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JOURNALS // Dal'nevostochnyi Matematicheskii Zhurnal // Archive

Dal'nevost. Mat. Zh., 2023 Volume 23, Number 2, Pages 161–177 (Mi dvmg515)

The problem of vibrations in a harmonic chain with damping and anti-damping on the boundaries

A. I. Gudimenkoa, A. V. Lihosherstovb

a Institute for Applied Mathematics, Far Eastern Branch, Russian Academy of Sciences, Vladivostok
b Far Eastern Federal University, Vladivostok

Abstract: The problem of oscillations in a finite homogeneous chain of coupled harmonic oscillators is considered under special boundary conditions that ensure a stable flow of energy from one end of the chain to the other. The problem covers, as special cases, the classical problem of oscillations in a chain with free and fixed ends, as well as the problem of oscillations in a chain with absorbing and anti-absorbing boundaries. Absorbing boundary conditions are used in numerical simulation of wave propagation to minimize the influence of non-physical boundaries.
An exact analytical solution to the considered problem is obtained. The dynamical system of the problem is studied. In particular, a description of the invariant subspaces of the system is given. The oscillatory properties of the system are investigated. The phenomenon of a significant increase in the amplitude of low-frequency oscillations has been discovered and studied.
The problem is solved in Schrödinger variables. The solution is presented both in terms of infinite series of Bessel functions and in terms of natural oscillations (eigenmodes) of the system.

Key words: harmonic chain, absorbing boundary conditions, exactly solvable dynamics, Schrödinger variables, heat flow.

UDC: 517.933,517.938

MSC: Primary 39A21; Secondary 39A27

Received: 09.09.2022
Accepted: 14.11.2023

DOI: 10.47910/FEMJ202314



© Steklov Math. Inst. of RAS, 2024