vestifmИзвестия Национальной академии наук Беларуси. Серия физико-математических наукProceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series1561-24302524-2415The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-2430-2026-62-3-193-210vestifm-915Research ArticleМАТЕМАТИКАMATHEMATICSРешение смешанной задачи для волнового уравнения с динамическим граничным условием и разрывными начальными данными в первом квадранте плоскостиSolution of a mixed problem for the wave equation with a dynamic boundary condition and discontinuous initial data in the first quadrant of the planeКорзюкВ. И.KorzyukV. I.

Корзюк Виктор Иванович – академик Национальной академии наук Беларуси, доктор физико-математических наук, профессор

ул. Сурганова, 11, 220072, Минск

пр. Независимости, 4, 220030, Минск

Viktor I. Korzyuk – Academician of the National Academy of Sciences of Belarus, Dr. Sc. (Physics and Mathematics), Professor

11, Surganov Str., 220072, Minsk

4, Nezavisimosti Ave., 220030, Minsk

korzyuk@bsu.byhttps://orcid.org/0000-0002-1482-9106РудькоЯ.RudzkoJ. V.

Рудько Ян Вячеславович – магистр (математика и компьютерные науки), младший научный сотрудник

ул. Сурганова, 11, 220072, Минск

пр. Независимости, 4, 220030, Минск

Jan V. Rudzko – Master of Mathematics and Computer Sciences, Junior Researcher

11, Surganov Str., 220072, Minsk

4, Nezavisimosti Ave., 220030, Minsk

janycz@yahoo.comИнститут математики Национальной академии наук Беларуси; Белорусский государственный университетБеларусьInstitute of Mathematics of the National Academy of Sciences of Belarus; Belarusian State UniversityBelarus202629092026623193210Copyright © Корзюк В.И., Рудько Я.V., 20262026Корзюк В.И., Рудько Я.Korzyuk V.I., Rudzko J.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://vestifm.belnauka.by/jour/article/view/915

Рассматривается смешанная задача для одномерного волнового уравнения в первом квадранте плоскости с разрывными начальными и граничными условиями, возникающими в теории продольного удара по упругому полубесконечному стержню с упругим закреплением на левом конце. Предлагается новый подход к решению задачи, строящий решение как предел классических решений. Предложенный метод не использует никаких дополнительных условий, которые требовались в предыдущих методах, обеспечивая более прямое и строгое построение. Решение отыскивается в явной аналитической форме как кусочно-непрерывная функция. Для рассматриваемой задачи доказывается единственность решения и устанавливаются необходимые и достаточные условия согласования, при которых решение существует.

We consider a mixed problem for the one-dimensional wave equation in the first quarter of the plane with discontinuous initial and boundary conditions arising from the theory of longitudinal impact on an elastic semi-infinite rod with elastic fastening at the left end. We propose a new approach to solve the problem, which constructs the solution as a limit of classical solutions. The proposed method does not use any additional conditions that were required in previous methods providing a more direct and rigorous construction. We find the solution in an explicit analytical form as a piecewise continuous function. For the problem under consideration, the uniqueness of the solution is proved and the necessary and sufficient matching conditions are established under which the solution exists. 

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