1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations 23

Since la 1 + 1 I< l under conditions (39), formulae ( 40) imply that

yielding, in turn,

N

( 41) IYil< L l/3kl,

k=i+l

Substitution /i = ai /Ji leads to the useful relations

i z

11;+11 < l1il+ l<tJil < 1111+ 2= l<tJkl = 2= l<tJkl

k=l k=l

and, consequently,

i = 2, 3, ... , N,

Combination of ( 41) with the foregoing allows us to establish (38).

Before going further, observe that a solution to the problem

(42) Ay;=-<pi, i = l, 2, ... , 1V - 1;

can be majorized by

As stated in Corollary 3, a solution to the problem

A Yi = 0) i= 1,2, ... ,N-1;

with ai > 0 and Ci > ai + a1+1 can be most readily evaluated by the

quantities

( 43)

The solution of problem (42) arranges itself as a sum Yi =Yi+ vi, where
the second summand is a solution of problem (37). Applying estimates (38)
and (43), and the inequality IYil < IYil+ lvil we come to the desired result.