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Mathematics and Mechanics

Russian Academy of Sciences
in January 1936
(Translated from 1958)
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IssuesArchive of Issues2015-5pp.506-513

Archive of Issues

Since 2018, translations of PMM's articles have been published in special issues of the journals Mechanics of Solids and Fluid Dynamics:

Total articles in the database: 1923
In Russian (): 1124
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Yu.A. Chirkunov, "Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model," J. Appl. Math. Mech. 79 (5), 506-513 (2015)
Year 2015 Volume 79 Issue 5 Pages 506-513
DOI 10.1016/j.jappmathmech.2016.03.012
Title Non-linear longitudinal oscillations of a viscoelastic rod in Kelvin's model
Author(s) Yu.A. Chirkunov (Novosibirsk State Technical University, Novosibirsk, Russia,
Abstract Non-linear longitudinal oscillations of a viscoelastic rod are investigated in Kelvin's model. All the second-order conservation laws for the differential equation describing these oscillations see obtained, which, using non-local variables, generate two non-linear systems of differential equations, equivalent to this equation. A group analysis of these systems is carried out. All their essentially different invariant solutions (unconnected with point transformations) are obtained, which are either found in explicit form or the search for them is reduced to solving non-linear integro-differential equations, which opens up new possibilities for analytical and numerical investigations. The presence of arbitrary constants in these equations enables them to be used to investigate various boundary value problems. With additional conditions, the existence and uniqueness of the solutions of certain boundary value problems, describing non-linear longitudinal oscillations of a visco-elastic rod in Kelvin's model, are established.
Received 10 November 2014
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