About one initial-boundary problem for equation of bending beam vibrations

This research is devoted for the inhomogeneous equation of bending vibrations of a beam , investigated mixed problem which non-decaying boundary conditions, higher-order time derivatives than in the equation. Differential expression and edge forms isn't contain only the main parts. The mixed problem is associated with the spectral for the fourth order equation with incommensurable degrees of the parameter in the boundary conditions and the Cauchy problem for the second order equation with the spectral parameter with respect to the time variable. The solution of the initial- boundary-value problem is constructed the form of complete integral residue from the solutions of the one-dimensional spectral problem and Cauchy problem. The existence of a classical solution to the studied initial-boundary-value problem is proved which under certain smoothness conditions for the initial data, vanishes together with all derivatives to a certain order at the ends of the variation interval of the spatial variable.