Chapter 9 : 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 – In this section we will do a partial derivation of the heat equation that can be solved to give the temperature in a one dimensional bar of length \(L\). In addition, we give several possible boundary conditions that can be used in this situation. We also define the Laplacian in this section and give a version of the heat equation for two or three dimensional situations.
The Wave Equation – In this section we do a partial derivation of the wave equation which can be used to find the one dimensional displacement of a vibrating string. In addition, we also give the two and three dimensional version of the wave equation.
Terminology – In this section we take a quick look at some of the terminology we will be using in the rest of this chapter. In particular we will define a linear operator, a linear partial differential equation and a homogeneous partial differential equation. We also give a quick reminder of the Principle of Superposition.
Separation of Variables – In this section show how the method of Separation of Variables can be applied to a partial differential equation to reduce the partial differential equation down to two ordinary differential equations. We apply the method to several partial differential equations. We do not, however, go any farther in the solution process for the partial differential equations. That will be done in later sections. The point of this section is only to illustrate how the method works.
Solving the Heat Equation – In this section we go through the complete separation of variables process, including solving the two ordinary differential equations the process generates. We will do this by solving the heat equation with three different sets of boundary conditions. Included is an example solving the heat equation on a bar of length \(L\) but instead on a thin circular ring.
Heat Equation with Non-Zero Temperature Boundaries – In this section we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature. Note that this is in contrast to the previous section when we generally required the boundary conditions to be both fixed and zero.
Laplace’s Equation – In this section we discuss solving Laplace’s equation. As we will see this is exactly the equation we would need to solve if we were looking to find the equilibrium solution (i.e. time independent) for the two dimensional heat equation with no sources. We will also convert Laplace’s equation to polar coordinates and solve it on a disk of radius \(a\).
Vibrating String – In this section we solve the one dimensional wave equation to get the displacement of a vibrating string.
Summary of Separation of Variables – In this final section we give a quick summary of the method of separation of variables for solving partial differential equations.