Some variants of the elimination method 35
making it possible to rearrange the problem statement and boundary con-
ditions (61)-(62) as(66) w; - w;+1 - di Yi = -fi, i = 1, 2, ... , N - l,
Substituting Yi = Z./i+l + w;+ 1 /a;+ 1 fron1 (65) into the first formula (64)
yields(68)To make ou1: exposition more transparent, we introduce Cl'; = a; (1 - ii';)
and /i = a; {3;. With these, equation (8) admits the simplified form(69) Cl; Yi + w; = Ii ,
showing the new notations to be sensible ones. Having completed the elim-
ination of Yi from (66) and (69), we arrive at
(70)In so doing Cl'i and /i are recovered from the recurrence relations
or(71)or
(72)(. ) a;+ 1 [a;(l-il';)+d;]
Cl'i+l = ai+l 1 - Cti+l = a,+1. + a,. ( 1 - Cl'z. ·) + d· ·z
1 +(Cl;+ d;) I a;+1 '
· · a;+1 (Ii+ f;)
/i+1 = a;+1 f3;+1 = 6'i+1 ( ai f3i + f;) = ---------
a;+1 +a; (1 - ii';)+ d;/i + f;
By comparing the first boundary condition (67) with (69) for i = 1 we find
that(73)