1549301742-The_Theory_of_Difference_Schemes__Samarskii

Some properties of difference elliptic operators 273

Observe that ih and, hence, wh contain no nodes (0, 0), (0, N 2 ), (N 1 , 0),

(N 1 , N 2 ) being the vertices of the rectangle.

We seek a solution of problem (1) by the method of separation of

variables as a product

Substitution of this expression into the equation

yields

Because we are interested only in nontrivial solutions of problem (1), the

division of both sides of equation (2) by μ( x 1 ) 17( x 2 ) -::/- 0 is meaningful. As

a final result we get Px, x, / μ + 1Jx 2 ,c 2 /17 +A = 0 or

(3) T/x2x2 - A=_;,(!)

T/

with A (l) = const being independent neither of x 1 nor of x 2 • In view of

this, we might set up for p(x 1 ) the eigenvalue problem on the grid

the statement of which is

A iμ+/\ (1) P=Pi: 1 :r 1 +/\ (1) μ='^0

The conditions μ 0 = μN, = 0 follow immediately from the relations
μ(O) TJ(x 2 ) = 0, μ(1 1 ) TJ(x 2 ) = 0 and TJ(x 2 ) '!- 0. As we learn from Chap-
ter 2, Section 3, a solution of this problem acquires the form