Comptes Rendus
no. 9
On the vibrations of a string with a concentrated mass and rapidly oscillating density
Timur R. Gadyl'shin1
1Ufa State Aviation Technical University, Karl Marx st., 12, Ufa 450000, Russia
Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 476-481

The paper is devoted to the vibrations of a string I with a concentrated mass ε−1ϰQ(ε−1x) and rapidly oscillating density q(x,μ−1x), where q(x,ζ) is a 1-periodic in ζ function, Q(ξ) is a function with compact support, the integral of which is equal to one, 0∈I, μ,ε are small positive parameters, ϰ∈R. By combining homogenization and the method of matched asymptotic expansions, we construct solutions to the problems up to O(ε+μ).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crme.2015.08.001
Keywords: Vibration of a string, Rapidly oscillating density, Concentrated mass, Homogenization, Method of matched asymptotic expansions

Timur R. Gadyl'shin  1

1 Ufa State Aviation Technical University, Karl Marx st., 12, Ufa 450000, Russia 
Timur R. Gadyl'shin. On the vibrations of a string with a concentrated mass and rapidly oscillating density. Comptes Rendus. Mécanique, Volume 343 (2015) no. 9, pp. 476-481. doi: 10.1016/j.crme.2015.08.001
@article{CRMECA_2015__343_9_476_0,
     author = {Timur R. Gadyl'shin},
     title = {On the vibrations of a string with a concentrated mass and rapidly oscillating density},
     journal = {Comptes Rendus. M\'ecanique},
     pages = {476--481},
     year = {2015},
     publisher = {Elsevier},
     volume = {343},
     number = {9},
     doi = {10.1016/j.crme.2015.08.001},
     language = {en},
}
TY  - JOUR
AU  - Timur R. Gadyl'shin
TI  - On the vibrations of a string with a concentrated mass and rapidly oscillating density
JO  - Comptes Rendus. Mécanique
PY  - 2015
SP  - 476
EP  - 481
VL  - 343
IS  - 9
PB  - Elsevier
DO  - 10.1016/j.crme.2015.08.001
LA  - en
ID  - CRMECA_2015__343_9_476_0
ER  - 
%0 Journal Article
%A Timur R. Gadyl'shin
%T On the vibrations of a string with a concentrated mass and rapidly oscillating density
%J Comptes Rendus. Mécanique
%D 2015
%P 476-481
%V 343
%N 9
%I Elsevier
%R 10.1016/j.crme.2015.08.001
%G en
%F CRMECA_2015__343_9_476_0

[1] N.S. Bakhvalov; G.P. Panasenko Homogenization: Averaging Processes in Periodic Media, Kluwer, London, 1989

[2] O.A. Oleinik; A.S. Shamaev; G.A. Yosifian Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam, 1992

[3] G.A. Chechkin; A.L. Piatnitski; A.S. Shamaev Homogenization: Methods and Applications, Amer. Math. Soc., Providence, 2007

[4] O.A. Oleinik Homogenization problems in elasticity: spectrum of singularly perturbed operators, Non-Classical Continuum Mechanics, Lecture Note Series, vol. 122, 1987, pp. 188-205

[5] Yu.D. Golovaty; S.A. Nazarov; O.A. Oleinik; T.S. Soboleva Eigenoscillations of a string with an additional mass, Sib. Math. J., Volume 29 (1988) no. 5, pp. 744-760

[6] V. Maz'ya; S. Nazarov; B. Plamenevsky Asymptotic Theory of Elliptic Boundary Value Problems in Singularly Perturbed Domains, vols. 1, 2, Birkhäuser Verlag, Basel, Switzeland, 2000

[7] E. Sánchez-Palencia Perturbation of eigenvalues in thermo-elasticity and vibration of systems with concentrated masses, Trends and Applications of Pure Mathematics to Mechanics, Lecture Notes in Physics, Springer-Verlag, 1984, pp. 346-368

[8] S. Albeverio; F. Gesztezy; R. Høegh-Krohn; H. Holden On point interactions in one dimension, J. Oper. Theory, Volume 12 (1984), pp. 101-126

[9] A.M. Il'in Matching of Asymptotic Expansions of Solutions of Boundary-Value Problems, American Mathematical Society, Providence, RI, USA, 1992

[10] V.P. Mikhailov Partial Differential Equations, Mir Publishers, Moscow, 1978

[11] T. Kato Perturbation Theory for Linear Operators, Springer-Verlag, Heidelberg, Germany, 1966

[12] R.R. Gadyl'shin; I.Kh. Khusnullin Perturbation of the Shrödinger operator by a narrow potential, Ufa Math. J., Volume 3 (2011) no. 3, pp. 54-64 (translated from Ufimsk. Mat. Zh., 3, 3, 2011, pp. 55-66)

Cited by Sources:

Comments - Policy