(untitled) (Scientific journal of the Fergana State University, 2023)
In this article, for a partial differential equation of high even order with a Bessel operator in a rectangle, an initial boundary value problem has been formulated. Applying the method of separation of variables to the considered problem, a spectral problem was obtained for an ordinary differential equation of high even order. A self-adjointness of the last problem was proved, which implies the existence of the system of its eigen-functions, as well as the orthonormality and completeness of this system. The solution of the considered problem study was found as the sum of a Fourier series with respect to the system of eigen-functions of the spectral problem. The uniform convergence of this series, as well as the series obtained from it by term-by-term differentiation, have been proved. The uniqueness of the solution of the problem has been proved by the method of energy of integrals. An estimate for solution of the problem was obtained, from which follows its continuous dependence on the given functions.