102 Basic Concepts of the Theory of Difference Schemeswhere E > 0 is an arbitrary number, which is sometimes referred to as the
1::-inequality, permitting us to establish(12)1
l(u, v)I < II u II · II v II < E II u 11
2
+ 4 E II v 11
2
·- The search for eigenfunctions and eigenvalues in the example of the
simplest difference problem. The method of separation of variables being
involved in the apparatus of mathematical physics applies equelly well to
difference problems. Employing this method enables one to split up an
original problem with several independent variables into a series of more
simpler problems with a smaller number of variables. As a rule, in this
situation eigenvalue problems with respect to separate coordinates do arise.
Difference problems can be solved in a quite similar manner.
In this section we consider the problem of searching for eigenvalues of
the simplest difference operator.
Results we are going to obtain here will be needed in the sequel because
applying the method of separation of variables leads to problems just of the
same type. In the next chapters we give various examples of employing this
method for the discovery of stability and convergence of concrete difference
schemes.
Before giving further motivations, we would like to recall the basic as-
pects concerned with the elementary problem of determining eigenfunctions
and eigenvalues for the differential equation
(13) u"(x) + .\u(x) = 0, 0 < x < l, u(O) = u(l) = 0.
The nontrivial solutions of this problem, that is, the eigenfunctions uk and
the appropriate eigenvalues .\k are expressed by
fi. k7rx
uk(x) = y l sm -
1
-,
The eigenfunctions Hk constitute what is called an orthonormal
system:
1
J 1Lk(x) um(x) clx = likm,
0where likm = {
0,
1,k of m,
k =in.- If f ( x) is twice differentiable and satisfies the nonhomogeneous
boundary conditions f(O) = J(l) = 0, it arranges itself into a uniformly
convergent series
00
f(x) = L f1; uk(x)
k=l