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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 

 

> Typesetting:-mrow(Typesetting:-mi(
 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

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. 

 

Steady-State Temperatures in a Cylinder 

 

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

 

If the temperature on the curved wall of the cylinder is fixed at Typesetting:-mrow(Typesetting:-mi(, 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 Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( evaluated at Typesetting:-mrow(Typesetting:-mi(. 

 

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

 

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 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(u(r, z), r)), `*`(r)), diff(diff(u(r, z), r), r), diff(diff(u(r, z), z), z)) = 0
 

Assuming the separated solution 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`*`(R(r), `*`(Z(z)))
 

leads to 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r))), `/`(`*`(diff(diff(Z(z), z), z)), `*`(Z(z)))) = 0
 

and  

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r)))) = `+`(`-`(`/`(`*`(diff(diff(Z(z), z), z)), `*`(Z(z)))))
 

as the variable-separated equation.  Introducing the Bernoulli separation constant Typesetting:-mrow(Typesetting:-mi(, we have the ordinary differential equation 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r)))) = lambda
 

for the radial component Typesetting:-mrow(Typesetting:-mi(.   Manipulating this to the form 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(r)), diff(diff(R(r), r), r)) = `*`(R(r), `*`(lambda))
 

and then 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(r)), diff(diff(R(r), r), r), `-`(`*`(R(r), `*`(lambda)))) = 0
 

we obtain the solution as 

 

> Typesetting:-mrow(Typesetting:-mi(
 

R(r) = `+`(`*`(_C1, `*`(BesselJ(0, `*`(`^`(`+`(`-`(lambda)), `/`(1, 2)), `*`(r))))), `*`(_C2, `*`(BesselY(0, `*`(`^`(`+`(`-`(lambda)), `/`(1, 2)), `*`(r))))))
 

The "self-adjoint" form of the equation for the radial component is  

 

Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( 

 

The radical in the solution suggests the substitution 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(r)), diff(diff(R(r), r), r), `*`(R(r), `*`(`^`(mu, 2)))) = 0
 

in which case the solution is given by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

R(r) = `+`(`*`(_C1, `*`(BesselJ(0, `*`(mu, `*`(r))))), `*`(_C2, `*`(BesselY(0, `*`(mu, `*`(r))))))
 

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 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(`^`(mu, 2), `*`(diff(y(x), x))), `*`(x)), `*`(`^`(mu, 2), `*`(diff(diff(y(x), x), x))), `*`(y(x), `*`(`^`(mu, 2)))) = 0
 

Factoring Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( leads to Bessel's equation of order zero, that is, to 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(diff(y(x), x)), `*`(x)), diff(diff(y(x), x), x), y(x)) = 0
 

The solution of this equation is 

 

> Typesetting:-mrow(Typesetting:-mi(
 

y(x) = `+`(`*`(_C1, `*`(BesselJ(0, x))), `*`(_C2, `*`(BesselY(0, x))))
 

so that Typesetting:-mrow(Typesetting:-mi(  Figure 1 contains a graph of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( (in black) and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( (in red).   

 

> Typesetting:-mrow(Typesetting:-mi(
 

Plot_2d
 

Figure 1   In black, Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(, the Bessel function of the first kind, and in red, Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(the Bessel function of the second kind. 

Consistent with the graph in Figure 1, we have 

 

> Typesetting:-mrow(Typesetting:-mi(
 

Limit(BesselY(0, x), x = 0, right) = `+`(`-`(infinity))
 

so that the Typesetting:-mrow(Typesetting:-mi(-axis in a vertical asymptote for Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(, and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is unbounded in any interval that includes Typesetting:-mrow(Typesetting:-mi(. 

 

The function Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is well approximated by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`*`(sin(`+`(x, `*`(`/`(1, 4), `*`(Pi)))), `*`(`^`(2, `/`(1, 2)), `*`(`^`(`/`(1, `*`(Pi, `*`(x))), `/`(1, 2)))))
 

as we confirm in Figure 2, where Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is graphed in black, and Typesetting:-mrow(Typesetting:-mi( is in red. 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Plot_2d
 

Figure 2   In black, Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(and in red, Typesetting:-mrow(Typesetting:-mi( 

Ignoring Maple's automatically simplifications of Typesetting:-mrow(Typesetting:-mi( to Typesetting:-mrow(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-mi( to Typesetting:-mrow(Typesetting:-mi(, we see from  

 

> Typesetting:-mrow(Typesetting:-mi(
 

Limit(`+`(BesselJ(0, x), `-`(`*`(sin(`+`(x, `*`(`/`(1, 4), `*`(Pi)))), `*`(`^`(2, `/`(1, 2)), `*`(`^`(`/`(1, `*`(Pi, `*`(x))), `/`(1, 2))))))), x = infinity) = 0
 

that Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is asymptotic to Typesetting:-mrow(Typesetting:-mi(. 

