1549301742-The_Theory_of_Difference_Schemes__Samarskii

104 Basic Concepts of the Theory of Difference Schemes

Observe, that the boundary condition at the point x = 0 is automatically

fulfilled for any O'.. At the point x = l we have

sin O'./ = 0,

giving O'. = O'.k = k7r/l, k = 1, 2, ... , N -1.

Thus, we have obtained the eigenfunctions and eigenvalues of problem

(14). A brief survey of their properties is presented below.

(16)

1.

k7rx

y( kl ( x) = sin

l

' 4. 2 k7rh

"'k = h2 sm 2/ J k = 1, 2, ... , N - 1.

2. The eigenvalues ,\k are enumerated in increasing order and for the
   whole collection { ,\k} the estimates holds:

(17) Q < Al ' = h 4. 2 Sin^2 7rh '

21 < "'2 <

, 4. 2 7rh (N - 1)

< "'N-1 = h2 Sin 2/

4 2 7rh 4

= h 2 cos 2/ < h 2.

In particular, it follows from (17) that all the eigenvalues of problem (14)

are positive.

3. Eigenfunctions of problem ( 14) y(k), y(m), corresponding to distinct
   eigenvalues, are orthogonal in the sense of the inner product (5):

(18) k of m.

To prove this fact, we make use of the second Green formula for the

homogeneous boundary conditions (10'):

In light of our supposition, yU) and y(m) are eigenfunctions corresponding
to distinct eigenvalues, that is, Ak -::J Am, the orthogonality of y(k) and y(m)
follows from the preceding equality:

(y(k) 1 y(m)) = 0.

4. The norm of an eigenfunction y(k)(x) is 11 y(k) II = /lfi. Recall
   that the norm is understood in the sense of the inner product (5):
   N-1
   llvll2=(u,vl= 2= v,2h.
   i=l