On the solution of a mixed problem for the equation of vibrations of a moving viscoelastic web

A model initial boundary value problem of small transverse oscillations of a viscoelastic moving web with a hinged condition of fastening is considered. The vibrations of such a canvas are described by a linear differential equation of the 5th order in a spatial variable with constant coefficients. It is worth noting that the equation includes mixed derivatives of the desired function both with respect to the spatial variable and with respect to time. The paper describes a technique for constructing a solution in the form of a functional series based on a system of basis functions. To solve the initial-boundary value problem under the additional condition of conservation of energy, a condition is obtained that ensures the uniqueness of the solution. A special class of functions for which the uniqueness theorem holds is explicitly described.