1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 19

The solution of problem (32) can be 1nost readily evaluated with the aid of

the relation llYc < llYc, valid an account of Theorem 2. Let the 111aximu1n

of the function Yi be arrived at a point i = ·i 0. Then

and (32) implies the double inequality

In the case Di 0 > 0, it follows from the foregoing that

(33)

For D; 0 = 0 we deduce from ( 32) that

With the relations Y; 0 > Y; 0 _1 and }'.; 0 > Y; 0 +1 in view, we find that

it being understood that the same maximal value is attained at the adjacent
to i 0 points.
By merely setting i = i 1 = i 0 +1 (or i 1 = i 0 - 1) we follow established
practice and obtain, as a final result, the inequality

giving either inequality (33) or the equality Y; 1 + 1 = Yi 1 -1 = Y; 1. As
Di ::j. 0, we get D;o: > 0 and inequality (33) for some i = io:, thereby
completing the proof.

Theorem 4 The solution y; of problem (25) with coefficients (22) admits
the estimate

(34 ,)

0

where Y is the solution of problem (28).