Geodesic
Representation of solutions of a certain integro-differential equation and applications
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 2, Tome 171 (2019), pp. 78-93.

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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 bundle 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.
Keywords: integro-differential equation, spectral analysis, operator bundle, $p$-base. 
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D. A. Zakora. Representation of solutions of a certain integro-differential equation and applications. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh Winter Mathematical School "Modern Methods of Function Theory and Related Problems." January 28 – February 2, 2019. Part 2, Tome 171 (2019), pp. 78-93. http://geodesic.mathdoc.fr/item/INTO_2019_171_a6/

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