1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference Green's function 203

as required.

Formula (11) means that Gik > 0 for i, k f 0, N and Gik = Gki·

Moreover, Gik as a function of k for fixed i satisfies the conditions

A

,, {jik

(k) \Jik = --h ' Gia= G;N = 0.

In the case of the boundary conditions Yo = x 1 y 1 and YN = x 2 YN-l

the Green function can be constructed in a similar way. Of interest is the

special case where d(x) - 0. Here the functions CY= ~(x) and j3 = fJ(x)

can be determined frorn equations (10) in the explicit form:

( 13) ~-~!:_ 1-L...,; '

s = 1 a s

and the Green function related to the problen1

(14) Yo= 0, YN = 0,

reduces to

(15)

i h N hi N h

2=~2=~ 2=~,

s=l s s=k+l s s=l s

k h

L a 3

s=l s=i+l

N

2=

• h IN 2=-h
  as s=l as '

i > k.

For the best scheme (15), (17) ansmg in Section 2 Gik coincides on

the grid wh with Green's function for a differential equation.

2. A priori esti1nates. The explicit representation (8) of a. solution of
   problem (6)-(7) in tenns of Green's function lies in the background of
   various a priori estimates of a solution in tenns of the right-hand side. It
   is easily seen from (8) that

(16)

N-1
IY;l<(G;k,l\Okl)= L G;kl\Oklh
k=l
and, therefore, a {iniform estimate of Yi can be obtained by estimating
max;, k G;k. In this line, \Oi arranges itself as
i-1
(17) \0 = T),,. ' T/1 = 0 ' TJi = 2= h \Os , i = 2, 3, ... , N.
s=l
Upon substituting these expressions into formula (8) we deduce by the
summation by parts formula that

(18) y(x) = (G(x,~), TJ() = -(G[(x,~), TJ(~)],

The next step is connected with I G[ ( x, ~)I·