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Chapter 8 : Boundary Value Problems & Fourier Series

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 boundary conditions (as opposed to initial conditions which we should already be familiar with at this point) and the boundary value problem. We will also work a few examples illustrating some of the interesting differences in using boundary values instead of initial conditions in solving differential equations.

Eigenvalues and Eigenfunctions – In this section we will define eigenvalues and eigenfunctions for boundary value problems. We will work quite a few examples illustrating how to find eigenvalues and eigenfunctions. In one example the best we will be able to do is estimate the eigenvalues as that is something that will happen on a fairly regular basis with these kinds of problems.

Periodic Functions and Orthogonal Functions – In this section we will define periodic functions, orthogonal functions and mutually orthogonal functions. We will also work a couple of examples showing intervals on which \(\cos\left(\frac{n \pi x}{L}\right)\) and \(\sin\left(\frac{n \pi x}{L}\right)\) are mutually orthogonal. The results of these examples will be very useful for the rest of this chapter and most of the next chapter.

Fourier Sine Series – In this section we define the Fourier Sine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the odd extension for a function and work several examples finding the Fourier Sine Series for a function.

Fourier Cosine Series – In this section we define the Fourier Cosine Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} \). We will also define the even extension for a function and work several examples finding the Fourier Cosine Series for a function.

Fourier Series – In this section we define the Fourier Series, i.e. representing a function with a series in the form \(\sum\limits_{n = 0}^\infty {{A_n}\cos \left( {\frac{{n\pi x}}{L}} \right)} + \sum\limits_{n = 1}^\infty {{B_n}\sin \left( {\frac{{n\pi x}}{L}} \right)} \). We will also work several examples finding the Fourier Series for a function.

Convergence of Fourier Series – In this section we will define piecewise smooth functions and the periodic extension of a function. In addition, we will give a variety of facts about just what a Fourier series will converge to and when we can expect the derivative or integral of a Fourier series to converge to the derivative or integral of the function it represents.