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2.2 A Finite Difference Approximation
45
In order to state the next property, we introduce a norm on the set

C [0, 1] . For any function f ∈ C [0, 1] , let
||f|| ∞ = sup |f(x)|.
x∈[0,1]
The scalar value ||f|| ∞ , which we will refer to as the sup-norm of f,
measures, in some sense, the size of the function f. Let us look at some
examples clarifying this concept.
√
x
. The sup-
Example 2.4 Let f(x)= x, g(x)= x(1 − x), and h(x)= e
norm of these functions, considered on the unit interval [0, 1], are given by
f ∞ =1, g ∞ =1/4, and finally h ∞ = e.
The following result relates the size of the solution u of the problem (2.1)
to the size of the corresponding data given by the right-hand side f.

Proposition 2.2 Assume that f ∈ C [0, 1] and let u be the unique solu-
tion of (2.1). Then
||u|| ∞ ≤ (1/8)||f|| ∞ .
Proof: Since G is nonnegative, it follows from (2.9) that for any x ∈ [0, 1],

1
|u(x)|≤ G(x, y)|f(y)| dy.
0
From the definition of ||f|| ∞ above, it therefore follows that
1

1
G(x, y) dy = ||f|| ∞ x(1 − x),
|u(x)|≤||f|| ∞
2
0
and hence
||u|| ∞ = sup |u(x)|≤ (1/8)||f|| ∞ .
x∈[0,1]

2.2 A Finite Difference Approximation
The basic idea of almost any numerical method for solving equations of the
form (2.1) is to approximate the differential equation by a system of alge-
braic equations. The system of algebraic equations is set up in a clever way
such that the corresponding solution provides a good approximation of the
solution of the differential equation. The simplest way of generating such
a system is to replace the derivatives in the equation by finite differences.