### 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.

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