1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 21

b) if i 0 = N, that is, max; y, = yN = M 0 > 0, but YN-l < M 0 , then

for 0 < X 2 < 1.

In both cases we came to a contradiction with the condition[, [Yi] > 0

for all i = 0, 1, 2, ... , N. It is therefore concluded that Yi < 0, since if

Yi > 0 at least at one point i = i*, its maximal positive value should be

attained at smne point i = i 0 (for example, at i 0 = i*) that in principle is

impossible.

Corollary 2a Under conditions (22*) the problem

£[Yi] = 0 , i=O,l, ... ,N

has only the trivial solution.

There is no difficulty to reformulate the remaining assertions from the

previous sections, but we do not dwell on precise statements.

10. Solution estimation for difference boundary-value problems by the elim-
    ination method. In tackling the first boundary-value problem difference
    equation (21) has the tridiagonal matrix of order N - 1

-C\ B 1 0

A2 -C2 B2

0 0 0

0 0 0

0 0

0 0

AN-2 -CN-2

0 AN-1

0

0

being symmetric for the case

(35) Bi = A;+1.

Difference equations with a symmetric matrix are typical in numerical
solution of boundary-value problems associated with self-adjoint differential
equations of second order. In what follows we will show that the condition
Bi =Ai+ I is necessary and sufficient for the operator[, [Yi] be self-adjoint.
As can readily be observed, any difference equation of the form

(36) Bi Yi+l - C; Yi+ A; Yi-1 = -F;,

A;# 0, B; # 0, i = 1, 2, ... , N - 1,