On model solutions of the initial boundary value problem for the equation of vibrations of thermoelastic plates

This work is devoted to the study and analysis of a model initial boundary value problem for the oscillations of a thermoelastic plate. The main problem is the construction of exact solutions for the given problem. The mathematical model of oscillations is a linear partial differential equation of the third order on the time variable and of the sixth order on the spatial variable with constant coefficients. It should be noted that for the spatial variable the problem is one-dimensional, but the equation contains a mixed derivative for the time and spatial variables. The integral identity is obtained, with the help of which in the general case it is possible to show the violation of the energy conservation law. An algorithm for obtaining explicit solutions of this problem in the form of a functional series is presented. The paper contains illustrations showing the deviations of the surface from the equilibrium position and the dynamics of oscillations at different moments of time.