SELF-AD JOINT EIGENVALUE PROBLEMS

AND

THEORY

It was mentioned in Section 4.3 that if an eigenvalue problem is self-adjoint, then the eigenfunctions are orthogonal (with the proper weighting function) and the eigenvalues are real. It was also stated that eigenvalue problems based on Eq. (4.3-4) are self-adjoint. To understand such statements and to extend the method to a wider variety of problems, including conduction or diffusion in cylindrical or spherical coordinates, it is necessary to define what is a self-adjoint eigenvalue problem. An eigenvalue problem involving the independent variable ordinarily consists of a second-order differential equation in the form of along homogeneous boundary conditions of the types given in Eq. (4.3-5). That problem is said to be if the operator is such that

where and are functions which satisfy the boundary conditions of the value problem, but not necessarily the differential equation. Implicit in Eq. (4.6-1) is the specification of a certain interval [a, b] and weighting function

To illustrate the application of Eq. consider the familiar operator for which = Starting with the left-hand side of 1) and integrating

by parts, we find that

which confirms that problems involving this operator are self-adjoint. The boundary terms vanish as a consequence of the homogeneous boundary conditions satisfied

This is obvious when the boundary conditions are of the or

say, there is a Robin boundary condition at = then

= and

Thus, = any combination of the usual three types of boundary conditions gives a self-adjoint eigenvalue problem.

It is shown now that a broad class of differential equations yields self-adjoint eigenvalue problems. Consider the general, second-order, linear, homogeneous equation,

and Eq. (4.6-4) is rewritten as

Motivated by the form of Eq. (4.6-5), we choose the operator associated with some weighting function as

where it is assumed that p, q, and w are all continuous in the interval of interest, a b, and that and w for b. Assuming again that and satisfy any combination of the three types of homogeneous boundary conditions, the left-hand

side of becomes

The boundary terms vanish as before, provided that and are finite. Evaluating the right-hand side of (4.6-1) gives

The solution for the (doubly) transformed temperature is

sinh (

+

(4.5-84) sinh (

+

Using and recognizing from (4.5-84) that only the odd values of n and m will

contribute, the overall solution is found to be

In general, if there are k independent variables in a partial differential equation, then finite Fourier transforms are needed to obtain the solution,

4.6

SELF-AD JOINT EIGENVALUE PROBLEMS

AND

THEORY

It was mentioned in Section 4.3 that if an eigenvalue problem is self-adjoint, then the eigenfunctions are orthogonal (with the proper weighting function) and the eigenvalues are real. It was also stated that eigenvalue problems based on Eq. (4.3-4) are self-adjoint. To understand such statements and to extend the method to a wider variety of problems, including conduction or diffusion in cylindrical or spherical coordinates, it is necessary to define what is a self-adjoint eigenvalue problem. An eigenvalue problem involving the independent variable ordinarily consists of a second-order differential equation in the form of along homogeneous boundary conditions of the types given in Eq. (4.3-5). That problem is said to be if the operator is such that

where and are functions which satisfy the boundary conditions of the value problem, but not necessarily the differential equation. Implicit in Eq. (4.6-1) is the specification of a certain interval [a, b] and weighting function

To illustrate the application of Eq. consider the familiar operator for which = Starting with the left-hand side of 1) and integrating

by parts, we find that

which confirms that problems involving this operator are self-adjoint. The boundary terms vanish as a consequence of the homogeneous boundary conditions satisfied

This is obvious when the boundary conditions are of the or

say, there is a Robin boundary condition at = then

= and

Thus, = any combination of the usual three types of boundary conditions gives a self-adjoint eigenvalue problem.

It is shown now that a broad class of differential equations yields self-adjoint eigenvalue problems. Consider the general, second-order, linear, homogeneous equation,

and Eq. (4.6-4) is rewritten as

Motivated by the form of Eq. (4.6-5), we choose the operator associated with some weighting function as

where it is assumed that p, q, and w are all continuous in the interval of interest, a b, and that and w for b. Assuming again that and satisfy any combination of the three types of homogeneous boundary conditions, the left-hand

side of becomes

The boundary terms vanish as before, provided that and are finite. Evaluating the right-hand side of (4.6-1) gives

with boundary conditions as in are self-adjoint with respect to the weighting

function problems of this form are called problems

[see, for example, (1963) and Greenberg

The conditions given in (4.3-5) are not the only ones which yield self-adjoint eigenvalue problems. Also leading to such problems is the periodicity re- quirement,

Inspection of (4.6-7) indicates that if u and v satisfy Eq. (4.6-lo), then the boundary terms will vanish in the integration by parts, as required for a self-adjoint problem. Equation (4.6- 10) arises, for example, with problems in cylindrical coordinates involving the angle in which the domain is a complete circle. One other way a self-adjoint eigenvalue problem is obtained is if vanishes at one or both ends of the interval. If, for example, and and are both finite, then the corresponding bound-

ary term in (4.6-7) will again vanish. This type of condition arises in certain prob- lems in cylindrical or spherical coordinates (see Sections 4.7 and 4.8).

All Sturm-Liouville problems yield eigenvalues which are real and eigenfunctions which are orthogonal. A proof that A is real is given, for example, in Churchill (1963). The orthogonality of the eigenfunctions is a direct consequence of the fact that the Sturm-Liouville operator is as is shown now. Considering two different eigenfunctions and self-adjointness implies that

From the definition of the eigenvalue problem and the properties of the inner product (Section 4.3) we obtain

Subtracting (4.6-13) from leads to

Because A, this implies that

In other words, the solutions to problems are

In Sturm-Liouville problems with boundary conditions from or with at one of the endpoints, the eigenfunctions are unique. That is, there is only one linearly independent corresponding to a given eigenvalue. A proof of this, with a discussion of the exception which occurs with periodic boundary condi- tions, given by Churchill (1963).

applying the method, the most awkward term to in the

equation is that involving the second-order operator in the basis function ,

general (1803-1855) and J. Liouville ( 1 French

folowing Fourier.

coordinate. Taking advantage of the integration by parts performed for the

operator in (4.6-7), the of that is written generally as

The eigenvalue problems introduced in Section 4.3 and applied in Section 4.5 correspond to w = p = and q = The operators and eigenfunctions in some of the more problems in cylindrical and spherical coordinates are discussed in Sec- tions 4.7 and 4.8. Eigenvalue problems not encompassed by classical

theory, such as those where w and p are discontinuous, are discussed in and

Amundson 1985) and et al. (1979). A few such situations are examined in the problems at the end of this chapter.