Abstract
In this paper we consider a spectral problem for ordinary differential equation of fourth order with a spectral parameter in the boundary conditions. This problem arises when variables are separated in the dynamical boundary value problem describing bending vibrations of a homogeneous rod, in cross-sections of which the longitudinal force acts, the both ends of which are fixed elastically and on these ends the masses are concentrated. We investigate locations, multiplicities of eigenvalues, study the oscillation properties of eigenfunctions and establish sufficient conditions for the subsystems of root functions of this problem to form a basis in the space \(L_p,\,1< p < \infty \).
Similar content being viewed by others
References
Aliev, Z.S.: Basis properties in \(L_p\) of systems of root functions of a spectral problem with spectral parameter in a boundary condition. Differ. Equ. 47, 766–777 (2011)
Aliev, Z.S.: On basis properties of root functions of a boundary value problem containing a spectral parameter in the boundary conditions. Dokl. Math. 87, 137–139 (2013)
Aliev, Z.S., Dunyamalieva, A.A.: Defect basis property of a system of root functions of a Sturm–Liouville problem with spectral parameter in the boundary conditions. Differ. Equ. 51, 1249–1266 (2015)
Aliev, Z.S., Dunyamalieva, A.A., Mehraliyev, Y.T.: Basis properties in \(L_p\) of root functions of Sturm–Liouville problem with spectral parameter-dependent boundary conditions. Mediterr. J. Math. 14, 1–23 (2017)
Aliyev, Z.S., Guliyeva, S.B.: Properties of natural frequencies and harmonic bending vibrations of a rod at one end of which is concentrated inertial load. J. Differ. Equ. 263, 5830–5845 (2017)
Aliyev, Z.S., Kerimov, N.B.: Spectral properties of the differential operators of the fourth-order with eigenvalue parameter dependent boundary condition. Int. J. Math. Math. Sci. 2012, 456517 (2012)
Aliyev, Z.S., Namazov, F.M.: Spectral properties of a fourth-order eigenvalue problem with spectral parameter in the boundary conditions. Electron. J. Differ. Equ. 2017(307), 1–11 (2017)
Aliyev, Z.S., Namazov, F.M.: On the spectral problem arising in the mathematical model of bending vibrations of a homogeneous rod. Complex Anal. Oper. Theory (2019). https://doi.org/10.1007/s11785-019-00924-z
Azizov, T.Y., Iokhvidov, I.S.: Linear operators in Hilbert spaces with \(G\)-metric. Russ. Math. Surv. 26, 45–97 (1971)
Amara, J.B., Vladimirov, A.A.: On oscillation of eigenfunctions of a fourth-order problem with spectral parameters in the boundary conditions. J. Math. Sci. 150, 2317–2325 (2008)
Banks, D.O., Kurowski, G.J.: A Prufer transformation for the equation of a vibrating beam subject to axial forces. J. Differ. Equ. 24, 57–74 (1977)
Binding, P.A., Browne, P.J.: Application of two parameter eigencurves to Sturm–Liouville problems with eigenparameter dependent boundary conditions. Proc. R. Soc. Edinb. Sect. A 125, 1205–1218 (1995)
Bolotin, B.B.: Vibrations in technique: handbook in 6 volumes, The vibrations of linear systems. I. Engineering Industry, Moscow (1978) (in Russian)
Fulton, T.: Two-point boundary value problems with eigenvalue parameter in the boundary conditions. Proc. R. Soc. Edinb. Sect. A 77, 293–308 (1977)
Il’in, V.A.: Unconditional basis property on a closed interval of systems of eigen-and associated functions of a second-order differential operator. Dokl. Akad. Nauk SSSR 273, 1048–1053 (1983) (in Russian)
Kapustin, N.Y.: On a spectral problem arising in a mathematical model of torsional vibrations of a rod with pulleys at the ends. Differ. Equ. 41, 1490–1492 (2005)
Kapustin, N.Y., Moiseev, E.I.: On a spectral problem in theory of parabolic–hyperbolic heat equation. Dokl. Math. 352, 451–454 (1997)
Kapustin, N.Y., Moiseev, E.I.: On spectral problems with a spectral parameter in the boundary condition. Differ. Equ. 33, 115–119 (1997)
Kapustin, N.Y., Moiseev, E.I.: On the basis property in the space \(L_p\) of systems of eigenfunctions corresponding to two problems with spectral parameter in the boundary condition. Differ. Equ. 36, 1357–1360 (2000)
Kerimov, N.B., Aliyev, Z.S.: The oscillation properties of the boundary value problem with spectral parameter in the boundary condition. Trans. Acad. Sci. Azerb. Ser. Phys. Tech. 25(7), 61–68 (2005)
Kerimov, N.B., Aliev, Z.S.: On the basis property of the system of eigenfunctions of a spectral problem with spectral parameter in a boundary condition. Differ. Equ. 43, 905–915 (2007)
Kerimov, N.B., Poladov, R.G.: Basis properties of the system of eigenfunctions in the Sturm–Liouville problem with a spectral parameter in the boundary conditions. Dokl. Math. 85, 8–13 (2012)
Krylov, A.N.: Some differential equations of mathematical physics having applications to technical problems. Academy Sci. USSR, Moscow (1932) (in Russian)
Moiseev, E.I., Kapustin, N.Y.: On singularities of the root space of a spectral problem with spectral parameter in a boundary condition. Dokl. Math. 385, 20–24 (2002)
Naimark, M.A.: Linear Differential Operators. Ungar, New York (1967)
Poisson, S.D.: Mémoire sur la Manière d’exprimer les Fonctions par des Séries de quantités périodiques, et sur l’Usage de cette Transformation dans la Resolution de differens Problèmes. École Polytech. 18, 417–489 (1820)
Ragusa, M.A., Russo, G.: ODEs approaches in modeling brosis Comment on “Towards a unified approach in the modeling of fibrosis: a review with research perspectives” by Martine Ben Amar and Carlo Bianca. Phys. Life Rev. 17, 112–113 (2016)
Roseau, M.: Vibrations in Mechanical Systems, Analytical Methods and Applications. Springer, Berlin (1987)
Russakovskii, E.M.: Operator treatment of boundary problems with spectral parameters entering via polynomials in the boundary conditions. Funct. Anal. Appl. 9, 358–359 (1975)
Schneider, A.: A note on eigenvalue problems with eigenvalue parameter in the boundary conditions. Math. Z. 136, 163–167 (1974)
Timoshenko, S.P.: Strength and vibrations of structural members (Collection of paper; E. I. Grigolyuk, editor). Nauka, Moscow (1975) (in Russian)
Walter, J.: Regular eigenvalue problems with eigenvalue parameter in the boundary condition. Math. Z. 133, 301–312 (1973)
Acknowledgements
We are very grateful to the referees for their valuable comments and suggestions.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Maria Alessandra Ragusa.
Rights and permissions
About this article
Cite this article
Aliyev, Z.S., Namazov, F.M. Spectral properties of the equation of a vibrating rod at both ends of which the masses are concentrated. Banach J. Math. Anal. 14, 585–606 (2020). https://doi.org/10.1007/s43037-019-00009-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s43037-019-00009-1
Keywords
- Ordinary differential equations of fourth order
- Bending vibrations of a homogeneous rod
- Pontryagin space
- Basis property of eigenfunctions