Abstract
The main purpose of this paper is to present high order discretization methods for a two point boundary problem,
y″ =
f(
x,
y,
y′);
y(
a) =
y
a
,
y(
b) =
y
b
where
f
y
> 0. Two methods are given. The former is a four step method with error ∥
e∥
∝ =
Ch
4, the latter is a six step method with error ∥
e∥
∝ =
Ch
6. We show that the error estimates are sharp and are less accurate by a factor of
O(
h
2) than the corresponding methods constructed for the problem
y″ =
f(
x,
y);
y(
a) =
y
a
,
y(
b) =
y
b
. As a by-product we show that these results can be applied to find the necessary conditions to determine the extremal value of ∝
a
b
g(
x,
y,
y′)
dx. Computer results are given to illustrate our results. We also give “start up” methods which are required for these algorithms and show they satisfy the errors described above.