Page 52 -

2
Two-Point Boundary Value Problems

In Chapter 1 above we encountered the wave equation in Section 1.4.3
and the heat equation in Section 1.4.4. These equations occur rather fre-
quently in applications, and are therefore often referred to as fundamental
equations. We will return to these equations in later chapters. Another
fundamental equation is Poisson’s equation, given by

n
2
∂ u
− = f,
∂x 2
j
j=1
where the unknown function u is a function of n spatial variables x 1 ,... ,x n .
The main purpose of this chapter is to study Poisson’s equation in one
space dimension with Dirichlet boundary conditions, i.e. we consider the
two-point boundary value problem given by
−u (x)= f(x), x ∈ (0, 1), u(0) = u(1)=0. (2.1)

Although the emphasis of this text is on partial differential equations, we
must first pay attention to a simple ordinary differential equation of second
order, since the properties of such equations are important building blocks
in the analysis of certain partial differential equations. Moreover, the tech-
niques introduced for this problem also apply, to some extent, to the case
of partial differential equations.
We will start the analysis of (2.1) by investigating the analytical proper-
ties of this problem. Existence and uniqueness of a solution will be demon-
strated, and some qualitative properties will be derived. Then we will turn