5.1.1 Boundary value problems
We have encountered several different eigenvalue problems such as:
with different boundary conditions
For example for the insulated wire, Dirichlet conditions correspond to applying a zero temperature at the ends, Neumann means insulating the ends, etc…. Other types of endpoint conditions also arise naturally, such as the Robin boundary conditions
for some constant . These conditions come up when the ends are immersed in some medium.
Boundary problems came up in the study of the heat equation when we were trying to solve the equation by the method of separation of variables. In the computation we encountered a certain eigenvalue problem and found the eigenfunctions . We then found the eigenfunction decomposition of the initial temperature in terms of the eigenfunctions
Once we had this decomposition and found suitable such that and were solutions, the solution to the original problem including the initial condition could be written as
We will try to solve more general problems using this method. First, we will study second order linear equations of the form
Essentially any second order linear equation of the form can be written as (5.1) after multiplying by a proper factor.
Example 5.1.1 (Bessel): Put the following equation into the form (5.1):
Multiply both sides by to obtain
The so-called Sturm-Liouville problem1 is to seek nontrivial solutions to
In particular, we seek s that allow for nontrivial solutions. The s that admit nontrivial solutions are called the eigenvalues and the corresponding nontrivial solutions are called eigenfunctions. The constants and should not be both zero, same for and .
Suppose , , and are continuous on and suppose and for all in . Then the Sturm-Liouville problem (5.2) has an increasing sequence of eigenvalues
and such that to each there is (up to a constant multiple) a single eigenfunction .
Moreover, if and , then for all .
Problems satisfying the hypothesis of the theorem are called regular Sturm-Liouville problems and we will only consider such problems here. That is, a regular problem is one where , , and are continuous, , , , and . Note: Be careful about the signs. Also be careful about the inequalities for and , they must be strict for all !
When zero is an eigenvalue, we usually start labeling the eigenvalues at 0 rather than 1 for convenience.
Example 5.1.2: The problem , , , and is a regular Sturm-Liouville problem. , , , and we have and . The eigenvalues are and eigenfunctions are . All eigenvalues are nonnegative as predicted by the theorem.
Exercise 5.1.1: Find eigenvalues and eigenfunctions for
Identify the . Can you use the theorem to make the search for eigenvalues easier? (Hint: Consider the condition )
Example 5.1.3: Find eigenvalues and eigenfunctions of the problem
These equations give a regular Sturm-Liouville problem.
Exercise 5.1.2: Identify in the example above.
First note that by Theorem 5.1.1. Therefore, the general solution (without boundary conditions) is
Let us see if is an eigenvalue: We must satisfy and , hence (as ), therefore, 0 is not an eigenvalue (no nonzero solution, so no eigenfunction).
Now let us try . We plug in the boundary conditions.
If , then and vice-versa, hence both are nonzero. So , and . As we get
Now use a computer to find . There are tables available, though using a computer or a graphing calculator is far more convenient nowadays. Easiest method is to plot the functions and and see for which they intersect. There is an infinite number of intersections. Denote by the first intersection, by the second intersection, etc…. For example, when , we get that , , …. That is , , …. A plot for is given in Figure 5.1. The appropriate eigenfunction (let for convenience, then ) is
When we get (approximately)
We have seen the notion of orthogonality before. For example, we have shown that \(\sin(nx)\) are orthogonal for distinct on . For general Sturm-Liouville problems we will need a more general setup. Let be a weight function (any function, though generally we will assume it is positive) on . Two functions , are said to be orthogonal with respect to the weight function when
\[ \int_a^b f(x)g(x)r(x)\, dx =0.\]
In this setting, we define the inner product as
and then say and are orthogonal whenever . The results and concepts are again analogous to finite dimensional linear algebra.
The idea of the given inner product is that those where is greater have more weight. Nontrivial (nonconstant) arise naturally, for example from a change of variables. Hence, you could think of a change of variables such that .
We have the following orthogonality property of eigenfunctions of a regular Sturm-Liouville problem.
|Suppose we have a regular Sturm-Liouville problem |
Let \(y_i\) and \(y_k\) be two distinct eigenfunctions for two distinct eigenvalues \(\lambda_j\) and \(\lambda_k\). Then
\[ \int_a^b y_i(x)y_k(x)r(x)\, dx =0.\]
that is,\(y_i\) and \(y_k\) are orthogonal with respect to the weight function \(r\).
Proof is very similar to the analogous theorem from § 4.1.
We also have the Fredholm alternative theorem we talked about before for all regular Sturm-Liouville problems. We state it here for completeness.
Theorem 5.1.3 (Fredholm Alternative)
Suppose that we have a regular Sturm-Liouville problem. Then either
has a nonzero solution, or
has a unique solution for any \(f(x)\) continuous on \([a,b]\).
This theorem is used in much the same way as we did before in § 4.4. It is used when solving more general nonhomogeneous boundary value problems. The theorem does not help us solve the problem, but it tells us when a unique solution exists, so that we know when to spend time looking it. To solve the problem we decompose and in terms of the eigenfunctions of the homogeneous problem, and then solve for the coefficients of the series for .
where the eigenfunctions. We wish to find out if we can represent any function in this way, and if so, we wish to calculate (and of course we would want to know if the sum converges). OK, so imagine we could write as (5.3). We will assume convergence and the ability to integrate the series term by term. Because of orthogonality we have
Note that are known up to a constant multiple, so we could have picked a scalar multiple of an eigenfunction such that (if we had an arbitrary eigenfunction , divide it by ). When we have the simpler form as we did for the Fourier series. The following theorem holds more generally, but the statement given is enough for our purposes.
Theorem 5.1.4. Suppose is a piecewise smooth continuous function on . If are the eigenfunctions of a regular Sturm-Liouville problem, then there exist real constants given by (5.4) such that (5.3) converges and holds for .
Example 5.1.4:Take the simple Sturm-Liouville problem
The above is a regular problem and furthermore we actually know by Theorem 5.1.1 that .
Suppose , then the general solution is , we plug in the initial conditions to get , and , hence is not an eigenvalue. The general solution, therefore, is
Plugging in the boundary conditions we get and . cannot be zero and hence . This means that must be an odd integral multiple of , i.e. . Hence
We can take . Hence our eigenfunctions are
Finally we compute
So any piecewise smooth function on can be written as
Note that the series converges to an odd -periodic (not -periodic!) extension of .
Exercise 5.1.3 (challenging): In the above example, the function is defined on , yet the series converges to an odd -periodic extension of . Find out how is the extension defined for .
This problem is not a Sturm-Liouville problem, but the idea is the same.
Hint: First write the system as a constant coefficient system to find general solutions. Do note that Theorem 5.1.1 guarantees .
Exercise 5.1.102: Put the following problems into the standard form for Sturm-Liouville problems, that is, find , , , , , , and , and decide if the problems are regular or not.
a) for , , ,
b) for , , .