this work, a nonlocal analogue of the perturbed biharmonic operator is introduced using orthogonal matrices. Spectral problems of two boundary value problems are studied for this operator. Dirichlet boundary conditions are considered in the first problem, and Dirichlet-type conditions in the second. When studying the first problem, we use the complete system of eigenfunctions of the Dirichlet problem for the perturbed biharmonic operator. Using the properties of these systems, as well as the properties of the image with orthogonal matrices, we find the eigenfunctions and eigenvalues of the main problem. In the second task, we use the eigenfunctions and eigenvalues of the Dirichlet problem for the Laplace operator. Using the explicit form, as well as the properties of these systems, we construct the eigenfunctions and eigenvalues of the second task. The proof of the theorem on the completeness of the system of eigenfunctions of the considered tasks in space is provided.

ортогональная матрицанелокальный бигармонический операторзадача Дирихлезадача типа Дирихлесобственные значениясобственные функцииполнота системыOrthogonal matrixnonlocal biharmonic operatorDirichlet problemDirichlet-type problemeigenvalueseigenfunctionssystem completenessБұл жұмыс Қазақстан Республикасы Ғылым және жоғары білім министрлігінің Ғылым комитетінің гранты аясында (грант № AP19677926) қаржыландырылды.ReferencesKirane M., Sadybekov M.A., Sarsenbi A.A. On an inverse problem of reconstructing a subdiffusion process from nonlocal data // Mathematical Methods in the Applied Sciences. – 2019. – Vol. 42. – No. 6. – P. 2043–2052. https://doi.org/10.1002/mma.5498.Kirane M., Sadybekov M.A., Sarsenbi A.A. On an inverse problem of reconstructing a subdiffusion process from nonlocal data // Mathematical Methods in the Applied Sciences. – 2019. – Vol. 42. – No. 6. – P. 2043–2052. https://doi.org/10.1002/mma.5498.Torebek B.T, Tapdigoglu R. 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The authors declare that there are no conflicts of interest present.