IMAGE CREDIT: NASA/JPL: This shows Jupiter’s Great Red Spot; a storm that has been occurring for over 300 years now. Quite recently, however, observations show that the Spot appears to be shrinking in size.
About a week ago, I was looking through my notebooks and came across an unfinished problem posed by one of my professors. Unfortunately, I was not able to solve the problem during the semester. However, I thought it might be something interesting to consider. I did a quick search and found that the problem he gave us was to prove the Sturm-Liouville Theorem.
The main brute-force method to analytically solving a given partial differential equation is the separation of variables. This method is heavily used by physicists and in doing so transforms the initial boundary value problem (IBVP) into a Sturm-Liouville problem in which we have an ordinary differential equation and linear homogeneous boundary conditions.
A quick statement of the problem:
Consider the following Sturm-Liouville Problem:
In the context of the problem, let . About the , I am aware that the original differential equation had a single . However, I am not sure of the implications to the problem by removing one of the , so I am including the problem as stated by my professor. So, Eq.(1.1) becomes
So the problem is to prove that if there exists two unique non-trivial eigenfunctions, such that and that for , then on the interval
The problem also has the properties
- An infinite sequence of eigenvalues: .
- For each eigenvalue there exists a unique, nontrivial (i.e. non-zero) solution , not including multiples of said solution.
Much of the background information (e.g. the properties and general form of the problem) was obtained from the following sources:
Farlow, S.J., Partial Differential Equations for Scientists and Engineers. Dover. 1993.
“Proof”*. Recall from above the Eqs. (1.4), (1.2), and (1.3). Let the following equations be indexed for :
Similarly, let the following equations be indexed for :
Consider Eqs.(i) and (iv.), let us multiply Eq.(i) by and the same with Eq.(iv.), however we multiply by . Taking the difference yields
Let us now integrate Eq.(2) over the closed interval to give
Let us focus on the first integral on the left-hand side. We can separate each term to give
We can integrate by parts for the first term
and doing the same for the second term we arrive at
Finding the difference, we see that the remaining integrals in (5) and (6) cancel leaving the terms in the braces. If we evaluate those terms over the interval we get the following
Let us suppose now that we equate Eqs.(ii) and (v.) and Eqs.(iii) and (vi) and multiply equations (ii) and (v) by and , respectively, and multiply equations (iii) and (vi) by and . For Eqs.(ii) and (v) it follows that
From which we observe that if we assume that then
The first term in the sum above vanishes which leads us to conclude that
By similar logic we find that for the second boundary condition
Therefore, returning (all the way back) to Eq.(7), we arrive at
where we assume that the eigenvalues satisfy , for . As a check, suppose that , in which case the integral (Eq.(10)) becomes
Furthermore, if , then the entire equation vanishes. Thus, this “proof”is complete. Q.E.D.
* Once again, I hesitate to call this a proof since it is not what a mathematician would call rigorous.