2 Preli1ninariesMost of the well-developed methods available for solving such a system
falls within the categories of direct or "exact-fitted" methods and iterative
or successive-approximate methods which are gaining increasing popularity.
The starting point in more a detailed exploration is the simplest sys-
tems of linear algebraic equations, namely, difference equations with special
matrices in simplified form, for example, with tridiagonal matrices.
The history of difference equations contains plenty of good examples
when such equations emerged during the course of direct descriptions of real
processes in science and technology. Equations of mathematical physics,
that is, partial differential equations are sources for a broad class of differ-
ence equations approximating integral and differential equations and give a
substantial contribution to the continuing development of the theory of dif-
ference schemes. The main feature of difference equations is stipulated by
the fact that the matrices of the high-order corresponding systems (about
104 -10^5 ) are sparse.
It is natural from the viewpoint of applications to treat various dif-
ference equations regardless of the initial differential equations which have
induced them. All of the resulting schemes and properties are invariant to
a concrete differential equation.
In the present section a direct method for solving the boundary-value
problems associated with second-order difference equations will be the sub-
ject of special investigations.- Examples of difference equations. Undoubtedly, the reader has already
encountered the simplest examples of first-order difference equations in con-
nection with the formulae for the terms of an arithmetic or a geometric
progressions: ak+l = ak + d or ak+l - 2 ak + ak-l = 0 and ak+l = q ak,
respectively, where the argmnent of the members ak = a(k) takes only
positive integer values.
We briefly touch upon the basic concepts of the grid methods. The
basic notions such as grids and grid functions will be studied in more de-
tail in Chapter 2. A discrete set of points (nodes) is called a grid or
lattice. Let a grid function Yi = y(i) of the integer-valued argument
i = 0, ±1, ±2, ... , be given. The right and left differences are defined
at a point i to be:
b. Yi:= y(i + 1) - y(i) = Yi+l - Yi,
\ly; := y(i) - y(i - 1) =Yi - Yi-l ·By definition, b. Yi-l = Yi - Yi-l = \1 Yi· It should be noted that the
expressions above may be viewed as formal analogs of the first derivative