References

vestifm

Известия Национальной академии наук Беларуси. Серия физико-математических наук

Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series

1561-24302524-2415

The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"

10.29235/1561-2430-2024-60-2-95-105

vestifm-777

Research Article

МАТЕМАТИКА

MATHEMATICS

Классическое решение смешанных задач из теории продольного удара по упругому полубесконечному стержню в случае отделения ударившего тела после удара

Classical solution to mixed problems from the theory of longitudinal impact on an elastic semi-infinite rod in the case of separation of the impacting body after the collision

Корзюк

В. И.

Korzyuk

V. 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.by

https://orcid.org/0000-0002-1482-9106

Рудько

Я. В.

Rudzko

J. V.

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

ул. Сурганова, 11, 220072, Минск

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

11, Surganov Str., 220072, Minsk

janycz@yahoo.com

Институт математики Национальной академии наук Беларуси; Белорусский государственный университетInstitute of Mathematics of the National Academy of Sciences of Belarus; Belarusian State University

Институт математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus

2024

09072024

60295105

2024

Корзюк В.И., Рудько Я.В.

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/777

Рассматриваются две связанные начально-краевые задачи, которые моделируют процесс продольного удара в полубесконечном стержне на основе теории Сен-Венана. Математическая постановка задачи представляет собой две смешанные задачи для одномерного волнового уравнения с условиями сопряжения. Условия Коши задаются на пространственной полупрямой. Начальное условие для частной производной по временной переменной имеет разрыв первого рода в одной точке. На временной полупрямой задается граничное условие, содержащее неизвестную функцию и ее частные производные первого и второго порядка. Решение строится методом характеристик в явном аналитическом виде. Доказана единственность и установлены условия существования кусочно-гладкого решения. Рассмотрено классическое решение смешанной задачи с условиями сопряжения.

In this work, we consider two coupled initial-boundary value problems, which, based on the Saint-Venant theory, model the longitudinal impact phenomena in a semi-infinite rod. The mathematical formulation of the problem is two mixed problems for the one-dimensional wave equation with conjugation conditions. The Cauchy conditions are specified on the spatial half-line. The initial condition for the partial derivative with respect to the time variable has a discontinuity of the first kind at one point. The boundary condition, which includes the unknown function and its first- and second-order partial derivatives, is specified on the time half-line. The solution is constructed by the method of characteristics in an explicit analytical form. The uniqueness of the solution is proved, and the conditions under which a piecewise-smooth solution exists are established. The classical solution to a mixed problem with matching conditions is considered.

продольный ударволновое уравнениесмешанная задачаклассическое решениеметод характеристикразрывные начальные условияразрывные граничные условияусловия согласованияусловия сопряжения

longitudinal impactwave equationmixed problemclassical solutionmethod of characteristicsdiscontinuous initial conditionsdiscontinuous boundary conditionsmatching conditionsconjugation conditions

Статья опубликована при финансовой поддержке Министерства науки и высшего образования Российской Федерации в рамках реализации программы Московского центра фундаментальной и прикладной математики по соглашению № 075-15-2022-284.

The article was financially supported by the Ministry of Science and Higher Education of the Russian Federation in the framework of implementing the program of the Moscow Center for Fundamental and Applied Mathematics by Agreement no. 075-15-2022-284.

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