 

Homogeneous Dirichlet Condition 

 

The solution of the singular Sturm-Liouville eigenvalue problem 

 

Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(  

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( bounded on Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn( 

 

consists of the eigenvalues Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( (the zeros of Typesetting:-mrow(Typesetting:-mi() and the eigenfunctions Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

 

The numbers Typesetting:-mrow(Typesetting:-mi(are represented symbolically by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

BesselJZeros(0, 1 .. 10)
 

and in floating-point form by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

2.404825558, 5.520078110, 8.653727913, 11.79153444, 14.93091771, 18.07106397, 21.21163663, 24.35247153, 27.49347913, 30.63460647
 

That Maple recognizes the eigenvalues as exact is shown by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

0, 0, 0, 0, 0, 0, 0, 0, 0, 0
 

If, for example Typesetting:-mrow(Typesetting:-mi(, the eigenvalues Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( would be 

 

> Typesetting:-mrow(Typesetting:-mi(
 

[1.202412779, 2.760039055, 4.326863956, 5.895767220, 7.465458855, 9.035531985, 10.60581832, 12.17623576, 13.74673956, 15.31730324]
 

Figure 3 shows the first three eigenfunctions for the case Typesetting:-mrow(Typesetting:-mi( 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Plot_2d
 

Figure 3   For Typesetting:-mrow(Typesetting:-mi(the first three eigenfunctions in black, red, and green, respectively 

The eigenfunctions are orthogonal with respect to the weight function Typesetting:-mrow(Typesetting:-mi(, as can be seen from the integral 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Int(`*`(r, `*`(BesselJ(0, `*`(mu[j], `*`(r))), `*`(BesselJ(0, `*`(mu[k], `*`(r)))))), r = 0 .. c) = `/`(`*`(c, `*`(`+`(`-`(`*`(BesselJ(0, `*`(mu[j], `*`(c))), `*`(BesselJ(1, `*`(mu[k], `*`(c))), `*`(m... (4.2.1)
 

which vanishes if Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( are distinct eigenvalues for the interval Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(  Making use of Maple's built-in representation of the zeros of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(we also have 

 

> Typesetting:-mrow(Typesetting:-mi(
 

Int(`*`(r, `*`(BesselJ(0, `/`(`*`(BesselJZeros(0, n), `*`(r)), `*`(c))), `*`(BesselJ(0, `/`(`*`(BesselJZeros(0, m), `*`(r)), `*`(c)))))), r = 0 .. c) = 0
 

Unfortunately, Maple does not yet detect that this result is not valid if Typesetting:-mrow(Typesetting:-mi(in which case the integral evaluates to 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`*`(`/`(1, 2), `*`(`^`(c, 2), `*`(`^`(BesselJ(1, `*`(mu[k], `*`(c))), 2)))), `*`(`/`(1, 2), `*`(`^`(c, 2), `*`(`^`(BesselJ(0, `*`(mu[k], `*`(c))), 2)))))
 

Since Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(this result simplifies to Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-msup(Typesetting:-mi(  Thus, under suitable conditions, a function Typesetting:-mrow(Typesetting:-mi( can be represented by the Fourier-Bessel series Typesetting:-mrow(Typesetting:-munderover(Typesetting:-mo(where  

 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(  

 

and the eigenvalue Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is the Typesetting:-mrow(Typesetting:-mi(th zero of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(. 

 

Homogeneous Neumann Condition 

 

The solution of the singular Sturm-Liouville eigenvalue problem 

 

Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(  

Typesetting:-mrow(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-mi( bounded on Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn( 

 

consists of the eigenvalues Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( (the zeros of Typesetting:-mrow(Typesetting:-mi() and the eigenfunctions Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

 

The numbers Typesetting:-mrow(Typesetting:-mi(are represented symbolically by 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

0, BesselJZeros(1, 1 .. 10)
 

and in floating-point form by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

0., 3.831705970, 7.015586670, 10.17346814, 13.32369194, 16.47063005, 19.61585851, 22.76008438, 25.90367209, 29.04682854, 32.18967991
 

>
 

where we have used the relationship Typesetting:-mrow(Typesetting:-mi(  Note carefully that Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(but Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( for positive integers Typesetting:-mrow(Typesetting:-mi(.   

