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.