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2. Two-Point Boundary Value Problems
40
our attention to numerical methods for solving this simple problem, and we
will carefully study how well the numerical solutions mimic the properties
of the exact solutions. Finally, we will study eigenvalue problems associated
with the boundary value problem (2.1). The results of this analysis will be
a fundamental tool in later chapters.
Although the equations investigated in this chapter are very simple and
allow analytical solution formulas, we find it appropriate to start our study
of numerical methods by considering these problems. Clearly, numerical
values of the solutions of these problems could have been generated without
the brute force of finite difference schemes. However, as we will encounter
more complicated equations later on, it will be useful to have a feeling for
how finite difference methods handle the very simplest equations.
2.1 Poisson’s Equation in One Dimension
In this section we will show that the problem (2.1) has a unique solution.
Moreover, we will find a representation formula for this solution.
We start by recalling a fundamental theorem of calculus: There is a
constant c 1 such that
x
u(x)= c 1 + u (y) dy, (2.2)

0
and similarly, there is a constant c 2 such that
y

u (y)= c 2 + u (z) dz. (2.3)
0
This is true for any twice continuously differentiable function u. Suppose
now that u satisfies the differential equation (2.1). Then (2.3) implies that
y

u (y)= c 2 − f(z) dz. (2.4)
0
Then, inserting this equation into (2.2), we obtain

x y
u(x)= c 1 + c 2 x − f(z) dz dy. (2.5)
0 0
In order to rewrite this expression in a more convenient form, we define

y
F(y)= f(z) dz,
0