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Differential Equations - Notes

 Partial Differential Equations

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.

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