 

Figure 4 shows the first three eigenfunctions for the case Typesetting:-mrow(Typesetting:-mi( 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Plot_2d
 

Figure 4   For Typesetting:-mrow(Typesetting:-mi(the first three eigenfunctions in black, red, and green, respectively 

The eigenfunctions are orthogonal with respect to the weight function Typesetting:-mrow(Typesetting:-mi(, as can be seen from the integral 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Int(`*`(r, `*`(BesselJ(0, `*`(sigma[j], `*`(r))), `*`(BesselJ(0, `*`(sigma[k], `*`(r)))))), r = 0 .. c) = `/`(`*`(c, `*`(`+`(`*`(BesselJ(0, `*`(sigma[j], `*`(c))), `*`(BesselJ(1, `*`(sigma[k], `*`(c))... (4.3.1)
 

which vanishes if Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( are distinct eigenvalues for the interval Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(  Of course, Maple has used Typesetting:-mrow(Typesetting:-mo( for Typesetting:-mrow(Typesetting:-mi(  

 

Making use of Maple's built-in representation of the zeros of Typesetting:-mrow(Typesetting:-mi(we also have 

 

> Typesetting:-mrow(Typesetting:-mi(
 

Int(`*`(r, `*`(BesselJ(0, `/`(`*`(BesselJZeros(1, n), `*`(r)), `*`(c))), `*`(BesselJ(0, `/`(`*`(BesselJZeros(1, m), `*`(r)), `*`(c)))))), r = 0 .. c) = 0
 

As seen earlier, Maple does not yet detect that this result is not valid if Typesetting:-mrow(Typesetting:-mi(in which case the integral evaluates to 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`*`(`/`(1, 2), `*`(`^`(c, 2), `*`(`^`(BesselJ(1, `*`(sigma[k], `*`(c))), 2)))), `*`(`/`(1, 2), `*`(`^`(c, 2), `*`(`^`(BesselJ(0, `*`(sigma[k], `*`(c))), 2)))))
 

Since Typesetting:-mrow(Typesetting:-mo(this result simplifies to Typesetting:-mrow(Typesetting:-mfrac(Typesetting:-msup(Typesetting:-mi(  Thus, under suitable conditions, a function Typesetting:-mrow(Typesetting:-mi( can be represented by the Fourier-Bessel series Typesetting:-mrow(Typesetting:-munderover(Typesetting:-mo(where  

 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(  

 

Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( 

 

and the eigenvalue Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( is the Typesetting:-mrow(Typesetting:-mi(th zero of Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(. 

 

Axial Asymmetry 

Separation of Variables 

 

If the temperature prescribed on top of the cylinder is given by Typesetting:-mrow(Typesetting:-mi(then Typesetting:-mrow(Typesetting:-mi( obeys Laplace's equation in the form 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(diff(u(r, theta, z), r)), `*`(r)), diff(diff(u(r, theta, z), r), r), `/`(`*`(diff(diff(u(r, theta, z), theta), theta)), `*`(`^`(r, 2))), diff(diff(u(r, theta, z), z), z)) = 0
 

The separation assumption 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`*`(R(r), `*`(Theta(theta), `*`(Z(z))))
 

leads to 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r))), `/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta), `*`(`^`(r, 2)))), `/`(`*`(diff(diff(Z(z)...
 

Moving the term in Typesetting:-mrow(Typesetting:-mi( to the right-hand side leaves us with the equation 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r))), `/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta), `*`(`^`(r, 2))))) = `+`(`-`(`/`(`*`(diff...
 

Introducing the Bernoulli separation constant Typesetting:-mrow(Typesetting:-mi( then gives  

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

 

`+`(`/`(`*`(diff(R(r), r)), `*`(R(r), `*`(r))), `/`(`*`(diff(diff(R(r), r), r)), `*`(R(r))), `/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta), `*`(`^`(r, 2))))) = `+`(`-`(lambda))
`+`(`-`(`/`(`*`(diff(diff(Z(z), z), z)), `*`(Z(z))))) = `+`(`-`(lambda))
 

The first of these equations is of immediate interest.  We first multiply through by Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( so that the term containing Typesetting:-mrow(Typesetting:-mi( contains only the independent variable Typesetting:-mrow(Typesetting:-mi(  This gives the equation 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(r, `*`(diff(R(r), r))), `*`(R(r))), `/`(`*`(`^`(r, 2), `*`(diff(diff(R(r), r), r))), `*`(R(r))), `/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta)))) = `+`(`-`(`*`(`^`(r, 2...
 

