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Classroom Tips and Techniques:

Eigenvalue Problems for ODEs - Part 3

Robert J. LopezEmeritus Professor of Mathematics and Maple Fellow

Maplesoft

Initializations

Introduction

In Part 1 of this series of articles on solving eigenvalue problems for ODEs, we discussed equationsfor which the general solution readily yielded eigenvalues and eigenfunctions without the need fordetailed knowledge of any of the special functions of applied mathematics. In Part 2 of this series,we examined the solution of Laplace's equation in a cylinder. Separation of variables in cylindricalcoordinates leads to a singular Sturm-Liouville eigenvalue problem whose differential equation is theBessel equation.

In Part 3 of this series, we will examine the solution of Laplace's equation in a sphere. Separation ofvariables in spherical coordinates leads to a singular Sturm-Liouville eigenvalue problem in whichthe differential equation is Legendre's equation. Reasoning from a general solution of Legendre'sequation to the bounded solutions needed to solve the eigenvalue problem is a significantly greaterchallenge than it was for the parallel case of Bessel's equation. Our discussion will highlight thecontributions Maple can make to this process.


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Steady-State Temperatures in a Sphere

At steady state, the temperature in a sphere satisfies Laplace's equation and some conditionson the boundary of the sphere, which we describe in spherical coordinates by

In addition to the conditions prescribed on the surfacethe physical properties of the system demand the solution be continuous. This requirement will

be the most important, and most difficult condition to impose.

If the temperature on the surface of the sphere is prescribed, we say that a Dirichlet condition has been imposed. If the prescribed temperature on the surface is a function of alone, thetemperature in the sphere will exhibit azimuthal symmetry so thatAlternatively, if this prescribed temperature is then the temperature in the sphere willexhibit azimuthal asymmetry so that

If the surface of the sphere is insulated so the net heat flux across this surface is zero, we say that ahomogeneous Neumann condition has been imposed. The flux across the surface is the normal

derivative given by evaluated at where is the radius of the sphere. The net flux

would be the surface integral of this derivative. However, if the net heat flux across the surface of ahomogeneous sphere is zero, the steady-state temperature in the sphere will be constant.

Spherical Coordinates in Maple

From our statement of the problem above, our definition of spherical coordinates can be inferred.However, because there are two different usages prevalent in the literature, we will explicitly defineour system according to the notation in most mathematics texts. In such texts, is the distance fromthe origin; measured from the positive -axis and around the -axis, lies in the range ;and measured downward from the positive -axis, lies in the range The equationsconnecting these spherical coordinates with Cartesian coordinates appear on the left in Table 1.

Spherical coordinates in texts for physics, engineering, and the applied sciences tend to interchangethe names and The equations connecting these spherical coordinates with Cartesian coordinatesappear on the right in Table 1.

Math Texts

Angle measured downfrom -axis

Science Texts

Angle measured downfrom -axis

Table 1 Spherical coordinates as defined in math


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texts (left) and science texts (right)

Finally, note that in Maple's VectorCalculus package, commands that use spherical coordinatesassume that the "middle" coordinate in the triple is the angle measured down from the -axis.Unfortunately, in a number of plot commands in the plots package, this convention is not respected.Maple is currently struggling with this quandary, especially so, given its commitment to backward

compatibility.

We set the ambient coordinate system via the command

Laplace's Equation in Spherical Coordinates

In a sphere, the steady-state temperature satisfied Laplace's equation Thisequation is given in Maple as

Maple can determine if the partial differential equation is variable separable:

0

The return of "0" indicates that the equation is indeed separable because the separability conditionsare identically satisfied.

Azimuthal Symmetry

Separation of Variables

Under the assumption that the steady-state temperatures are symmetric about the -axis,dependence on angle can be dispensed with. Hence, and a Maple-generatedvariable-separation is obtained with


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(6.1.1)(6.1.1)

Equation shows that a variable separation solution of the form

exists, and provides the ordinary differential equations the functions and must satisfy. We now proceed to obtain these same results from first principles.

Under the separation assumption, Laplace's equation assumes the simpler form

Moving all terms in to the right, we then have

Introduction of Bernoulli's separation constant then leads to the ordinary differential equations


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We are primarily interested in the second of these equations - it will become Legendre's equationafter a mild rearrangement and change of variables. First, write the equation in the form

and then

Now, make the change of variables with becoming Thisis done in Maple with

Further simplifying, we have

which is the standard form of Legendre's equation, the self-adjoint form of which would be

The Sturm-Liouville Eigenvalue Problem

The eigenvalue problem that embeds Legendre's equation is singular. The boundary conditionsare simply that must be continuous on the interval Passage from the generalsolution

to the eigenfunctions is surprisingly more difficult than it was for Bessel's equation. Because weare in extended typesetting mode, the functions and aredisplayed as and respectively. (Were we in extended typesetting mode during ourearlier discussion of Bessel's equation, Maple would have displayed as )


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When solving Laplace's equation in the cylinder, it was relatively easy to use continuity to restrictthe general solution to just , the Bessel function bounded on the interval and to

determine the eigenvalues from the zeros of We began the process by ruling out the

Bessel function of the second kind because we could tell from a graph that all such functions wereunbounded at the origin.

