About one method for replacing variables for a wavean equation describing vibrations of systems with moving boundaries

An analytical method for solving the wave equation describing the oscillations of systems with moving boundaries is considered. By replacing variables that set boundaries and leave the equation invariant, the original boundary value problem is reduced to a system of functional – difference equations that can be solved using forward and reverse methods. The inverse method is described, which allows us to apply sufficiently diverse laws of boundary motion to the laws obtained from the solution of the inverse problem. New partial solutions for a fairly wide range of boundary motion laws are obtained. A direct asymptotic method for approximating the solution of a functional equation is considered. The errors of the approximate method are estimated depending on the speed of the border movement.

Oscillation of nonlinear third order perturbed functional difference equations

This paper deals with oscillatory and asymptotic behavior of all solutions of perturbed nonlinear third order functional difference equation\Delta {\left( {{b_n}\Delta ({a_n}(\Delta {x_n}} \right)^\alpha })) + {p_n}f\left( {{x_{\sigma \left( n \right)}}} \right) = g\left( {n,{x_n},{x_{\sigma (n)}},\Delta {x_n}} \right),\,\,\,n \ge {n_0}.By using comparison techniques we present some new sufficient conditions for the oscillation of all solutions of the studied equation. Examples illustrating the main results are included.