Problem Description:

Let’s illustrate many of the concepts and statements associated with typical eigenvalue problems with an example that demonstrates several important items:

1. Analytical technique for finding eigenvalues and eigenfunctions

2. Normalization of the eigenfunctions

3. Orthogonality of eigenfunctions for this particular case

4. Method for expanding arbitrary functions in sets of orthonormal eigenfunctions - an example of Generalized Fourier Series Eigenfunction expansions

In particular, consider the following homogeneous equation with homogeneous boundary conditions (this just happens to represent the physics of a vibrating elastic string):

with

Problem Solution:

I. Show that this is a special case of a Sturm-Liouville Problem:

We first let . With these substitutions, eqn. (9.1) becomes

or

which is the desired ODE for this problem.

Now with boundary points , the coefficients in the general boundary condition equation [see eqn. (9.2)] become

With these substitutions, the BCs in the general equation can be written simply as

which again are the desired conditions for this problem. Thus, the current situation is indeed a special case of the general Sturm-Liouville problem.

II. Find the eigenvalues and eigenfunctions for this problem:

In trying to determine the solution to this problem, there are three possibilities - the eigenvalues might be negative, zero, or positive (note that, since the coefficient functions are real, we already know that the eigenvalues will be real). Let’s try each possibility:

This is just a simple, constant coefficient, second-order ODE with characteristic equation

and roots

Thus, the general solution for the negative eigenvalue assumption is

Applying the first boundary condition gives

Similarly, the second boundary condition gives

Therefore, for real , , and the only solution is the trivial solution, . Thus, for this problem, letting be negative was not a good choice.

Applying the first boundary condition gives

Using the same logic, the second boundary condition gives

Therefore, as for Case 1, the only solution for the case of zero eigenvalues is the trivial solution, .

This time the characteristic equation is

with roots

With pure imaginary roots, the general solution can be written as

or, more commonly, one writes

Using this latter expression and the boundary condition at x = 0 gives

Therefore, the general solution reduces to

Now using the boundary condition at gives

Thus, the eigenvalues for this problem are or, in terms of the parameter,, in the original equation, we have . Finally, the eigenfunction corresponding to the n^th eigenvalue is simply

III. Show that the eigenfunctions for this problem are orthogonal:

For this problem, the definition of orthogonality requires the following equality for :

From a set of integral tables (for ), we have

and for our case, this becomes

However, for all integer p (including negative values). Therefore,

which proves the above expression.

IV. Find the set of normalized eigenfunctions [i.e. ] for this problem:

Therefore, if we let

then an orthonormal set of eigenfunctions, g_n(x), results. This is shown explicitly by the normalization expression,

Clearly, for the normalized functions, the norm, , is unity (as designed).

V. Finally, as an example of using a Fourier series expansion, let’s expand a few simple functions, f(x), using the orthonormal basis functions defined above. With the Fourier series, the function f(x) can be written as

Case 1: The Constant Function

For example, if f(x) is unity over the interval , we have

where , since the g_n(x) have already been normalized.

To find the expansion coefficients we multiply f(x) by and integrate over the domain of interest, giving

However, since the eigenfunctions are orthonormal functions, the RHS reduces to a_m and we have,

But , therefore,

and

or

Note also that this can be written as

where now the indexing is simply incremented by unity and only the nonzero terms are included.

This particular evaluation of the Fourier series is not very elegant or efficient, but it is easy to see exactly how the computations are performed. As the number of terms used in the expansion increases, we would expect to see better and better agreement with the desired function (a constant line at a numerical value of unity for this case). This is the general trend that is observed in Fig. 9.1 in the center of the desired interval but, at the end points, this can never occur because all the expansion functions are identically zero at the interval boundary points. Thus, in this case, we can never have an exact representation over the entire interval, even with an infinite number of terms in the expansion.

Fig. 9.1 Some partial sums for the Fourier series for constant f(x).

Case 2: The Quadratic Function

As another example, let’s assume that f(x) varies quadratically over the interval with zero endpoint values. In particular, letting , we have

where again , since the g_n(x) have already been normalized.

As before, to find the expansion coefficients we multiply f(x) by g_m(x), integrate over the domain of interest, and use the fact that the eigenfunctions are orthonormal to give

Using the equalities that and for integer m, evaluation of the above expressions at the endpoints gives

Finally, the finite Fourier series representation can be written as

where the indexing properly treats only the nonzero coefficients.

For the Case 2 quadratic relationship, , 15 terms were needed for convergence to within a relative tolerance of 0.001, and the final converged expansion function after 15 terms is plotted in Fig. 9.2. Note that, this time, the Fourier expansion agrees exactly with the desired f(x). This was expected because the function endpoints and the eigenfunction endpoint are identical and, with enough terms in the convergent series, we expect to get an exact match.

In summary, the two examples given here show the two most common situations that occur with Generalized Fourier Series expansions. If the boundary points of the desired function match the eigenfunctions, then convergence over the whole domain is expected. However, if the endpoint values do not match, complete convergence can never be achieved. Also, concerning an algorithm for evaluating the series, the one illustrated in Table 9.2 for the Case 2 example is clearly the better method - since it only uses as many terms as necessary for the desired level of accuracy.

Fig. 9.2 The converged Fourier series for quadratic f(x).