1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 17

The problem of auxiliary character such as

,C [ y;] = 0, O<i<N· ,

will complernent our stucliea, for which the comparison theorem gives II Ye <

llYc· On the same grounds, Theorem 1 implies ll:iJc < jl, since the function

Yi > 0 takes its maximal positive value only at a boundary point: either

for i = 0 or for i = N.

Theorem 3 Let the conditions

(24) IAil > 0, IBil > 0,

hold for all i = 1, 2, ... , N - 1. A solution of the problem

(25) L'.[y;]=-Fi,

admits the estimate

(26)

i = 1, 2, ... , N - 1;

F

D c

Yo= 0,

To prove the desired esti1nate, the intention is to use an alternative
fonn of equation (21)

(27) C; Yi = A; Yi -l + B; Yi+ i + Fi.

Let IYil take its maximal value IYiol > 0 at a point i = i 0 , 0 < i 0 < N,

so that IYi 0 I > I Yi I for all i = 0, 1, ... , N. We are led by equation (27) for

i = i 0 to

Because of this