Constructing a Basis from Systems of Eigenfunctions of one not Strengthened Regular Boundary Value Problem

Abstract

We investigate a nonlocal boundary value spectral problem for an
ordinary differential equation in an interval. Such problems arise
in solving the nonlocal boundary value for partial equations by
the Fourier method of variable separation. For example, they arise
in solving nonstationary problems of diffusion with boundary
conditions of Samarskii-Ionkin type. Or they arise in solving
problems with stationary diffusion with opposite flows on a part
of the interval. The boundary conditions of this problem are
regular but not strengthened regular. The principal difference of
this problem is: the system of eigenfunctions is comlplete but not
forming a basis. Therefore the direct applying of the Fourier
method is impossible. Based on these eigenfunctions there is
constructed a special system of functions that already forms the
basis. However the obtained system is not already the system of
the eigenfunctions of the problem. We demonstrate how this new
system of functions can be used for solving a nonlocal boundary
value equation on the example of the Laplace equation.