Abstract:
We study the problem with boundary conditions of the first and second kind on the boundary of the rectangular area for an equation with two internal perpendicular lines of change of a type. With the use of spectral method we prove the uniqueness and the existence of a solution. Obtained in the process of separation of variables, the eigenvalue problem for an ordinary differential equation is not self-adjoint, and the system of root functions is not orthogonal. We construct corresponding biorthogonal system of functions and prove its completeness, based on which we establish a criterion for the uniqueness of the problem. A solution to the problem is constructed as a sum of biorthogonal series.
Keywords:mixed type equation, mixed boundary-value problem, biorthogonal system functions, completeness, existence and uniqueness of solution.