A particular solution of a problem for a system of equations from mechanics with nonsmooth Cauchy condition-Reference-Cited by-同舟云学术

A particular solution of a problem for a system of equations from mechanics with nonsmooth Cauchy condition

Published:2022-10-12 Issue:3 Volume:58 Page:300-311
ISSN:2524-2415
Container-title:Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series
language:
Short-container-title:Vescì Akademìì navuk Belarusì. Seryâ fizika-matematyčnyh navuk

Author:

Korzyuk V. I.¹,Rudzko J. V.²^ORCID

Affiliation:

1. Belarusian State University;Institute of Mathematics of the National Academy of Sciences of Belarus

2. Belarusian State University

Abstract

In this article, we study a mixed problem in a quarter-plane for a system of differential equations, which describes vibrations in a string from viscoelastic material, which corresponds to Maxwell material. At the bottom of the boundary, the Cauchy conditions are specified, and one of them has a discontinuity of the first kind at one point. A smooth boundary condition is set at the side boundary. The Klein – Gordon – Fock equation is derived for one of the system’s functions. We find a particular solution in two ways. The first method builds it in an explicit analytical form (with a continuation of one function), and the second one constructs it as a solution of an integral equation using the method of characteristics (without continuation of one function). Conditions are established under which the solution has sufficient smoothness.

Publisher

Publishing House Belorusskaya Nauka

Subject

Computational Theory and Mathematics,General Physics and Astronomy,General Mathematics

Reference30 articles.

1. Lazaryan V. A. On dynamic forces in harness devices of homogeneous trains with resistance to relative movements of carriages. Trudy Dnepropetrovskogo instituta inzhenerov zheleznodorozhnogo transporta = Proceedings of the Dnepropetrovsk Institute of Railway Engineers, 1950, no. 20, pp. 3–32 (in Russian).

2. Mavrin A. I. To the theory of shock piling. Izvestiya vuzov (stroitel’stvo i arkhitektura) = News of Higher Educational Institutions (Building and Architecture), 1967, no. 8, pp. 24–28 (in Russian).

3. Boussinesq J. Du choc longitudinal d’une barre élastique prismatique fixée à un bout et heurtée à l’autre. Comptes Rendus, 1883, vol. 97, no. 2, pp. 154–157 (in French).

4. Gayduk S. I. On some problems related to the theory of transverse impact on rods. Differentsial’nye uravneniya = Differential Equations, 1977, vol. 13, no. 7, pp. 1233–1243 (in Russian).

5. Gayduk S. I., Zayats G. M. On the uniqueness of the solution of one problem from the wave theory of mechanical shock. Differentsial’nye uravneniya = Differential Equations, 1989, vol. 25, no. 5, pp. 833–839 (in Russian).

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1. Problem of Longitudinal Vibrations of a Viscoelastic Rod of Maxwell Type;Прикладная математика и механика;2023-05-01

2. Classical solution of a problem for a system of equations from mechanics with nonsmooth Cauchy conditions;Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics Series;2023-04-03