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Let
be the approximations of a
solution y of (2.1) at .Let the (exact)solution y be twice
continually differentiable on [a,b],
. Further let
and
where L and M are positive constants. Then
|
(2.3) |
Proof:
By the mean value we have,
we also know,
Substraction, now leads to,
|
(2.4) |
Again, by mean value theorem
where d lies between and Substituting in (2.4)we
get
L |
(2.5) |
Let be the solution of the difference equation
|
(2.6) |
Claim:
The claim follows by induction. Also by induction
The theorem now is proved once we notice
Remark: Inequality (2.3) implies that the error is .
Theorem also given an upper band for the error.
Next: 3. Runge-Kutta Method
Up: 2. Error Estimates and
Previous: 2. Error Estimates and
root
2006-02-16