kaz29Вестник Казахстанско-Британского технического университетаHerald of the Kazakh-British Technical University1998-66882959-8109Казахстанско-Британский Технический Университет10.55452/1998-6688-2026-23-3-124-143kaz29-3180Research ArticleМАТЕМАТИЧЕСКИЕ НАУКИMATHEMATICAL SCIENCESО РАЗРЕШИМОСТИ ОДНОЙ КРАЕВОЙ ЗАДАЧИ ДЛЯ ПОЛИГАРМОНИЧЕСКОГО УРАВНЕНИЯ В МНОГОМЕРНОМ ШАРЕON THE SOLVABILITY OF ONE BOUNDARY VALUE PROBLEM FOR THE POLYHARMONIC EQUATION IN A MULTIDIMENSIONAL BALLhttps://orcid.org/0000-0002-0784-5183КошановБ. Д.KoshanovB. D.

Д.ф.-м.н., профессор, главный научный сотрудник

Алматы

Doctor of Physical and Mathematical Sciences, Professor, Chief Researcher

Almaty

koshanov@math.kzhttps://orcid.org/0009-0000-5894-1194ТурганбекУ.TurganbekU.

Магистрант

Алматы

Master's student

Almaty

Ulzya03@mail.ruhttps://orcid.org/0009-0001-5552-8908ЕрханА. Ж.YerkhanA. Zh.

Магистрант

Алматы

Master's student

Almaty

ainur.erkhan@icloud.comИнститут математики и математического моделированияРоссияInstitute of Mathematics and Mathematical ModelingRussian FederationКазахский национальный университет имени аль-ФарабиРоссияAl-Farabi Kazakh National UniversityRussian Federation202625092026233124143Copyright © Кошанов Б.Д., Турганбек У., Ерхан А.Ж., 20262026Кошанов Б.Д., Турганбек У., Ерхан А.Ж.Koshanov B.D., Turganbek U., Yerkhan A.Z.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/3180

Необходимость исследования краевых задач для эллиптических уравнений продиктована с многочисленными практическими приложениями при теоретическом изучении процессов гидродинамики, электростатики, механики, теплопроводности, теории упругости, квантовой физики. Например, распределения потенциала электростатического поля описываются с по­мощью уравнения Пуассона, колебания тонких пластин малых прогибов описываются бигармоническими уравнениями. В данной работе исследуется разрешимость одной краевой задачи для полигармонического уравнения в единичном шаре. В этой задаче последовательно задаются нормальные произ­водные искомой функции, начиная с первой нормальной производной до производной по­рядка (m 1), а в качестве последнего условия дополнительно задается производная поряд­ка (m + 1). Для решения данной задачи исходная задача с помощью специального интегродифференциального оператора сводится к задаче Неймана. Затем задача Неймана с помощью интегро-дифференциального оператора сводится к задаче Дирихле. Используется тот факт, что решение задачи Дирихле представляется с помощью функции Грина и формулы Альманси. В результате этих представлений получено необходимое и достаточное условие разрешимости исходной задачи.

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