Model solutions of a fourth order equation with a mixed derivative

The paper analyzes a model initial boundary value problem that describes the propagation of vibrations in a moving elastic web. To find an exact solution to this problem, we use the Galerkin method of basis expansion, which is determined by using an auxiliary initial boundary value problem that describes the vibrations of a beam with different types of boundary conditions (fastening conditions). The main task is to study and find exact solutions in the form of a functional series. For various initial disturbances, a rule was formulated that allows to calculate the coefficients of this functional series. The main characteristics of this problem determine the constant speed of movement of the web, its hinged fastening, and the peculiarity of the equation associated with the equality of the speed of the moving wave and the speed of propagation of vibrations in the stationary web. This feature affects the final form of solutions and the methodology for solving the problem itself. The equation describing the oscillations is a fourth-order linear partial differential equation with constant coefficients and contains a mixed derivative with respect to space and time variables. For the considered oscillatory process and equation, the law of conservation of energy is established and the uniqueness theorem for the solution of the initial-boundary value problem is proven.