6.1. The method of "transfer of boundary value conditions" to any point of the interval of integration.

The complete solution of the system of differential equations has the form

.

Or you can write:

.

We substitute this expression for into the boundary conditions of the left edge and obtain:

,

,

.

Or we get the boundary conditions transferred to the point :

,

where and .

Further, we write similarly

And substitute this expression for into the transferred boundary conditions of the point :

,

,

.

Or we get the boundary conditions transferred to the point :

,

where and .

And so we transfer the matrix boundary condition from the left edge to the point and in the same way transfer the matrix boundary condition from the right edge.

Let us show the steps of transferring the boundary conditions of the right edge.

We can write:

We substitute this expression for in the boundary conditions of the right edge and obtain:

,

,

Or we get the boundary conditions of the right edge, transferred to the point :

,

where and .

Further, we write similarly

And substitute this expression for in the transferred boundary conditions of the point :

,

,

.

Or we get the boundary conditions transferred to the point :

,

where and .

And so at the inner point of the integration interval we transfer the matrix boundary condition, as shown, and from the left edge and in the same way transfer the matrix boundary condition from the right edge and obtain:

,

.

From these two matrix equations with rectangular horizontal coefficient matrices, we obviously obtain one system of linear algebraic equations with a square matrix of coefficients:

.

6.2. The case of "stiff" differential equations.

In the case of "stiff" differential equations, it is proposed to apply a line orthonormalization of the matrix boundary conditions in the process of their transfer to the point under consideration. For this, the orthonormalization formulas for systems of linear algebraic equations can be taken in [Berezin, Zhidkov].

That is, having received

we apply a line orthonormation to this group of linear algebraic equations and obtain an equivalent matrix boundary condition:

.

And in this line orthonormal equation is substituted

.

And we get

,

.

Or we get the boundary conditions transferred to the point :

,

where and .

Now we apply linear orthonorming to this group of linear algebraic equations and obtain an equivalent matrix boundary condition:

And so on.

And similarly we do with intermediate matrix boundary conditions carried from the right edge to the point under consideration.

As a result, we obtain a system of linear algebraic equations with a square matrix of coefficients, consisting of two independently stepwise orthonormal matrix boundary conditions, which is solved by Gauss’ method with the separation of the main element for obtaining the solution at the point under consideration:

.