Next, we would like to move the term Typesetting:-mrow(Typesetting:-mo( to the left, and the term containing Typesetting:-mrow(Typesetting:-mi( to the right.  The first transformation is given by 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(r, `*`(diff(R(r), r))), `*`(R(r))), `/`(`*`(`^`(r, 2), `*`(diff(diff(R(r), r), r))), `*`(R(r))), `/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta))), `*`(`^`(r, 2), `*`(lam...
 

and the second, by 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`/`(`*`(r, `*`(diff(R(r), r))), `*`(R(r))), `/`(`*`(`^`(r, 2), `*`(diff(diff(R(r), r), r))), `*`(R(r))), `*`(`^`(r, 2), `*`(lambda))) = `+`(`-`(`/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`...
 

Introducing a new separation constant Typesetting:-mrow(Typesetting:-mi(we have the two ordinary differential equations 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

 

`+`(`/`(`*`(r, `*`(diff(R(r), r))), `*`(R(r))), `/`(`*`(`^`(r, 2), `*`(diff(diff(R(r), r), r))), `*`(R(r))), `*`(`^`(r, 2), `*`(lambda))) = nu
`+`(`-`(`/`(`*`(diff(diff(Theta(theta), theta), theta)), `*`(Theta(theta))))) = nu
 

The second equation can be put into the form 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(diff(diff(Theta(theta), theta), theta), `*`(Theta(theta), `*`(nu))) = 0
 

and has general solution 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

Theta(theta) = `+`(`*`(_C1, `*`(sin(`*`(`^`(nu, `/`(1, 2)), `*`(theta))))), `*`(_C2, `*`(cos(`*`(`^`(nu, `/`(1, 2)), `*`(theta))))))
 

Continuity of Typesetting:-mrow(Typesetting:-mi( implies the periodic boundary conditions 

 

Typesetting:-mrow(Typesetting:-mi(  

Typesetting:-mrow(Typesetting:-mi( 

 

Imposing these conditions leads to 

 

> Typesetting:-mrow(Typesetting:-mi(
Typesetting:-mrow(Typesetting:-mi(
 

 

`+`(`-`(`*`(2, `*`(_C1, `*`(sin(`*`(`^`(nu, `/`(1, 2)), `*`(Pi)))))))) = 0
`+`(`*`(2, `*`(_C2, `*`(sin(`*`(`^`(nu, `/`(1, 2)), `*`(Pi))), `*`(`^`(nu, `/`(1, 2))))))) = 0
 

from which we see that Typesetting:-mrow(Typesetting:-mi(  Notice that for Typesetting:-mrow(Typesetting:-mi(, the eigenspace has dimension 1 and basis Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(, but for Typesetting:-mrow(Typesetting:-mi( the eigenspaces have dimension 2 and bases Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(. 

 

The equation for the radial component Typesetting:-mrow(Typesetting:-mi( now takes the form 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(diff(R(r), r), `*`(r, `*`(diff(diff(R(r), r), r))), `*`(r, `*`(R(r), `*`(lambda)))) = `/`(`*`(R(r), `*`(`^`(n, 2))), `*`(r))
 

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

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`*`(`+`(`*`(r, `*`(lambda)), `-`(`/`(`*`(`^`(n, 2)), `*`(r)))), `*`(R(r))), diff(R(r), r), `*`(r, `*`(diff(diff(R(r), r), r)))) = 0
 

The solution of this equation is given by 

 

> Typesetting:-mrow(Typesetting:-mi(
 

R(r) = `+`(`*`(_C1, `*`(BesselJ(n, `*`(`^`(lambda, `/`(1, 2)), `*`(r))))), `*`(_C2, `*`(BesselY(n, `*`(`^`(lambda, `/`(1, 2)), `*`(r))))))
 

suggesting first, the substitution Typesetting:-mrow(Typesetting:-mi( so that the equation for the radial component becomes 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`*`(`+`(`*`(r, `*`(`^`(mu, 2))), `-`(`/`(`*`(`^`(n, 2)), `*`(r)))), `*`(R(r))), diff(R(r), r), `*`(r, `*`(diff(diff(R(r), r), r)))) = 0
 

