1549301742-The_Theory_of_Difference_Schemes__Samarskii

16 Preliminaries

Corollary 2 Under conditions (22) the problem

(23) [, [Yi] = 0 , i = 1, 2, ... , N - l; Yo = 0, YN = 0,

has the unique solution Yi _ 0, thereby justi[yingg that problem (21) JS

uniquely solvable for any ingredients Fi, μ 1 and μ 2.

In fact, assuming that the solution Yi of problem (23) becomes nonzero

at least at one point i = i, we come to a contradiction with the maximum

principle: if Yi. > 0, then Yi attains its maximal positive value at some

point i 0 , 0 < i 0 < N, violating with Theorem 1; the case Yi. < 0 may be

viewed on the same footing.

Theorem 2 (Comparison theorem). Let conditions (22) hold, Yi be the

solution of problem (21) and Yi be the solution to the following proble1n:

£[iii]= -F;, i = 1, 2, ... ,N-1;

Yo= μ1' YN = μ2'

with

IFil<Fi, i=l,2, ... ,N-1; lμ1l<f11, IP2I < f12.

Then the relations occur:

i=O,l, ... ,N.

Proof Due to Corollary 1 we have Yi > 0 for 0 < i < N, since [, [Yi] =
-F; < 0 and y 0 > 0, yN > 0. Observe that the functions ui = Yi - Yi
and Vi = Yi +Yi satisfy equation (21) with the right parts Fi - Fi > 0 and
Fi+ F; > 0, and boundary values u 0 = j1 1 - μ 1 > 0, uN = j1 2 - μ 2 > 0 and
v 0 = j1 1 + μ 1 > 0, vN = j1 2 + μ 2 > 0, respectively. By applying Corollary 1
to such a setting we get Ui > 0, Vi > 0 or -yi < Yi < Yi, meaning I Yi I < Yi.
The function Yi is called a major ant for the solution of problem
(21). A first step towards the solution of problem (21) is connected with a

major ant Yi: I I Ye < 11 Ye.

Corollary 3 The solution to the problem

[, [Yi] = 0 , O<i<N;

can be majorized as