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 Mat. Sb., 1992, Volume 183, Number 3, Pages 38–54 (Mi msb1040)

On the spectrum of the discrete inhomogeneous wave equation, and vibrations of a discrete string

L. D. Pustyl'nikov

Abstract: Explicit analytic expressions are found for the spectrum and solutions of the discrete, inhomogeneous wave equation
$${d^2 q_n \over d t^2}-a_n(q_{n+1}-2q_n+q_{n-1})+\delta_n q_n=0$$
with boundary conditions $q_0(t) = q_N(t) = 0$, where $n=0, 1, …, N$, $a_n>0$, and $\delta_n \geqslant 0$. As a corollary a solution is given of the classical problem of finding an explicit analytic expression describing the vibrations of a string all the mass of which is concentrated at a finite number of equidistant points, which was the object of detailed study by Euler, D'Alembert, D. Bernoulli, Lagrange, Sturm, Routh, and others, who gave a solution of it in the particular case where the masses of all points are the same. The general solution of the problem turns out to be connected with a generalized quaternion algebra and properties of certain of its ideals, and this connection is used in an essential way in the proofs of the theorems.

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English version:
Russian Academy of Sciences. Sbornik. Mathematics, 1993, 75:2, 317–331

Bibliographic databases:

UDC: 517.927.25+534.11
MSC: Primary 34B10, 34L05; Secondary 35L05

Citation: L. D. Pustyl'nikov, “On the spectrum of the discrete inhomogeneous wave equation, and vibrations of a discrete string”, Mat. Sb., 183:3 (1992), 38–54; Russian Acad. Sci. Sb. Math., 75:2 (1993), 317–331

Citation in format AMSBIB
\Bibitem{Pus92} \by L.~D.~Pustyl'nikov \paper On the spectrum of the~discrete inhomogeneous wave equation, and vibrations of a~discrete string \jour Mat. Sb. \yr 1992 \vol 183 \issue 3 \pages 38--54 \mathnet{http://mi.mathnet.ru/msb1040} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1180917} \zmath{https://zbmath.org/?q=an:0782.34085|0770.34055} \adsnasa{http://adsabs.harvard.edu/cgi-bin/bib_query?1993SbMat..75..317P} \transl \jour Russian Acad. Sci. Sb. Math. \yr 1993 \vol 75 \issue 2 \pages 317--331 \crossref{https://doi.org/10.1070/SM1993v075n02ABEH003387} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=A1993LT65700002} 

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This publication is cited in the following articles:
1. Kh. P. Kulterbaev, A. Ya. Dzhankulaev, “Smeshannaya sistema differentsialnykh uravnenii kak matematicheskaya model kolebanii kontinualno-diskretnykh mekhanicheskikh sistem”, Vladikavk. matem. zhurn., 3:4 (2001), 28–35
2. L. D. Pustyl'nikov, “Discrete Wave Equations with Random Parameters and a Discrete String with Random Masses”, Theory Probab. Appl, 47:2 (2003), 257
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