SOLUTION OF THE FIRST BOUNDARY PROBLEM FOR A THIRD ORDER EQUATION WITH MINOR TERMS, A METHOD FOR CONSTRUCTING THE GREEN'S FUNCTION

Abstract

The paper considers the first boundary value problem in a rectangular domain for an inhomogeneous partial differential equation of the third order with constant coefficients at lower terms. The uniqueness of the solution of the stated problem is proved by the method of energy integrals. Using the method of separation of variables, the solution of the problem is sought as a product of two functions X(x) and Y(y). To determine X(x), we obtain a third-order ordinary differential equation with three boundary conditions on the boundary of the segment [a, b], and for Y(y), we obtain a second-order ordinary differential equation with two boundary conditions on the boundary of the segment [c, d]. The Green's function method is used to construct solutions to these problems. Estimates for the resolvent and Green's function are obtained. When substantiating the uniform convergence of the solution, the non-zero “small denominator” is used.