We consider an integro-differential fractional order equation describing one-dimensional eigenoscillations of a homogeneous viscoelastic medium. We show that the spectrum of this equation consists of N sequences of eigenvalues, where N = 4 or N = 5. Elements of these sequences are squared roots of equations of degree N. We study their limit behavior. We prove that there are only two sequences consisting of points with negative real part.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. N. Rabotnov, Elements of Hereditary Solid Mechanics [in Russian], Nauka, Moscow (1977).
M. A. Zhuravkov and N. S. Romanova, Perspectives of Applications of Fractional Differential Calculus to Mechanics [in Russian], Beloruss. State University Press, Minsk (2013).
A. S. Shamaev and V. V. Shumilova, “Effective acoustic equations for a layered material described by the fractional Kelvin-Voigt model,” J. Sib. Fed. Univ., Math. Phys. 14, No 3, 351–359 (2021).
M. V. Shitikova, “Fractional operator viscoelastic models in dynamic problems of mechanics of solids: a review,” Mech. Solids 57, No. 1, 1–33 (2022).
A. S. Shamaev and V. V. Shumilova, “On the spectrum of an integro-differential equation arising in viscoelasticity theory,” J. Math. Sci. 185, No. 2, 751–754 (2012).
V. V. Shumilova, “Spectral analysis of integro–differential equations in viscoelasticity theory,” J. Math. Sci. 196, No. 3, 434–440 (2014).
V. V. Vlasov and N. A. Rautian, “Investigation of operator models arising in viscoelasticity theory,” J. Math. Sci. 260, No. 4, 456–468 (2022).
L. Suarez and A. Shokooh, “Response of systems with damping materials modeled using fractional calculus,” Appl. Mech. Rev. 48, No 11, 118–126 (1995).
Yu. A. Rossikhin and M. V. Shitikova, “Application of fractional derivatives to the analysis of damped vibrations of viscoelastic single mass systems,” Acta Mech. 120, No. 1–4, 109–125 (1997).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 125, 2023, pp. 173-179.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Shamaev, A.S., Shumilova, V.V. Spectrum of an Integro-Differential Equation of Fractional Order. J Math Sci 276, 191–198 (2023). https://doi.org/10.1007/s10958-023-06733-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-023-06733-2