Understanding Sturm-Liouville Problems: Implications and Solutions

[thelema418]

I'm trying to understand how to solve PDEs that are Sturm Liouville problems. I've read a couple of presentations about this, but I'm lost as to the implications for the solution process.

Most descriptions discuss putting the second order differential equation into a Sturm-Liouville form. From this there are a set of implications that I can prove about ordered eigenvectors, the sign of the eigenvectors, the orthogonality of eigenfunctions, etc. While I understand these implications, I don't understand how this helps me solve a Sturm-Liouville problem.

Additionally, the examples I have seen are the most basic (like the trivial case where p=1 q=0 and r=1), and they all begin with solving the ODE. I don't understand if this is how you solve EVERY Sturm-Liouville problem, or if this is just done for the sake of demonstration of a connection?

I'm concerned with how to do cases like p=x, q=0, r=1/x and boundary conditions u(1)=u'(e)=0.

Should I be considering the ODE solution when solving this, or do something else?

 

[TachyonRunner]

I can't speak much of the implications of the spectral theorems themselves, but one of the ways that I think the Sturm-Liouville theory helps you solve problems is in demonstrating the orthogonality of the eigenfunctions. Say you find that after separation of variables the problem reduces to a Sturm-Liouville problem in one of the parameters. This suggests you do a Fourier expansion in the eigenfunctions and then just solve the infinite system of ODEs in the other variable (basically a finite Fourier Transform).

As far as how the theorems help you solve the resulting ODEs themselves, I'm not quite sure how that comes in, other than perhaps telling you the existence of solutions, or if you already know the eigenfunctions, allowing you to do Fourier expansions.