Since the solution of this form of the equation will be 

 

> Typesetting:-mrow(Typesetting:-mi(
 

R(r) = `+`(`*`(_C1, `*`(BesselJ(n, `*`(mu, `*`(r))))), `*`(_C2, `*`(BesselY(n, `*`(mu, `*`(r))))))
 

we make the change of variables Typesetting:-mrow(Typesetting:-mi( so that Typesetting:-mrow(Typesetting:-mi(  This change is implement in Maple via 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`*`(`+`(`*`(x, `*`(mu)), `-`(`/`(`*`(mu, `*`(`^`(n, 2))), `*`(x)))), `*`(y(x))), `*`(mu, `*`(diff(y(x), x))), `*`(x, `*`(mu, `*`(diff(diff(y(x), x), x))))) = 0
 

Under the assumption that Typesetting:-mrow(Typesetting:-mi(this equation simplifies to 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

`+`(`*`(`+`(x, `-`(`/`(`*`(`^`(n, 2)), `*`(x)))), `*`(y(x))), diff(y(x), x), `*`(x, `*`(diff(diff(y(x), x), x)))) = 0
 

a Bessel equation of order Typesetting:-mrow(Typesetting:-mi(, and has general solution  

 

> Typesetting:-mrow(Typesetting:-mi(
 

y(x) = `+`(`*`(_C1, `*`(BesselJ(n, x))), `*`(_C2, `*`(BesselY(n, x))))
 

Homogeneous Dirichlet Condition 

 

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

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

 

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

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

Int(`*`(r, `*`(BesselJ(n, `*`(mu[j], `*`(r))), `*`(BesselJ(n, `*`(mu[k], `*`(r)))))), r = 0 .. c)
 

where the superscript Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi( has been dropped from the distinct eigenvalues Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi(  For this integral Maple gives 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`/`(`*`(c, `*`(`+`(`*`(mu[k], `*`(BesselJ(n, `*`(mu[j], `*`(c))), `*`(BesselJ(`+`(n, `-`(1)), `*`(mu[k], `*`(c)))))), `-`(`*`(mu[j], `*`(BesselJ(`+`(n, `-`(1)), `*`(mu[j], `*`(c))), `*`(BesselJ(n, `*`...
 

which clearly vanishes by the definition of the eigenvalues  Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( and Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi(  

 

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

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(  

 

which we enter into Maple as 

 

> Typesetting:-mrow(Typesetting:-mi(
 

Int(`*`(r, `*`(`^`(BesselJ(n, `*`(BesselJZeros(n, m), `*`(r))), 2))), r = 0 .. c)
 

We call this the "Norm Relation" since it is related to the Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(-norm of the eigenfunction. 

 

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

 

> Typesetting:-mrow(Typesetting:-mi(
 

`/`(`*`(`^`(2, `+`(`-`(2), `-`(`*`(2, `*`(n))))), `*`(`^`(c, `+`(1, `*`(2, `*`(n)))), `*`(`^`(Pi, `/`(1, 2)), `*`(`^`(BesselJZeros(n, m), `+`(`-`(1), `*`(2, `*`(n)))), `*`(`^`(`*`(BesselJZeros(n, m), ...
 

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

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(  

 

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

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo( = Typesetting:-mrow(Typesetting:-mo(  

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo( =Typesetting:-mrow(Typesetting:-mo( 

 

There is one additional subtlety to confront.  Since Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(the first eigenvalue in the sequence Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( is not zero, so Typesetting:-mrow(Typesetting:-mi( corresponds to the indexing available via Maple's BesselJZeros command.  Indeed, we see that 

 

> Typesetting:-mrow(Typesetting:-mi(
 

2.404825558
 

is the first zero for Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(.  However, Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( for Typesetting:-mrow(Typesetting:-mi(, so the "first zero" is Typesetting:-mrow(Typesetting:-mi(, 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 Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( starts with Typesetting:-mrow(Typesetting:-mi( for any value of Typesetting:-mrow(Typesetting:-mi(.  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 Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( implies Typesetting:-mrow(Typesetting:-mi(so that Typesetting:-mrow(Typesetting:-mi( must be a zero of Typesetting:-mrow(Typesetting:-mi(Typesetting:-mrow(Typesetting:-mo(  If we denote the Typesetting:-mrow(Typesetting:-mi(th zero of Typesetting:-mrow(Typesetting:-mi( by Typesetting:-mrow(Typesetting:-mi(then Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( represents a doubly-indexed set of eigenvalues for which the corresponding eigenfunctions are 

