An Inverse Method for Solving Problems about Oscillations of Mechanical Systems with Moving Boundaries

An analytical method for solving the wave equation describing the oscillations of systems with moving boundaries is considered. By changing the variables that stop the boundaries and leave the equation invariant, we reduce the original boundary value problem to a system of functional-difference equations, which can be solved using direct and inverse methods. An inverse method is described that allows approximating quite diverse laws of boundary motion by laws obtained from solving the inverse problem. New particular solutions are obtained for a fairly wide range of laws of boundary motion. A direct asymptotic method for the approximate solution of a functional equation is considered. The errors of the approximate method are estimated depending on the velocity of the boundary movement.