In this chapter we are going to take a very brief look at
one of the more common methods for solving simple partial differential
equations. The method we’ll be taking a
look at is that of Separation of Variables.
We need to make it very clear before we even start this
chapter that we are going to be doing nothing more than barely scratching the
surface of not only partial differential equations but also of the method of
separation of variables. It would take
several classes to cover most of the basic techniques for solving partial
differential equations. The intent of
this chapter is to do nothing more than to give you a feel for the subject and
if you’d like to know more taking a class on partial differential equations
should probably be your next step.
Also note that in several sections we are going to be making
heavy use of some of the results from the previous chapter. That in fact was the point of doing some of
the examples that we did there. Having
done them will, in some cases, significantly reduce the amount of work required
in some of the examples we’ll be working in this chapter. When we do make use of a previous result we
will make it very clear where the result is coming from.
Here is a brief listing of the topics covered in this
chapter.
The Heat Equation 
We do a partial derivation of the heat
equation in this section as well as a discussion of possible boundary values.
The Wave Equation Here we do a partial derivation of the wave
equation.
Terminology In this section we take a quick look at some
of the terminology used in the method of separation of variables.
Separation of Variables We take a look at the first step in the method
of separation of variables in this section.
This first step is really the step that motivates the whole process.
Solving the Heat Equation In this section we go through the complete
separation of variables process and along the way solve the heat equation with
three different sets of boundary conditions.
Heat Equation with Non-Zero Temperature
Boundaries Here we take a quick look at solving the heat
equation in which the boundary conditions are fixed, non-zero temperature
conditions.
Laplace’s Equation We discuss solving Laplace’s equation on both
a rectangle and a disk in this section.
Vibrating String Here we solve the wave equation for a
vibrating string.
Summary of Separation of Variables In this final section we give a quick summary
of the method of separation of variables.