 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( 

 

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

 

> Typesetting:-mrow(Typesetting:-mi(
 

`/`(`*`(c, `*`(`+`(`*`(mu[k], `*`(BesselJ(n, `*`(mu[j], `*`(c))), `*`(BesselJ(`+`(n, `-`(1)), `*`(mu[k], `*`(c)))))), `-`(`*`(mu[j], `*`(BesselJ(`+`(n, `-`(1)), `*`(mu[j], `*`(c))), `*`(BesselJ(n, `*`...
 

vanishes.  To this end, we seek to eliminate Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( and introduce Typesetting:-mrow(Typesetting:-mi(.  A generalization of the identity Typesetting:-mrow(Typesetting:-mi( is 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

Diff(BesselJ(n, z), z) = `+`(`-`(BesselJ(`+`(n, 1), z)), `/`(`*`(n, `*`(BesselJ(n, z))), `*`(z)))
 

Solving this for Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( leads to 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

BesselJ(`+`(n, 1), z) = `+`(`-`(Diff(BesselJ(n, z), z)), `/`(`*`(n, `*`(BesselJ(n, z))), `*`(z)))
 

Replacing Typesetting:-mrow(Typesetting:-mi( in this equation with its value from the previous equation gives 

 

> Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(
 

BesselJ(`+`(n, `-`(1)), z) = `+`(`/`(`*`(n, `*`(BesselJ(n, z))), `*`(z)), Diff(BesselJ(n, z), z))
 

Finally, substituting for Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( gives 

 

> Typesetting:-mrow(Typesetting:-mi(
 

`+`(`/`(`*`(c, `*`(mu[k], `*`(BesselJ(n, `*`(mu[j], `*`(c))), `*`(eval(Diff(BesselJ(n, z), z), {z = `*`(mu[k], `*`(c))}))))), `*`(`+`(`*`(`^`(mu[j], 2)), `-`(`*`(`^`(mu[k], 2)))))), `-`(`/`(`*`(c, `*`...
 

from which it is evident that the functions Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi( are orthogonal, even when Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( are zeros of Typesetting:-mrow(Typesetting:-mi( 

 

Since Typesetting:-mrow(Typesetting:-mi(but Typesetting:-mrow(Typesetting:-msub(Typesetting:-mi(then Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mi( is an eigenvalue and the indexing starts at Typesetting:-mrow(Typesetting:-mi(  The corresponding eigenfunction is 1. 

 

Maple computes Typesetting:-mrow(Typesetting:-mi(from the formula 

 

> Typesetting:-mrow(Typesetting:-mi(
 

Diff(BesselJ(n, x), x) = `+`(`-`(BesselJ(`+`(n, 1), x)), `/`(`*`(n, `*`(BesselJ(n, x))), `*`(x)))
 

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

 

> Typesetting:-mrow(Typesetting:-mi(
 

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

However, an examination of the series expansions for Typesetting:-mrow(Typesetting:-mi(shows that this error is spurious.  Indeed, we have 

 

> Typesetting:-mrow(Typesetting:-mo(
 

 

 

 

 

series(`+`(`/`(1, 2), `-`(`*`(`/`(3, 16), `*`(`^`(x, 2)))), `*`(`/`(5, 384), `*`(`^`(x, 4))))+O(`^`(x, 5)),x,5)
series(`+`(`*`(`/`(1, 4), `*`(x)), `-`(`*`(`/`(1, 24), `*`(`^`(x, 3)))))+O(`^`(x, 5)),x,5)
series(`+`(`*`(`/`(1, 16), `*`(`^`(x, 2))), `-`(`*`(`/`(5, 768), `*`(`^`(x, 4)))))+O(`^`(x, 5)),x,5)
series(`+`(`*`(`/`(1, 96), `*`(`^`(x, 3))))+O(`^`(x, 5)),x,5)
series(`+`(`*`(`/`(1, 768), `*`(`^`(x, 4))))+O(`^`(x, 5)),x,5)
 

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

 


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