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
https://doi.org/10.29235/1561-2430-2024-60-2-95-105
Abstract
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.
Keywords
longitudinal impact,
wave equation,
mixed problem,
classical solution,
method of characteristics,
discontinuous initial conditions,
discontinuous boundary conditions,
matching conditions,
conjugation conditions
Supporting agencies: 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.
About the Authors
V. I. Korzyuk
Institute of Mathematics of the National Academy of Sciences of Belarus;
Belarusian State University
Belarus
Belarus
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
J. V. Rudzko
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Jan V. Rudzko – Postgraduate Student, Master of Mathematics and Computer Sciences, Junior Researcher
11, Surganov Str., 220072, Minsk
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