danДоклады Национальной академии наук БеларусиDoklady of the National Academy of Sciences of Belarus1561-83232524-2431The Republican Unitary Enterprise Publishing House "Belaruskaya Navuka"10.29235/1561-8323-2025-69-4-271-278dan-1260Research ArticleМАТЕМАТИКАMATHEMATICSКлассическое решение смешанной задачи в криволинейной полуполосе для волнового уравнения с разрывными начальными условиямиClassical solution of a mixed problem for the wave equation with discontinuous initial conditions in a curvilinear half-stripКорзюкВ. И.KorzyukV. I.

Корзюк Виктор Иванович – академик, д-р физ.-мат. наук, профессор.

ул. Сурганова, 11, 220072, Минск

Korzyuk Viktor I. – Academician, Professor, D. Sc. (Phy- sics and Mathematics).

11, Surganov Str., 220072, Minsk

https://orcid.org/0000-0002-1482-9106РудькоЯ. В.RudzkoJ. V.

Рудько Ян Вячеславович – магистр (математика и компьютерные науки), аспирант.

ул. Сурганова, 11, 220072, Минск

Rudzko Jan V. – Master (Mathematics and Computer Sciences), Postgraduate student.

11, Surganov Str., 220072, Minsk

https://orcid.org/0000-0002-3773-1187КолячкоВ. В.KolyachkoV. V.

Колячко Владислав Владимирович – стажер мл. науч. сотрудника.

ул. Сурганова, 11, 220072, Минск

Kolyachko Vladislav V. – Research Intern.

11, Surganov Str., 220072, Minsk

Институт математики Национальной академии наук Беларуси; Белорусский государственный университетInstitute of Mathematics of the National Academy of Sciences of Belarus; Belarusian State UniversityИнститут математики Национальной академии наук БеларусиInstitute of Mathematics of the National Academy of Sciences of Belarus202528082025694271278Copyright © Корзюк В.И., Рудько Я.В., Колячко В.В., 20252025Корзюк В.И., Рудько Я.В., Колячко В.В.Korzyuk V.I., Rudzko J.V., Kolyachko V.V.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://doklady.belnauka.by/jour/article/view/1260

Для одномерного волнового уравнения рассматривается смешанная задача в криволинейной полуполосе. Начальные условия имеют разрыв первого рода в одной точке. Смешанная задача моделирует задачу о продольном ударе по конечному упругому стержню с подвижной границей. Решение строится методом характеристик в явном аналитическом виде. Для рассматриваемой задачи доказывается единственность решения и устанавливаются условия, при которых существует ее классическое решение. 

For a one-dimensional wave equation, we consider a mixed problem in a curvilinear half-strip. The initial conditions have a first-kind discontinuity at one point. The mixed problem models the problem of a longitudinal impact on a finite elastic rod with a movable boundary. We construct the solution using the method of characteristics in an explicit analytical form. For the problem in question, we prove the uniqueness of the solution and establish the conditions under which its classical solution exists.

