Geodesic
The problem of vibrations in a harmonic chain with damping and anti-damping on the boundaries
Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 2, pp. 161-177.

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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. 
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A. I. Gudimenko; A. V. Lihosherstov. The problem of vibrations in a harmonic chain with damping and anti-damping on the boundaries. Dalʹnevostočnyj matematičeskij žurnal, Tome 23 (2023) no. 2, pp. 161-177. http://geodesic.mathdoc.fr/item/DVMG_2023_23_2_a2/

[1] A. Maradudin, E. Montroll, and G. Weiss, Theory of lattice dynamics in the harmonic approximation, Academic Press, New York and London, 1963 | MR

[2] Thermal transport in low dimensions, Lecture Notes in Physics, 921, ed. S. Lepri, Springer International Publishing, 2016 | MR

[3] A. M. Krivtsov, “Rasprostranenie tepla v beskonechnom odnomernom garmonichekom kristalle”, Doklady akademii nauk, 464:2 (2015), 162–166 | DOI | MR

[4] O. S. Loboda, E. A. Podolskaya, D. V. Tsvetkov, and A. M. Krivtsov , “On the fundamental solution of the heat transfer problem in one-dimensional harmonic crystals”, Continuum Mech. Thermodyn., 33 (2021), 485–496 | DOI | MR

[5] M. A. Guzev, “Zakon Fure dlya odnomernogo kristalla”, Dalnevost. matem. zhurn., 20:1 (2018), 34–38 | MR

[6] V. V. Migulin, V. I. Medvedev, E. R. Mustel, V. N. Parygin, Osnovy teorii kolebanii, Nauka, M., 1978

[7] E. Schrödinger, “Zur Dynamik elastisch gekoppelter Punktsysteme”, Annalen der Physik, 44 (1914), 916–934 | DOI

[8] F. R. Gantmakher, M. G. Krein, Oscillation matrices and kernels and small vibrations of mechanical systems, AMS Chelsea Publishing, 2002 | MR | Zbl

[9] L. Halpern, “Absorbing Boundary Conditions for the Discretization Schemes of the One-Dimensional Wave Equation”, Mathematics of Computation, 38:158 (1982), 415–429 | DOI | MR | Zbl

[10] I. Alonso-Mallo and A. M. Portillo, “A proof of the well posedness of discretized wave equation with an absorbing boundary condition”, J. Numer. Math., 22:4 (2014), 271–287 | DOI | MR

[11] E. Takizawa and K. Kobayasi, “Heat Flow in a System of Coupled Harmonic Oscillators”, Chinese J. Phys., 1:2 (1963), 59–73

[12] K. Kobayasi and E. Takizawa, “Effect of a Light Isotopic Impurity on the Energy Flow in a System of One-Dimensional Coupled Harmonic Oscillators”, Chinese J. Phys., 2:2 (1964), 68–79

[13] A. I. Gudimenko, “Teplovoi potok v odnomernoi polubeskonechnoi garmonicheskoi reshetke s pogloschayuschei granitsei”, Dalnevost. matem. zhurn., 20:1 (2020), 38–50 | MR

[14] J. L. van Hemmen, Physics Reports, 65:2 (1980), 43–149 | DOI | MR

[15] V. I. Arnold, Matematicheskie metody klassicheskoi mekhaniki, Izd. 6, URSS, 2017 | MR

[16] C. Chicone, Ordinary Differential Equations with Applications, Texts in Applied Mathematics, 34, Springer, 2006 | MR | Zbl

[17] G. Teschl, Ordinary Differential Equations and Dynamical Systems, Graduate Studies in Mathematics, XXX, AMS, Providence, RI, 2011 | MR

[18] M. V. Fedoryuk, Obyknovennye differentsialnye uravneniya, Nauka, M., 1985

[19] W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, NY, 2007 | MR | Zbl

[20] S. Elaydi, An Introduction to Difference Equations, Springer, 2005 | MR | Zbl

[21] G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, 1944 | MR | Zbl