We will try to rule out the function in a similar way, but we will find the processmore difficult than it was for the Bessel function. For example, consider

from which it is clear that the function is unbounded at the endpoints because of thelogarithms. But this is obvious for , an integer. It is a bit more difficult to divine theendpoint behavior for general values of . For example, we can calculate the values

which suggest may indeed be unbounded at for general values of .Figure 1 contains graphs of the real and imaginary parts of with in the openinterval and in the interval .

(a) - Real Part(b) - Imaginary Part

Figure 1 Real and imaginary parts of forsuggesting is unbounded on

From Figure 1(a) especially, we conclude that is unbounded for general values


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of . On the basis of this conclusion, we set to zero in the general solution of Legendre'sequation, and turn our attention to the Legendre function of the first kind.

We first show that for general (real) values of is unbounded. Samplecalculations include

To illustrate this behavior for multiple values of , we define the following piecewise function.

If is large, then a graph of will show a point at for that value of .If is "not large" then a graph of will show the value of

We can control the evaluation points for a graph of if we define the uniform random variable via

then create a uniform but random sample of -values that includes the integers in the interval.

The graph of in Figure 2 shows that virtually all evaluations of are large inmagnitude.

0 1 2 3

1

2

Figure 2 Stylized graph of


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(6.2.1)(6.2.1)

for

However, it also suggests that for integer . For

noninteger , is unbounded so that the bounded solutions of Legendre's equation

will be the eigenfunctions with

an integer. Hence, the eigenvalues will be

that is, The first few eigenfunctions are

which are the Legendre polynomials normalized so that These polynomials aregraphed in Figure 3.


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0 1

1

Figure 3 The Legendre polynomials

That the function reduces to the polynomial for can be seenfrom the following calculations.

For noninteger , we first obtain the formal power series expansion of via

then extract the general term in the first series with

The pochhammer symbol

or "rising factorial" for complex generalizes to


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for complex If is a nonpositive integer, then

Making this transformation and setting in the general term of the first series for

gives the general coefficient

For large this coefficient is asymptotic to

suggesting that is unbounded since the series under consideration will behavelike the harmonic series at . We can confirm this behavior by comparing the general

coefficient with for large . In the limit we find the ratio tends to

which is finite for not an integer. To see that for integer the series for reducesto a polynomial, examine the recursion formula for its coefficients. This is most efficientlyobtained in Maple via

from which it becomes clear that when Hence,Therefore, is a polynomial of degree for .

Orthogonality of the Eigenfunctions


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The classical proof of the orthogonality of the eigenfunctions of Legendre's equation is based onintegration by parts. The self-adjoint form of the equation, namely,

is written once for an eigenfunction and once for The first equation is multiplied byand the second, by , and the difference of the two products is integrated over

. Integration by parts is applied to the terms containing the derivatives, which thenvanish as we can see from the following sketch. Integrals of the terms containing the derivativescan be written as

Integration by parts and subtraction then lead to

0

What remains is If the eigenvalues and are different, then

which implies orthogonality of and .

Thus, for as we see for via the matrix of evaluations

below.


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From this matrix we also infer that , a result Maple cannot show in

general, as we see from

Fourier-Legendre Series

An integrable function can be represented by the Fourier-Legendre series

where

The coefficients for the Fourier-Legendre series of the function

are


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A partial sum of the Fourier-Legendre series itself is given by

or better still, by

Figure 4 compares graphs of and the partial sum of its Fourier-Legendre series.

0 1

1

Figure 4 Graphs of (in black) and a partial sum of its Fourier-Legendre series (in red)


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An integrable function can be represented by the Fourier-Legendre series

where

The function has for its Fourier-Legendre coefficientsthe numbers

and for a partial sum of its Fourier-Legendre series, the polynomial

Figure 5 compares graphs of and the partial sum of its Fourier-Legendre series.


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0 1 2 30

1

Figure 5 Graphs of (in black)and a partial sum of its Fourier-Legendre series(in red)

Azimuthal Asymmetry

Separation of Variables

Without symmetry, so the separated form of the solution of Laplace's equationwould be

Maple provides the following ODEs governing these three functions.

