508 Difference Methods for Solving Nonlinear Equationsmaking it possible to set up the difference problen1 for the error z = y - u:
(3) zxx + J' (y) z = -1/J , x = ih, i = 1, 2, ... , N - 1,
Where y = 1l + B Z, 0 < B < 1, and 4• = H:r.c + f( ll) is the residual.
It seems clear that scheme (2) generates an approximation of order 2:If f'(y) < 0, then a solution of the difference problem (3) satisfies the
estimate(4) llzllc < 111/Jllc,
which serves to inotivate the uniform convergence of scheme (2) with the
rate O(h^2 ):
llzllc = llY - ullc = O(h^2 ).
What is more, a solution of the difference proble1n (2) is bounded, so that(5) llYllc < lf(O)I =Co
under the constraint f'(y) < 0. Indeed, si1nple algebra gives
f(y) = f(O) + (f(y) - f(O)) = f(O) + J'(y) y,
where y = ey,0 < e < 1, yielding
Yxx + f'(y) Y = -f(O), Yo:::: YN:::: 0.
Whence estimate (5) follows on account of (4).
In this regard; Newton's method suits us perfectly in connection with
solving the nonlinear difference equation (2). It is worth recalling here its
algorith1n:
k+l I !.: k+l k !.:
y xx+ f (y) ( y - y) = -f(y),
where k is the iteration number, k = 0, 1, 2, ... , leaving us with the three-
pomt. 1. mear d.,,.. iaerence equat10n. re 1 d ate to k+1 y :
(6) k+l YN = 0.