In this chapter we’ll be taking a quick and very brief look
at a couple of topics. The two main
topics in this chapter are Boundary Value Problems and Fourier Series. We’ll also take a look at a couple of other
topics in this chapter. The main point
of this chapter is to get some of the basics out of the way that we’ll need in
the next chapter where we’ll take a look at one of the more common solution
methods for partial differential equations.

It should be pointed out that both of these topics are far
more in depth than what we’ll be covering here.
In fact you can do whole courses on each of these topics. What we’ll be covering here are simply the
basics of these topics that well need in order to do the work in the next
chapter. There are whole areas of both
of these topics that we’ll not be even touching on.

Here is a brief listing of the topics in this chapter.

**Boundary Value Problems** In this section we’ll define the boundary
value problems as well as work some basic examples.

**Eigenvalues and Eigenfunctions** Here we’ll take a look at the eigenvalues and
eigenfunctions for boundary value problems.

**Periodic Functions and Orthogonal Functions**
We’ll take a look at periodic functions and
orthogonal functions in section.

**Fourier Sine Series** In this section we’ll start looking at Fourier
Series by looking at a special case : Fourier Sine Series.

**Fourier Cosine Series** We’ll continue looking at Fourier Series by
taking a look at another special case : Fourier Cosine Series.

**Fourier Series** Here we will look at the full Fourier series.

**Convergence
of Fourier Series** Here we’ll take a look at some ideas involved
in the just what functions the Fourier series converge to as well as
differentiation and integration of a Fourier series.