Solution of a boundary value problem for a third-order inhomogeneous equation with multiple characteristics with the construction of the Green’s function

Abstract

In the paper the second boundary value problem in a rectangular domain for an inhomogeneous thirdorder partial differential equation with multiple characteristics with constant coefficients was considered. The uniqueness of the solution to the problem posed is proven by the method of energy integrals. A counterexample is constructed in case when the uniqueness theorem’s conditions are violated. Using the method of separation of variables, the solution to the problem is sought in the form of a product of two functions X(x) and Y (y). To determine Y (y), we obtain a second-order ordinary differential equation with two boundary conditions at the boundaries of the segment [0; q]. For this problem, the eigenvalues and the corresponding eigenfunctions are found for n = 0 and n 2 N. To determine X(x), we obtain a third-order ordinary differential equation with three boundary conditions at the boundaries of the segment [0; p]. Using the Green’s function method, we constructed solution of the specified problem. A separate Green’s function for n = 0 and a separate Green’s function for the case when n is natural were constructed. The solution for X(x) is written in terms of the constructed Green’s function. After some transformations, an integral Fredholm equation of the second kind is obtained, the solution of which is written through the resolvent. Estimates for the resolvent and Green’s function are obtained. The uniform convergence of the solution and the possibility of its term-by-term differentiation under certain conditions on given functions are proven. When justifying the uniform convergence of the solution, the absence of a “small denominator” is proven.