1549301742-The_Theory_of_Difference_Schemes__Samarskii

106 Basic Concepts of the Theory of Difference Schemes

3. Eigenvalue problems with boundary conditions of the second and third
   kinds. We now turn to the second eigenvalue problen1

(23) u11 + >-u = 0, O<x<l, u^1 (0)=1/(l)=O, u(x)f=-0.

With the relations ux,o = 1t1(0)+ thu//(O)+O(h^2 ) = -th>-u(O)+O(h^2 ) and

ux,O + th>-u(O) = O(h^2 ) in view, we approximate the boundary conditions

with accuracy O(h^2 ), leading to the second boundary-value problem on

eigenvalues:

Yxx+>-y(x)=O, x -- ih ) i = 1, 2, ... , N - 1, h = ~,

(24)

Yx,o + t hAYo = 0,

In such a setting it is required to find the values of the parameter A

such that these homogeneous equations have nontrivial solutions y(x) f=- 0.

In contrast to the first boundary-value problem, here the parameter A enters

not only the governing equation, but also the boundary conditions. The

introduction of new sensible notations

2

• h vx for x=O '
  (25) Av= -Vix for x -- ih ' O<i<N, )
  2
  h ·ux for x = l )

is connected with setting problem (24) in the operator form

(26) Ay = >-y,

where the operator A acts in the space H = Q comprising all the functions

y(x) defined on the grid wh = {x; = ih, i = 0, 1, ... , N}.

We introduce in that space the inner product

N-1

[y,v]= L y;u;h+~(y 0 u 0 +yNvN)

i=l

and show that the operator A is self-adjoint and nonnegative. By definition,
this means that

[Ay, v] = [y, Av],

[Ay,y]>O for any y EH.