1549301742-The_Theory_of_Difference_Schemes__Samarskii

210 Homogeneous Difference Sche1nes

The function v~(x) admits an alternative form

x;+1

(11) v;(x) = j G(x,~)f(~) d~,

where G(x, ~) is the Green function related to problem (4) (see (11) m

Section 6):

(12) G(x, ~) =

v~(~) v~(x)

vf (xi+1) '

v;(x) v~(~)

vf (xi+1) '

By placing successively expression (12) and x =xi in (11) we obtain

(13)

x;+1

+v:(x;) j v;(<)f(<)d()

z

By virtue of relations (7), (9) and (13) we deduce from (:3) that

(14)

where

(15)

1

d; = -~­

hv;(x;)

tp; =

x;+1

j v;(~)q(~) d~,