A Boundary Problem for the Fourth Order Equation with a Singular Coefficient in a Rectangular Region

Abstract

In the work, considered an inhomogeneous equation with partial derivatives of the fourth order with a singular coefficient, for which a boundary problem with initial conditions was investigated in a rectangle. The study of the considered problem was carried out by the a method of spectral analysis. On the basis of the property of completeness of systems of eigenfunctions of one-dimensional spectral problem, i.e. the system of sinus functions, the uniqueness theorem was proved. The solution of the initial-boundary problem was constructed in the form of series with respect to the system of eigenfunctions of a one-dimensional spectral problem. For proving uniform convergence of the constructed series, it was used estimates for trigonometric functions and Bessel-Clifford functions. On the basis of them, estimates were obtained for each member of a series that made it possible to prove the convergence of the obtained series and its derivatives to the fourth order inclusive, as well as the existence theorem in the class of regular solutions.