Sturm-Liouville Boundary Value Problem

So this seems like a good time to bring up Sturm Liouville Boundary Value Problems.  In my previous post I derived the heat equation and specifically said that we will treat all the physical parameters such as thermal conductivity as uniform throughout the wire as a simplifying assumption.  But what if we choose not to do this?  Well then, picking up the equation from my previous post, we will have..

The last two terms on the right hand side combined are G(x,t) from the previous post.  The logic behind these terms is that the heat generated inside the wire is proportional to the temperature itself with a proportionality constant q(x), and all other types of heat sources are given by h(x,t).

Now in the previous post, we said that the boundary conditions at the ends of the wire could either be kept at zero temperature or insulated and kept at a constant temperature.  What can also happen is the wire could be partially insulated, we would therefore need to combine the two boundary value problems as a generalization.

These conditions are called separated boundary conditions since one set of boundary conditions does not relate in any way to the other.  We still need an initial heat distribution of u(x,0)=f(x).

We will now use the method of separation of variables to separate the equation in terms of t and x.  We need to assume that the solution is of the form u(x,t)=X(x)T(t).  I always thought the word "assume" was kind of misleading when learning things like this.  I always thought, "Why would you assume that?"  Instead of thinking of it as an assumption, think of it as an educated guess that we are rolling with until we get nonsense, at that point that will tell us our guess is wrong.  So, plugging this in and separating the variables we get..

We need to make a logical leap.  What exactly are we saying here?  We are saying that two functions of two different variables are equal to each other at every point of x and t imaginable.  The only way this could happen is if they are both equal to a constant value.  We will denote this value with the Greek letter lambda.  Setting each side equal to lambda, we obtain.. (We will only focus on the function of x as this is the most important.)

This equation and the boundary conditions is what is known as a Sturm Liouville boundary value problem.  Notice we did a little change of notation and called the function y instead of X.  The partial derivatives also changed into normal derivatives because the function is only of one variable.  These types of functions are very important in solving the heat equation and any other type of partial differential equation where they show up.  If one does a little expanding, you can see that the Sturm-Liouville problem is actually a normal second order differential equation.

So the obvious question comes to mind, does this mean that all second order homogeneous differential equations can be put into Sturm Liouville form?  The answer to that is yes!  In much the same way you use an integrating factor to solve first order differential equations, you use an integrating factor to combine the first two terms of the differential equation.

The Sturm Liouville problem  is what is known as an eigenvalue problem.  An eigenvalue problem says that if I define an operator L[y], what values of lambda does L[y]=λy.  The operator L is called a differential operator and can be defined any way you like.  In the case of the Sturm Liouville problem L is defined as..

The r(x) in the problem is a weight function and sometimes may also be thrown in.  The function r(x) in our case will be 1.  So we can write the Sturm Loiuville problem as L[y]=λy.  To those of you who have had linear algebra, eigenvalue problems come up as Ax=λx.  Where A is the operator in this case.  

It is important to note that you must have boundary values or some sort of initial condition to be able to find the eigenvalues.

At this point, I could delve into an entire discussion on corresponding eigenfunctions and ultimately the solution to the eigenvalue problem, and I will probably do this in the future, but at the moment I just don't have the time.

One thing I want to mention before I sign off, the Sturm Liouville eigenvalue problem shown has a lot of very special properties that make it nice to deal with.  I will again go into the details in another post. Until then, see yah!