волновое уравнениесмешанная задачаметод характеристикклассическое решениеусловия согласованияусловия сопряженияразрывные условиякриволинейная областьwave equationmixed problemmethod of characteristicsclassical solutionmatching conditionsconjugation conditionsdiscontinuous conditionscurvilinear domainИсследования авторов поддержаны Московским центром фундаментальной и прикладной математики МГУ им. М. В. Ломоносова по соглашению № 075-15-2025-345 и Национальной академией наук Беларуси в рамках выполнения НИР «Решение задач с негладкими граничными условиями для гиперболических уравнений» по соглашению № 2024-25-141.The authors’ research is supported by the Moscow Center of Fundamental and Applied Mathematics of Lomonosov Moscow State University under agreement No. 075-15-2025-345 and by the National Aca- demy of Sciences of Belarus in the framework of imple- menting the scientific research program “Solutions of prob- lems with non-smooth boundary conditions for hyperbolic equations” under agreement No. 2024-25-141.ReferencesGaiduk S. I. Some problems related to the theory of longitudinal impact on a rod. Differential Equations, 1977, vol. 12(1976), pp. 607–617.Gaiduk S. I. Some problems related to the theory of longitudinal impact on a rod. Differential Equations, 1977, vol. 12(1976), pp. 607–617.Korzyuk V. I., Rudzko J. V. A mathematical investigation of one problem of the longitudinal impact on an elastic rod with an elastic attachment at the end. Trudy Instituta Matematiki [Proceedings of the Institute of Mathematics], 2023, vol. 31, no. 1, pp. 81–87 (in Russian).Korzyuk V. I., Rudzko J. V. A mathematical investigation of one problem of the longitudinal impact on an elastic rod with an elastic attachment at the end. Trudy Instituta Matematiki [Proceedings of the Institute of Mathematics], 2023, vol. 31, no. 1, pp. 81–87 (in Russian).Koshlyakov N. S., Smirnov M. M., Gliner E. B. Differential Equations of Mathematical Physics. Amsterdam, North-Holland Publishing Co., 1964.Koshlyakov N. S., Smirnov M. M., Gliner E. B. Differential Equations of Mathematical Physics. Amsterdam, North-Holland Publishing Co., 1964.Naumavets S. N. Classical solution of the first mixed problem for the one-dimensional wave equation with a differential polynomial of the second order in the boundary conditions. Zbirnyk statei. Matematyka. Informatsiini tekhnologiї. Osvita [Collection of articles. Mathematics. Information technology. Education], 2018, no. 5, pp. 96–101 (in Russian).Naumavets S. N. Classical solution of the first mixed problem for the one-dimensional wave equation with a differential polynomial of the second order in the boundary conditions. Zbirnyk statei. Matematyka. Informatsiini tekhnologiї. Osvita [Collection of articles. Mathematics. Information technology. Education], 2018, no. 5, pp. 96–101 (in Russian).Korzyuk V. I., Kozlovskaya I. S., Naumavets S. N. Classical solution of the first mixed problem for the wave equation in a curvilinear half-strip. Differential Equations, 2020, vol. 56, pp. 98–108. https://doi.org/10.1134/s0012266120010115Korzyuk V. I., Kozlovskaya I. S., Naumavets S. N. Classical solution of the first mixed problem for the wave equation in a curvilinear half-strip. Differential Equations, 2020, vol. 56, pp. 98–108. https://doi.org/10.1134/s0012266120010115Korzyuk V. I., Stolyarchuk I. I. Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients. Differential Equations, 2017, vol. 53, pp. 74–85. https://doi.org/10.1134/ s0012266117010074Korzyuk V. I., Stolyarchuk I. I. Classical solution of the first mixed problem for second-order hyperbolic equation in curvilinear half-strip with variable coefficients. Differential Equations, 2017, vol. 53, pp. 74–85. https://doi.org/10.1134/ s0012266117010074Korzyuk V. I., Rudzko J. V. Classical solution of the first mixed problem for the telegraph equation with a nonlinear potential in a curvilinear quadrant. Differential Equations, 2023, vol. 59, pp. 1075–1089. https://doi.org/10.1134/s0012266123080062Korzyuk V. I., Rudzko J. V. Classical solution of the first mixed problem for the telegraph equation with a nonlinear potential in a curvilinear quadrant. Differential Equations, 2023, vol. 59, pp. 1075–1089. https://doi.org/10.1134/s0012266123080062Korzyuk V. I. Equations of Mathematical Physics. Moscow, 2021. 480 p. (in Russian).Korzyuk V. I. Equations of Mathematical Physics. Moscow, 2021. 480 p. (in Russian).Korzyuk V. I., Rudzko Ja. V., Kolyachko V. V. Solutions of problems with discontinuous conditions for the wave equation. Journal of the Belarusian State University. Mathematics and Informatics, 2023, vol. 3, pp. 6–18 (in Russian).Korzyuk V. I., Rudzko Ja. V., Kolyachko V. V. Solutions of problems with discontinuous conditions for the wave equation. Journal of the Belarusian State University. Mathematics and Informatics, 2023, vol. 3, pp. 6–18 (in Russian).

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