538 Difference Methods for Solving Nonlinear Equationsone succeeds in showing that equation ( 4 7) becomesk k
-(Jbr;bg - I 'since i - E + gC^01 l(~ -17) = 0 on account of (43). Because of this, we thus
have(50)
0 k+l. E + g k lo i I {J k+l TJ + (JI ( 7/ k - 1)) {J k+l !J = (J l 0 .h" 17 {J !J k.An alternative form of equation ( 48) is( 51)
k+l k k+l k k+l k+l k k
-b E +a g b r; +a 7J b g +av b v 8 = ab 7J b g ,Indeed, from ( 48) it follows thatwhere
k+l k k k k k·
+ a v ( b vs - b vs ) - ( b E + E - a g 7J - a v vs )[k+l k k+l k k+l k+l] k
= -b E +a g b 7J +a r; b g + av b vs - F 4 ,k " k k k h" • kk k
F 4 = a g b r; + a r; b g + a lJ b u 8 + E - a r; [/ - a IJ l'sk k k
Substituting here t = ag~ + avv 8 yields F 4 = ab 7/ b g.
Having completed the elimination of oktl from (50) and (51) both,
we arrive at
( 52) (a g k + g k (^01 ) ) b k+l 17 + ( (a + (J k ) k+l k+l
1 ) r/ - (J 1 7/ b [! + au b vs
k k:
=(a+(J 1 )017bg.