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# Classroom Tips and Techniques: Eigenvalue Problems for ODEs - Part 2

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Classroom Tips and Techniques:

Eigenvalue Problems for ODEs - Part 2

Robert J. Lopez

Emeritus Professor of Mathematics and Maple Fellow

Maplesoft

Initializations

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Introduction

In Part 1 of this series of articles on solving eigenvalue problems for ODEs, we discussed equations for which the general solution readily yielded eigenvalues and eigenfunctions without the need for detailed knowledge of any of the special functions of applied mathematics.  In Part 2 of this series, we concentrate on eigenvalue problems for Bessel's equation whose solution requires some knowledge of Bessel functions of the first and second kinds.

Since Bessel's equation readily arises when separating variables in Laplace's equation in cylindrical coordinates, we have allowed our discusion to be colored by references to calculations arising from boundary value problems posed in a cylinder.

At steady state, the temperature in a cylinder satisfies Laplace's equation and some conditions on the boundary of the cylinder.  For example, if the cylinder is described in cylindrical coordinates by we could impose the condition or on the top surface, or on the curved wall of the cylinder, and or on the bottom surface.  (Subscripts denote partial derivatives.)

If the temperature on the curved wall of the cylinder is fixed at , we say that a homogeneous Dirichlet condition has been imposed.  Alternatively, if the curved wall is insulated so the heat flux across this surface is zero, we say that a homogeneous Dirichlet condition has been imposed.  The flux across the curved wall is the normal derivative given by evaluated at .

If the prescribed temperature on the top surface is a function of alone, the temperature in the cylinder will be axially symmetric so that  Alternatively, if this prescribed temperature is then the temperature in the cylinder will be axially asymmetric so that

Axial Symmetry

Separation of Variables

Steady-state temperatures in a cylinder obey Laplace's equation.  Under the assumption of axial symmetry, Laplace's equation in cylindrical coordinates becomes

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Assuming the separated solution

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and

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as the variable-separated equation.  Introducing the Bernoulli separation constant , we have the ordinary differential equation

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for the radial component .   Manipulating this to the form

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and then

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we obtain the solution as

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The radical in the solution suggests the substitution

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in which case the solution is given by

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The solution is a linear combination of two Bessel functions, one of the "first kind" and one of the "second."

The radial equation can be transformed to Bessel's equation by the change of variables

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Factoring leads to Bessel's equation of order zero, that is, to

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The solution of this equation is

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so that  Figure 1 contains a graph of (in black) and (in red).

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Figure 1   In black, , the Bessel function of the first kind, and in red, the Bessel function of the second kind.

Consistent with the graph in Figure 1, we have

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so that the -axis in a vertical asymptote for , and is unbounded in any interval that includes .

The function is well approximated by

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as we confirm in Figure 2, where is graphed in black, and is in red.

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Figure 2   In black, and in red,

Ignoring Maple's automatically simplifications of to and to , we see from

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that is asymptotic to .

Homogeneous Dirichlet Condition

The solution of the singular Sturm-Liouville eigenvalue problem

bounded on

consists of the eigenvalues (the zeros of ) and the eigenfunctions

The numbers are represented symbolically by

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and in floating-point form by

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That Maple recognizes the eigenvalues as exact is shown by

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If, for example , the eigenvalues would be

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Figure 3 shows the first three eigenfunctions for the case

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Figure 3   For the first three eigenfunctions in black, red, and green, respectively

The eigenfunctions are orthogonal with respect to the weight function , as can be seen from the integral

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 (4.2.1)

which vanishes if and are distinct eigenvalues for the interval  Making use of Maple's built-in representation of the zeros of we also have

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Unfortunately, Maple does not yet detect that this result is not valid if in which case the integral evaluates to

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Since this result simplifies to  Thus, under suitable conditions, a function can be represented by the Fourier-Bessel series where

and the eigenvalue is the th zero of .

Homogeneous Neumann Condition

The solution of the singular Sturm-Liouville eigenvalue problem

bounded on

consists of the eigenvalues (the zeros of ) and the eigenfunctions

The numbers are represented symbolically by

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and in floating-point form by

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where we have used the relationship  Note carefully that but for positive integers .

