We study spectral properties of boundary value problems for integro-differential equations in viscoelasticity theory. The sum of N exponential functions with negative exponents is taken for the convolution kernel. We prove that the spectrum of the boundary value problem is the union of N sequences of real numbers and two sequences all elements, except for finitely many ones, are not real. In particular, we obtain conditions under which these two sequences contains no real numbers. Bibliography: 8 titles.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. Eremenko and S. Ivanov, “Spectra of the Gurtin–Pipkin type equations,” SIAM J. Math. Anal. 43, No 5, 2296–2306 (2011).
V. V. Vlasov and J. Wu, “Spectral analysis and solvability of abstract hyperbolic equations with aftereffect” [in Russian], Differ. Uravn. 45, No 4, 524–533 (2009); English transl.: Differ. Equ. 45, No 4, 539–548 (2009).
V. V. Vlasov, J. Wu, and G. R. Kabirova, “Well-defined solvability and spectral properties of abstract hyperbolic equations with aftereffect” [in Russian], Sovrem. Mat. Fundam. Napravl. 35, 44–59 (2010); English transl.: J. Math. Sci., New York 170, No. 3, 388–404 (2010).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Solvability and spectral analysis of integro-differential equations arising in the theory of heat transfer and acoustics” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 434, No. 1, 12–15 (2010); English transl.: Dokl. Math. 82, No. 2, 684–687 (2010).
V. V. Vlasov, N. A. Rautian, and A. S. Shamaev, “Investigation of operator models that arise in problems of hereditary mechanics” [in Russian], Sovrem. Mat. Fundam. Napravl. 45, 43–61 (2012).
A. A. Gavrikov and A. S. Shamaev, “On some problems in acoustics of emulsions” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 434, No. 1, 42–46 (2010); English transl.: Dokl. Phys. 55, No. 9, 450–454 (2010).
A. S. Shamaev and V. V. Shumilova, “On the spectrum of an integro-differential equation arising in viscoelasticity theory” [in Russian], Probl. Mat. Anal. 63, 189–192 (2012); English transl.: J. Math. Sci., New York 185, No. 2, 751–754 (2012).
A. S. Shamaev and V. V. Shumilova, “Spectrum of one-dimensional oscillations in the combined stratified medium consisting of a viscoelasticmaterial and a viscous compressible fluid” [in Russian], Izv. Akad. Nauk, Mekh. Zhidk. Gaza No. 1, 17–25 (2013); English transl.: Fluid Dyn. 48, No. 1, 14–22 (2013).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Problemy Matematicheskogo Analiza 73, October 2013, pp. 167–172.
Rights and permissions
About this article
Cite this article
Shumilova, V.V. Spectral Analysis of Integro–Differential Equations in Viscoelasticity Theory. J Math Sci 196, 434–440 (2014). https://doi.org/10.1007/s10958-014-1666-9
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10958-014-1666-9