1549301742-The_Theory_of_Difference_Schemes__Samarskii

204 Hon1ogeneous Difference Schen1es

Lemma For the Green function G(x,~) related to problem (6)-(7) the

uniform estimates

(19)

(20)

a.re valid for a.11 x, ~ E w h.

Proof 1) In Chapter 1, Section 1 we have obtained for a solution of problem

(6)-(7) the estimate

(21)

Replacing in (21) Yi by G;k and 1.fs by Dsk/h, we establish the relations

where

z

S;k = L Dsk , S;k = 0 for i < k, sik =^1 for i > k.

s=l

2. Let d( x) 0 and the Green function G = Go( x, ~) be specified by
   formulae (15). Assuming this to be the case, we find that

0 0

ex;;(x) {3(~)

0 for x:=;~,

ex( 1)

Go:r(x,~) =

0 0

ex(~) f3;;(J:)

0 for x > ~.

ex( 1)

By virtue of the relations

0 0 ° 0

ex(~)::; ex(l), {3(~)::; ex(l)