Let us investigate Green's function for the problem with nonlocal boundary conditions

We can write many problems with nonlocal boundary conditions (NBC) in this form, where
,
, is a classical part and
,
, is a nonlocal part of boundary conditions.

If

, then problem (6.1)–(6.3) becomes classical. Suppose that there exists Green's function

for the classical case. Then Green's function exists for problem (6.1)–(6.3) if

. For

,

, we derive

Since

,

, we can rewrite formula (5.26) as

Example 6.1.

Let us consider the differential equation with two nonlocal boundary conditions

We introduce a mesh

(see (5.8)). Denote

,

for

. Then problem (6.6) can be approximated by a finite-difference problem (scheme)

We suppose that the points
,
are coincident with the grid points, that is,
,
.

We rewrite (6.7) in the following form:

We can take the following fundamental system:

,

. Then

For a problem with the boundary conditions

we have

,

and we express Green's function

of the Dirichlet problem via Green's function

of the initial problem

We derive expressions for "classical" Green's function

or (see (5.7) and (5.13))

Remark 6.2.

Note that the index of
on the right-hand side of (6.9) is shifted (cf. (6.1)).

Green's function

is the same as in [

10], and it is equal to Green's function

for differential problem (6.6) at grid points in the case
.

For a "nonlocal" problem with the boundary conditions

,

,

It follows from (6.5) that

if

. Green's function does not exist for

. By substituting Green's function

for the problem with the classical boundary conditions into the above equation, we obtain Green's function for the problem with nonlocal boundary conditions

This formula corresponds to the formula of Green's function for differential problem (6.6) (see [

4])

Example 6.3.

Let us consider the problem

where
.

Problem (6.22) can be approximated by the difference problem

where
are approximations of the weight functions
in integral boundary conditions,
is a quadrature formula for the integral
approximation (e.g., trapezoidal formula
).

The expression of Green's function for the problem with the classical boundary conditions (

,

,

) is described in Example 6.1. The existence condition of Green's function for problem (6.23) is

, where

(such a condition was obtained for problem (6.23) in [

15,

16]) and Green's function is equal to (see Theorem 5.4)

where
is defined by (6.15).

Green's function for differential problem (6.22) was derived in [

8]. For this problem

if
, where
is defined by formula (6.17).

Remark 6.4.

We could substitute (6.15) into (6.25) and obtain an explicit expression of Green's function. However, it would be quite complicated, and we will not write it out. Note that, if
,
, then discrete problem (6.23) is the same as (6.7)-(6.8). For example, it happens if a trapezoidal formula is used for the approximation
,
and we take
. It is easy to see that we could obtain the same expression for Green's function (6.19) in this case.

Example 6.5.

Let us consider a difference problem

A condition for the existence of the Green's function (fundamental system

) is

We consider three types (

,

;

,

;

,

,

,

) of discrete boundary conditions

All the cases yield
. Consequently, Green's function for the three problems does not exist.