Hybrid Methods for Initial Value Problems in Ordinary Differential ...

Define
HYBRID METHODS FOR INITIAL VALUE PROBLEMS
X(6) = lnax / y'(x*) - y1(z)1.
1 s-s* 116
x,s*€ [cbl
X(6) exists and -+ 0 as 6 4 0, since dy/& = f(x, y(x)) exists and is con-
tinuous in the closed interval [a,b]. Hence
and
where
yl(xn-i) = y '(~n)+ OiX(ih),
y(~-i)= y(xn) - ih[yl(xn)+ 6i1z(ih)],
[ 0 1 1, 1 0,' [ 5 1, and x, , xn-i E [a,bl.
Now
LhP(xn, y) = ~Ll,n+l - Y(xn+l)
But
Therefore
Similarly
+ hX(kh)B,
where B is bounded as h -+ 0.
- y(xn> - h[y(xn)+ ~ LIx(~)].
LhP(x,, y) = O(h)4 0 as h -+ 0.
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