Figure 4 shows the first three eigenfunctions for the case

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Figure 4   For the first three eigenfunctions in black, red, and green, respectively

The eigenfunctions are orthogonal with respect to the weight function , as can be seen from the integral

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 (4.3.1)

which vanishes if and are distinct eigenvalues for the interval  Of course, Maple has used for

Making use of Maple's built-in representation of the zeros of we also have

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As seen earlier, Maple does not yet detect that this result is not valid if in which case the integral evaluates to

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Since this result simplifies to  Thus, under suitable conditions, a function can be represented by the Fourier-Bessel series where

and the eigenvalue is the th zero of .

Axial Asymmetry

Separation of Variables

If the temperature prescribed on top of the cylinder is given by then obeys Laplace's equation in the form

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The separation assumption

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Moving the term in to the right-hand side leaves us with the equation

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Introducing the Bernoulli separation constant then gives

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The first of these equations is of immediate interest.  We first multiply through by so that the term containing contains only the independent variable  This gives the equation

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Next, we would like to move the term to the left, and the term containing to the right.  The first transformation is given by

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and the second, by

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Introducing a new separation constant we have the two ordinary differential equations

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The second equation can be put into the form

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and has general solution

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Continuity of implies the periodic boundary conditions

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from which we see that  Notice that for , the eigenspace has dimension 1 and basis , but for the eigenspaces have dimension 2 and bases .

The equation for the radial component now takes the form

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After moving all terms to the left, we obtain the self-adjoint form as

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The solution of this equation is given by

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suggesting first, the substitution so that the equation for the radial component becomes

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Since the solution of this form of the equation will be

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we make the change of variables so that  This change is implement in Maple via

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Under the assumption that this equation simplifies to

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a Bessel equation of order , and has general solution

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Homogeneous Dirichlet Condition

Continuity on implies that where is constant.  The homogeneous Dirichlet condition implies so that must be a zero of  If we denote the th zero of by then represents a doubly-indexed set of eigenvalues for which the corresponding eigenfunctions are

To establish orthogonality of these eigenfunctions, we will have to evaluate the integral

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where the superscript has been dropped from the distinct eigenvalues and  For this integral Maple gives

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which clearly vanishes by the definition of the eigenvalues   and

The function will be given by a Fourier-Bessel series in which a double sum is taken over both and .  The coefficients in this series require us to evaluate the integral

which we enter into Maple as

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We call this the "Norm Relation" since it is related to the -norm of the eigenfunction.

Now it is indeed unfortunate that Maple incorrectly evaluates this integral to

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an error that has been corrected for the next release of the product.  The correct value of this integral is

as can be found, for example, in the text Boundary Value Problems, Ladis D. Kovach, Addison-Wesley Publishing Company, 1984.  Consequently, we have

=

=

There is one additional subtlety to confront.  Since the first eigenvalue in the sequence is not zero, so corresponds to the indexing available via Maple's BesselJZeros command.  Indeed, we see that

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is the first zero for .  However, for , so the "first zero" is , but this cannot be an eigenvalue because by definition, eigenvalues are numbers for which there are nontrivial solutions.  Consequently, for the homogeneous Dirichlet condition, each sequence starts with for any value of .  This indexing is consistent with Maple's BesselJZeros command where "1" in the second argument always gives the first positive zero.

Homogeneous Neumann Condition

The homogeneous Neumann condition implies so that must be a zero of  If we denote the th zero of by then represents a doubly-indexed set of eigenvalues for which the corresponding eigenfunctions are

To establish orthogonality of these eigenfunctions, we will have to show

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vanishes.  To this end, we seek to eliminate and introduce .  A generalization of the identity is

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Replacing in this equation with its value from the previous equation gives

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Finally, substituting for gives

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from which it is evident that the functions are orthogonal, even when are zeros of

Since but then is an eigenvalue and the indexing starts at  The corresponding eigenfunction is 1.

Maple computes from the formula

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Consequently, a division-by-zero error occurs in, for example

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 Error, (in VectorCalculus:-eval) numeric exception: division by zero

However, an examination of the series expansions for shows that this error is spurious.  Indeed, we have

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Each of these functions can be readily evaluated at .  Surprisingly, for but .  Therefore, is not even a candidate for being an eigenvalue when , and for greater values of , although is a zero, so that would not be an eigenvalue.  Consequently, for the homogeneous Neumann condition, zero is an eigenvalue only for .

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