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2. Two-Point Boundary Value Problems
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In fact, the basic idea of any finite difference scheme stems from a very
familiar definition; the definition of the derivative of a smooth function:
u(x + h) − u(x)

u (x) = lim
.
h
h→0
This indicates that in order to get good approximations, h must be suffi-
ciently small. Typically, the number of unknowns in the algebraic system is
of order O(1/h). Thus, in order to compute good approximations, we have
2
to solve very large systems of algebraic equations. From this point of view,
3
the differential equation may be regarded as a linear system of infinitely
many unknowns; the solution is known at the endpoints and determined
by the differential equation in the interior solution domain.
In this section we will introduce a finite difference scheme approximating
a two-point boundary value problem. We shall observe that such schemes
can provide quite accurate approximations, and that they are, in fact, very
simple to deal with on a computer. A more elaborate analysis of the ap-
proximation properties will be the topic of subsequent sections.
2.2.1 Taylor Series
In order to define the finite difference approximation of problem (2.1), we
first recall how Taylor’s theorem can be used to provide approximations of
derivatives. Assume that g = g(x) is a four-times continuously differentiable
function. For any h> 0wehave
h 2 h 3 h 4

g(x + h)= g(x)+ hg (x)+ g (x)+ g (3) (x)+ g (4) (x + h 1 ),

2 6 24
where h 1 is some number between 0 and h. Similarly,
h 2 h 3 h 4

g(x − h)= g(x) − hg (x)+ g (x) − g (3) (x)+ g (4) (x − h 2 ),

2 6 24
for 0 ≤ h 2 ≤ h. In particular, this implies that
g(x + h) − 2g(x)+ g(x − h)

= g (x)+ E h (x), (2.11)
h 2
where the error term E h satisfies
M g h 2
|E h (x)|≤ . (2.12)
12
2
The O notation is discussed in Project 1.1.
3
This is currently a very active field of research, and the advent of high-speed com-
puting facilities has dramatically increased the applicability of numerical methods. In
fact, the numerical solution of partial differential equations has been a major motivation
for developing high-speed computers ever since World War II. A thorough discussion of
this issue can be found in Aspray [2].