1549301742-The_Theory_of_Difference_Schemes__Samarskii

32 Preliminaries

which are called the first and the second Green formulae. Usually the

first formula can be modified into a more general form, namely

b b b

(50) ju Lv dx = -j k u' v' dx j q u v dx + k u v' 1:,

a a a

with Lv := (kv')' - qv.

Making here the mutual replacen1ents of u(x) and v(x) and subtracting

the resulting equation from (50), we establish the second Green formula in

a more general form

b

( 51) j ( u Lv - v Lu) dx k ( u v' - u' v) I~.

a

If, in addition, u and v vanish at the end points x = a and x = b, then all

preceding substitutions are equal to 0, thereby reducing formulae (50)-(51)

to

(52) (u, Lv) 0 = -(ku', v^1 ) 0 - (qu, u) 0 , (u, Lv) 0 = (v, Lu) 0

b

where as usually (u,v) 0 =fa uvdx.

In particular, we might have

(u, Lu) 0 = -(k, (u')^2 ) 0 - (q, u^2 ) 0.

Observe that the equality (u, Lv) 0 = (v, Lu) 0 means the self-adjointness
of the operator L.
In the further development of the difference analogs of formulae (50),
(51) and (52) v; = 6. u;-1 = v Ui is put together with (48):

(53) (y, 6. v u) = - (v u, v y] + y N vu N - Yo v u1.

Likewise, setting Vi = ai\Ju; one obtains instead of (53) one more useful

result

(54) (y, 6. (av u)) =-(a vu, vy] + YN aN \JUN - Yo al vu!.

Applying (54) to the difference operator

(55) A ui = 6. (a; v ui) - d; Ui = a;+1 (u;+1 - u;) - a; (u; - u;_i) - d; u;