**MSC2010** 35L20

### On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation

#### P. N. Burago^{1}, A. I. Egamov^{2}

Annotation | We consider the second initial boundary-value problem for a certain class of second-order integro-differential PDE with integral operator. The connection of its solution with the solution of the standard second linear initial boundary-value problem for the hyperbolic equation is shown. Thus, the nonlinear problem is reduced to a standard linear problem, whose numerical solution can be obtained, for example, by the Fourier method or Galerkin method. The article provides examples of five integro-differential equations for various integral operators as particular representatives of the class of integro-differential equations for a better understanding of the problem. The application of the main theorem to these examples is shown. Some simple natural requirement is imposed on the integral operator; so, in four cases out of five the problem’s solution satisfies some phase constraint. The form of these constraints is of particular interest for the further research. |
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Keywords | the second initial boundary value problem, integro-differential equation with PDE, phase constraint, hyperbolic equation |

^{1}**Pavel N. Burago**, Post-graduate student, Department of Differential Equations, Mathematical and Numerical Analysis, Institute of Information Technology, Mathematics and Mechanics Lobachevsky State University of Nizhni Novgorod (23 Prospekt Gagarina, Nizhni Novgorod, 603950, Russia), ORCID: https://orcid.org/0000-0002-8010-906X, burago.pasha@yandex.ru^{2}**Albert I. Egamov**, Associate Professor, Department of Differential Equations, Mathematical and Numerical Analysis, Institute of Information Technology, Mathematics and Mechanics Lobachevsky State University of Nizhni Novgorod (23 Prospekt Gagarina, Nizhni Novgorod, 603950, Russia), Ph.D. (Physics and Mathematics), ORCID: https://orcid.org/0000-0002-3630-7237, albert810@yandex.ru

**Citation**: P. N. Burago, A. I. Egamov, "[On the connection between solutions of initial boundary-value problems for a some class of integro-differential PDE and a linear hyperbolic equation]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,21:4 (2019) 413–429 (In Russian)

**DOI** 10.15507/2079-6900.21.201904.413-429