A PROBLEM ON LONGITUDINAL VIBRATION IN A SHORT BAR WITH DYNAMICAL BOUNDARY CONDITIONS



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Abstract

In this paper, we consider an initial-boundary problem with dynamical nonlocal boundary condition for a pseudohyperbolic fourth-order equation in a rectangular. Dynamical nonlocal boundary condition represents a relation between values of a required solution, its derivatives with respect of spacial variables, second-order derivatives with respect of time-variables and an integral term. This problem may be used as a mathematical model of longitudinal vibration in a thick short bar and illustrates a nonlocal approach to such processes. The main result lies in justification of solvability of this problem. Existence and uniqueness of a generalized solution are proved. The proof is based on the a priori estimates obtained in this paper, Galerkin’s procedure and the properties of the Sobolev spaces.

About the authors

A. B. Beylin

Samara State Technical University

Author for correspondence.
Email: morenov.sv@ssau.ru
Russian Federation

L. S. Pulkina

Samara National Research University

Email: morenov.sv@ssau.ru
Russian Federation

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