Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end

Authors

  • Viktor I. Korzyuk Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus; Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
  • Jan V. Rudzko Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus; Otkrytye informatsionnye sistemy, 143b Vialiki Hasciniec Street, Maladziečna 222310, Belarus

Keywords:

one-dimensional wave equation, inhomogeneous equation, mixed problem, non-smooth boundary conditions, longitudinal impact, method of characteristics

Abstract

In this article, we study the classical solution of the mixed problem in a quarter of a plane for a one-dimensional wave equation. This mixed problem models the propagation of displacement waves during a longitudinal impact on a bar, when the load remains in contact with the bar and the bar has a linear elastic element at the end. On the lower boundary, the Cauchy conditions are specified, and the second of them has a discontinuity of the first kind at one point. The boundary condition, including the unknown function and its first and second order partial derivatives, is set at the side boundary. The solution is built using the method of characteristics in an explicit analytical form. The uniqueness is proven and the conditions are established under which a piecewise-smooth solution exists. The problem with matching conditions is considered.

Author Biographies

  • Viktor I. Korzyuk, Institute of Mathematics, National Academy of Sciences of Belarus, 11 Surhanava Street, Minsk 220072, Belarus; Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus

    academician of the National Academy of Sciences of Belarus, doctor of science (physics and mathematics), full professor; chief researcher at the department of mathematical physics, Institute of Mathematics, National Academy of Sciences of Belarus; and professor at the department of mathematical cybernetics, faculty of mechanics and mathematics, Belarusian State University

  • Jan V. Rudzko, Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus; Otkrytye informatsionnye sistemy, 143b Vialiki Hasciniec Street, Maladziečna 222310, Belarus

    mathematician-programmer, Otkrytye informatsionnye sistemy

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Published

2022-08-01

Issue

No. 2 (2022): Journal of the Belarusian State University. Mathematics and Informatics

Section

Differential Equations and Optimal Control

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How to Cite

[1]
Korzyuk, V.I. and Rudzko, J.V. 2022. Classical solution of one problem of a perfectly inelastic impact on a long elastic semi-infinite bar with a linear elastic element at the end. Journal of the Belarusian State University. Mathematics and Informatics. 2 (Aug. 2022), 34–46. DOI:https://doi.org/10.33581/2520-6508-2022-2-34-46.