Abstract
In this paper, we consider a second-order integro-differential equation with unbounded operator coefficients in a Hilbert space, which is an abstract hyperbolic equation perturbed by an integral term with a difference-type kernel of a special form. This equation arises in the description of various viscoelastic systems. Using the system of generalized eigenvectors of the operator sheaf associated with the equation considered, we construct a p-basis in the orthogonal sum of Hilbert spaces. Using this basis, we find a representation of the solution of the equation. We also discuss possible applications to problems of viscoelasticity.
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 171, Proceedings of the Voronezh Winter Mathematical School “Modern Methods of Function Theory and Related Problems,” Voronezh, January 28 – February 2, 2019. Part 2, 2019.
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Zakora, D.A. Representation of Solutions of a Certain Integro-Differential Equation and Applications. J Math Sci 263, 675–690 (2022). https://doi.org/10.1007/s10958-022-05958-x
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DOI: https://doi.org/10.1007/s10958-022-05958-x