1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference Green's function

we arrive at

(22)

By the same token,

1

Go:v(x,~) < -.

C1

205

Let v(x,~) = G 0 (x,~) - G(x,~). The equations for Go and G imply

that

Aa v(x, ~) = -d(x) Ga(x, ~), v(O,~) = v(l,~) = 0.

We can calculate the left difference derivative of both sides of this equation

with respect to ~, whose use permits us to establish for w( x, ~) = V[

Axw(x,~) = (a(x)w;;(J:,~))x· -d(x)w(x,~) = -d(x)Go[(x,~),

w(0,~)=0, w(l,~)=0.

By the lemma from Chapter 1, Section 1,

(23)

1

rnax x I w(x, ~)I::; max x I Gof(x, ~)I::; -c ·

1

Using estimates (22) and (23) behind, we derive frorn the obvious inequality

IG[(x,~)I < IGo[(x,01+ jw(x,~)I

the desired estimate ( 20).

Theorem For a solution of problem (6)-(7) the estimates a.re valid:

N-1

(24) 2= hips

s=z

(25)

When 1P( x) happens to be of the form lp = T/x + lp*, a solution of problem
( 6 )-(7) satisfies the estimate

(26)

where f-li = I:t-::\ h 1P'k for i = 2, 3 ... , N and f-li = 0.