Summary: TWO-POINT BVP
Consider the two-point boundary value problem of a
second-order linear equation:
Y 00(x) = p(x) Y 0(x) + q(x) Y (x) + r(x)
a x b
Y (a) = g1; Y (b) = g2
Assume the given functions p, q and r are continuous
on [a; b]. Unlike the initial value problem of the equa-
tion that always has a unique solution, the theory of
the two-point boundary value problem is more com-
plicated. We will assume the problem has a unique
smooth solution Y ; a su cient condition for this is
q(x) > 0 for x 2 [a; b].
In general, we need to depend on numerical methods
to solve the problem.
FINITE DIFFERENCE METHOD
We derive a nite di erence scheme for the two-point
boundary value problem in three steps.
Step 1. Discretize the interval [a; b].
Let N be a positive integer, and divide the interval