SteadyState Temperatures 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
Steadystate 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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leads to
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and
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as the variableseparated 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 "selfadjoint" form of the equation for the radial component is
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 SturmLiouville 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 floatingpoint 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 builtin 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 FourierBessel series where
and the eigenvalue is the th zero of .
Homogeneous Neumann Condition
The solution of the singular SturmLiouville 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 floatingpoint 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 builtin 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 FourierBessel 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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leads to
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Moving the term in to the righthand 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
Imposing these conditions leads to
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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 selfadjoint 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 doublyindexed 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 FourierBessel 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, AddisonWesley 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 doublyindexed 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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Solving this for leads to
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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 divisionbyzero 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 .