Mathematical Methods (10/24.539)

IX. The Sturm-Liouville Problem

Orthogonality of the Eigenfunctions

The General Case

We have stated that the eigenfunctions of the Sturm-Liouville problem are orthogonal and, in Example 9.1, we have shown this property for a particular situation. However, since orthogonality is such an important property, it is important to examine its development for the general Sturm-Liouville problem. Thus, the present subsection treats the general case and much of the remainder of this section shows how Example 9.1, the Legendre polynomials, and the ordinary Bessel functions fit into the general development. These three specific illustrations cover the most common situations that occur in practical applications.

To start the proof of orthogonality for the general case, we simply rewrite the general Sturm-Liouville problem [see eqns. (9.1) and (9.2)] using a shorthand notation, or

with boundary conditions

We now let ym(x) and yn(x) be eigenfunctions for two different eigenvalues, and (i.e. ). Then the defining equations for ym(x) and yn(x) are:

Multiplying eqn. (9.3) by yn and eqn. (9.4) by ym and subtracting the resultant expressions give

or

The right hand side (RHS) of eqn. (9.5) can be written as

One can show this latter relationship by performing the indicated operations, or

and, since the first and third terms cancel, we are left with the expression in eqn. (9.6).

Now combining eqns. (9.5) and (9.6) and integrating over the interval give

Since , for orthogonality, the right hand side of eqn. (9.7) must vanish. Thus, we see that the boundary conditions usually play an important role in establishing orthogonality (as well as in defining the eigenfunctions originally). Note that many different combinations for the BCs at the two boundary points, a and b, will force the RHS of eqn. (9.7) to zero. However, for the special case of r(a) and r(b) both zero, the boundary conditions play no role in showing orthogonality - and, for this case, the solutions of the Sturm-Liouville problem are orthogonal independent of the boundary conditions imposed on the problem. Also note that, in general, orthogonality is with respect to the weight function p(x). These observations are very important - and the student should have a good understanding of the development and use of the general orthogonality conditions implied in eqn. (9.7). The following three cases illustrate its use to establish orthogonality for three different situations.

Example 9.1 Revisited

In Example 9.1, the ODE of interest was

This is a specific case of a general Sturm-Liouville problem with

Using eqn. (9.7) to establish orthogonality gives

and since the boundary conditions are y(0) = 0 and , every term on the right hand side vanishes identically. Therefore, , as shown previously for Example 9.1.

Legendre Polynomials

Recall that Legendre’s equation was written as

but this is equivalent to

Therefore, this is a Sturm-Liouville problem with

Also we note that the range of interest here is with a = -1 and b = 1 and that

Equation (9.7) shows that the solutions to Legendre’s equation are indeed orthogonal and that no specific boundary conditions are needed to force orthogonality [since r(a) and r(b) are already both zero].

Ordinary Bessel Functions

The ordinary Bessel equation is given as

but this is equivalent to (dividing by x)

Therefore, this is a Sturm-Liouville problem with

Thus, orthogonality will be with respect to the weight function p(x) = x and the boundary conditions must be such that the right hand side of eqn. (9.7) vanishes.

To elaborate a little, let’s limit our discussion to integer order Bessel functions, or let (this is the usual case). Then the eigenvalues, , represent the infinite number of values of for for the ordinary Bessel functions of order n that satisfies the specific boundary conditions for a given problem.

Writing , we see that eqn. (9.7), for this case, becomes

Clearly, since both boundary points, a and b, cannot be zero, orthogonality can only be specified by appropriate boundary conditions on the problem. As will be seen from Example 9.2, the specific values of that satisfy these conditions are the eigenvalues of the problem and they will be related to the zeros of the Jn(x) function - that is, is interpreted as the mth value of x such that Jn(x) = 0. The reader should see the previous discussion on Bessel functions in Section VIII for more information [for example, one should recall that Jn(x) has an infinite number of zeros, etc,]. Also, one should study Example 9.2 in detail as a specific application that illustrates orthogonality for the ordinary Bessel functions.

10/24.539 Lecture Notes by Dr. J. R. White, UMass-Lowell (updated November 1998).

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