1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference equations

where i^1 = i + l. For Yi = Yo+ (y 1 - y 0 ) =Yo + \7 Yi we arrive at

N

(y, 6. v) = YN VN - Yo Vi - L Vi V Yi = -( v, 6. Y] + YN VN - vi Yo·

i:=l

31

If Yi vanishes at the boundary grid nodes i = 0 and i = N, that is, either

Yo = 0, UN = 0 or u 0 = 0, vN = 0, then ( 48) can be rewritten in the form

( 49) (y, 6.v) = -(v, vy].

The identities obtained above are frequently encountered in difference trans-

formations and calculating various finite sums and series. We give below

some examples of such applications.

Example 1 It is necessary to calculate the sum SN

setting Yi = i and 6. Vi = 2i so that

Vi+l. -. + - v, 2i -- "\' L., ' qk L, + -Vo - 2i+l -^1 + Vo

k:=O

"\'N L.,i:=l z · 2 i ' By

and choosing v 0 in such a way that v N+i = U we are led by formula ( 48) to

N N N+l

L i 2i = L Yi 6. V; L Viv Yi+ YN+1 VN+1 - Yo V1

i=l i=l i= I

= ( N - 1) 2N +^1 + 2.

Example 2 Of interest is the sum SN = L;:~^1 i ai. In this case y,: = z,

6.vi = ai, v; = (ai -aN)/(a- l), vN =0 and

SN= (

1

a-l )^2 [aN(N(a-l)-a)+a].

13. Green's difference formulae. For the simplest operator Lu = u^11 the
    following identities are valid:

b b

ju v^11 clx = - j u^1 v^1 clx + u v^1 1:,

a (l

b

j ( u v^11 - u^11 v) dx ( u v^1 - u^1 u ) I ~ ,

a