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On small vibrations of a damped Stieltjes string

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Inverse problem of recovering masses, coefficients of damping and lengths of the intervals between the masses using two spectra of boundary value problems and the total length of the Stieltjes string (an elastic thread bearing point masses) is considered. For the case of point-wise damping at the first counting from the right end mass the problem of recovering the masses, the damping coefficient and the lengths of the subintervals by one spectrum and the total length of the string is solved.
Rocznik
Strony
143--159
Opis fizyczny
Bibliogr. 31 poz.
Twórcy
autor
  • South-Ukrainian National Pedagogical University Staroportofrankovskaya Str. 26 65020, Odessa, Ukraine
  • South-Ukrainian National Pedagogical University Staroportofrankovskaya Str. 26 65020, Odessa, Ukraine
Bibliografia
  • [1] T. Aktosun, M. Klaus, C. van der Mee, Wave scattering in one dimension with absorp­tion, J. Math. Phys. 39 (1998) 4, 1957-1992.
  • [2] D.Z. Arov, Realization of a canonical system with a dissipative boundary condition at one end of the segment in terms of the coefficient of dynamical compliance, Sibirsk. Math. Z. 16 (1975), 440-463 [in Russian]; English trans.: Sibirian Math. J. 16 (1975), 335-352.
  • [3] O. Boyko, V. Pivovarchik, Inverse problem for Stieltjes string damped at one end, Methods Funct. Anal. Topology 14 (2008) 1, 10-19.
  • [4] O. Boyko, V. Pivovarchik, The inverse three-spectral problem for a Stieltjes string and the inverse problem with one-dimensional damping, Inverse Problems 24 (2008) 1, 015 019, 13 pp.
  • [5] R. Courant, D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience, New York, 1953.
  • [6] S. Cox, M. Enibree, J. Hokanson, One can hear the composition of a string: experiments with an inverse eigenvalue problem, SIAM Rev. 54 (2012) 1, 157-178.
  • [7] S. Cox, E. Zuazua, The rate at which energy decays in a string damped at one end, Indiana Univ. Math. J. 44 (1995) 2, 545-573.
  • [8] R. Duffing, A Minimax theory for overdamped networks, Arch. Rat. Mech. Anal. 4 (1955), 221-233.
  • [9] F.R. Gantmakher, M.G. Krein, Oscillating Matrices and Kernels and Vibrations of Mechanical Systems, GITTL, Moscow-Leningrad, 1950 [in Russian]; German transl. Akademie Verlag, Berlin, 1960; Revised edition, AMS Chelsea Publishing, Providence, RI, 2002.
  • [10] G. Gubreev, V. Pivovarchik, Spectral analysis of T. Regge problem with parameters, Funktsional. Anal, i Prilozhen. 31 (1997) 1, 70-74 [in Russian]; English transl.: Funct. Anal. Appl. 31 (1997) 1, 54-57.
  • [11] C. Hermite, Extract d'une lettre de M.Ch. Hermite de Paris a' Mr. Borchardt de Berlin sur le nombre des racines d'une e'quation alge'brique comprises entre des limites donne'es, J. Reine Angew. Math. 52 (1856), 39-51; reprinted in his Oeuvres, Vol. 1, Gauthier-Villars, Paris, 1905, 397-414.
  • [12] M. Jaulent, Inverse scattering problems in absorbing media, J. Math. Phys. 17 (1976) 7, 1351-1360.
  • [13] I.S. Kac, M.G. Krein, On spectral functions of a string, [in:] F.V. Atkinson, Discrete and Continuous Boundary Problems (Russian translation), Moscow, Mir, 1968, 648-737 (Addition II); I.C. Kac, M.G. Krein, On the spectral function of the string, Amer. Math. Soc, Translations, Ser. 2, 103 (1974), 19-102.
  • [14] M.G. Krein, On some new problems of the theory of vibrations of Sturm systems, Prikladnaya Matematika i Mekhanika 16 (1952) 5, 555-568 [in Russian].
  • [15] M.G. Krein, A. A. Nudelman, On direct and inverse problems for frequencies of boundary dissipation of inhomogeneous string, Doklady AN SSSR 247 (1979) 5, 1046-1049 [in Russian].
  • [16] M.G. Krein, A.A. Nudelman, On some spectral properties of an inhomogeneous string with dissipative boundary condition, J. Operator Theory 22 (1989), 369-395 [in Russian].
  • [17] P. Lancaster, J. Maroulas, Inverse eigenvalue problems for damped vibrating Systems, J. Math. Anal. Appl. 123 (1987), 238-261.
  • [18] C.K. Law, V. Pivovarchik, W.C. Wang, A polynomial identity and its application to inverse problems in Stieltjes strings, Oper. Matrices 7 (2013) 3, 603-617.
  • [19] B.Ja. Levin, Distribution of Zeros of Entire Functions, Transl. Math. Monographs, Amer. Math. Soc, Providence, R.I., 1980.
  • [20] A.I. Markushevich, Theory of Analytic Functions, Vol. 1, Nauka, Moscow, 1968 [in Russian]; Revised English edition, translated and edited by R.A. Silverman, Prentice Hall Inc., Englewood Cliffs, N.Y., 1967.
  • [21] V. Pivovarchik, Inverse problem for a smooth string damped with damping at one end, J. Operator Theory 38 (1997), 243-263.
  • [22] V. Pivovarchik, Direct and inverse problems for a damped string, J. Operator Theory 42 (1999), 189-220.
  • [23] V. Pivovarchik, C. van der Mee, The inverse generalized Regge problem, Inverse Prob­lems 17 (2000), 1831-1845.
  • [24] V. Pivovarchik, Existence of a tree of Stieltjes strings corresponding to two given spectra, J. Phys. A: Math. Theor. 42 (2009), 375 213, 16 pp.
  • [25] V. Pivovarchik, N. Rozhenko, C. Tretter, Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings, Linear Algebra Appl. 439 (2013) 8, 2263-2292.
  • [26] V. Pivovarchik, O. Taistruk, On charateristic functions of operators on equilateral graph, Methods Funct. Anal. Topology 18 (2012) 2, 189-197.
  • [27] M.A. Shubov, Asymptotics of resonances and geometry of resonance states in the prob­lem of scattering of acoustic waves by a spherically symmetric inhomogeneity of the density, Differential Integral Equations 8 (1995) 5, 1073-1115.
  • [28] T.-L. Stieltjes, Recherches sur les fractions continues, Ann. Fac. Sci. Toulouse 8 (1894), 1-122, 9 (1895), 1-47.
  • [29] K. Veselic, On linear vibrational systems with one dimensional damping, Appl. Anal. 29 (1988), 1-18.
  • [30] K. Veselic, On linear vibrational systems with one dimensional damping II, Integral Equations Operator Theory 13 (1990), 883-897.
  • [31] M. Yamamoto, Inverse eigenvalue problem for a vibration of a string with viscous drag, J. Math. Anal. Appl. 152 (1990), 20-34.
Typ dokumentu
Bibliografia
Identyfikator YADDA
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