We proceed to obtain these results from first principles.

Upon division by and multiplication by Laplace's equation becomes


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If the terms containing are moved to the right, we have

Introducing the separation constant leads to the two equations

The resulting -equation can be put into the form


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Continuity requires the imposition of the periodic boundary conditions

thus forming a Sturm-Liouville eigenvalue problem for which the solution isThus, . Making this change in the companion

equation, and dividing by we have

Isolating the terms in yields the separated equation

and introduction of the separation constant leads to the two ODEs


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(7.1.1)(7.1.1)

The -equation can be manipulated to the form

from which we see that it is an Euler equation solvable in powers of .

The remaining ODE is the associated Legendre equation, which we cast in the form

by bringing all terms to the left and multiplying through by The same change ofvariables that was used for Legendre's equation is applied, leading to

and then

after suitable rearrangement. A slightly better form for this equation can be obtained with thecommand

but the form typically seen for Legendre's associated equation is


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The Sturm-Liouville Eigenvalue Problem

The general solution of Legendre's associated equation is

a linear combination of the two associated Legendre functions, and(In extended typesetting mode, Maple writes these functions as and

respectively.) These are imaginatively called associated Legendre functions of the first and second kinds , respectively. From Legendre's associated equation we can see that

and

In the complex plane, a branch cut for a function is a line or line segment across which thefunction has a jump discontinuity. In Maple, there are two cut-regimes for the Legendrefunctions. The default regime imposes a cut on the real line coincident with the intervalAlternatively, the real intervals and comprise a second cut regime. Theenvironment variable _EnvLegendreCut is used to fix the cut regime by assigning it either of theexpressions -1..1 or 1..infinity.

To solve Laplace's equation in the interior of a sphere, the associated Legendre functions that

arise must have their branch cut outside of the interval that is, opposite to the defaultregime. Hence, when working in Maple, the branch cut must be shifted via a proper assignmentto the environment variable. However, commands such as evalf or simplify , commands that willmost likely be invoked in the context of the solution process, have a remember table , which storesthe value assigned to the environment variable. Reassigning a new value to the environmentvariable will not change the cut regime unless something is done to modify the remember tablesin commands such as evalf and simplify . This is done by applying the forget command to theseoperators before changing the assignment to the environment variable.

For the sake of completeness, we illustrate these issues below.

With the default cut in place, the function is discontinuous across the linesegment coincident with the real interval a discontinuity we sample at with theevaluations

Another way to see the discontinuity across this cut is symbolically, with


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2

Careful inspection shows that for , the term will be the square root of a negativenumber, from whence the discontinuity

arises. Now, if we attempt to shift the branch cut with

it can appear that the behavior of simplify is erratic; simplify may or may not reflect the change in

the branch cut, depending on the internal state of Maple. Here, we see

2

Thus, it is possible that we could have obtained exactly the same result as when the branch cut isalong The reason for the uncertainty lies in the remember table attached to the simplify command. Although reassignment to the environment variable is immediate, because of theremember table simplify will not immediately access the new value unless an internal eventcauses the table to be cleared. To force the remember table to access the new setting, use

the effect of which we test via

In either event, notice that now the term is real for and there will not be a jumpacross as we see from

3 2

3 2

Relating Maple to the Literature


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(7.3.1)(7.3.1)

(7.3.2)(7.3.2)

The associated Legendre function of the first kind appears in the literature with two differentsymbols. For example, in the Handbook of Mathematical Functions by Abramowitz and Stegun(Dover Publications), we find the following two formulas relating these functions to Legendre

polynomials.

By the obvious experiment, we can conclude that in Maple

Indeed, we construct as

and compare it to in the form

On , the radicals appearing in both expressions are equivalent, as demonstrated by thegraphs in Figure 6.

0 1

20

40


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Figure 6 Graphs of(red) and (black)

Orthogonality of the Eigenfunctions

The orthogonality relation for the associated Legendre functions of the first kind is

a relation Maple can instantiate, but not easily establish from first principles. For example, Table2 lists some integrals for which while Table 3 lists some for which comparing thecomputed and formulaic values of the integral.


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Table 2 forTable 3 For

compared to

Fourier-Legendre Series


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The functions together with the functions

form a complete set on the rectangle

Consequently, a function can be expanded in a Fourier-Legendre seriesof the form

where

and

and

and

For example, take as


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a function whose graph is seen in Figure 7.

Figure 7 Graph of

It is also useful to define the function

whose values appear in the denominators of the expressions for the series coefficients. Thecomputation of these coefficients is slightly simplified by recognizing that all the are zero bysymmetry. Then, the first few are given by

and the first few are given by


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A partial sum of the Fourier-Legendre series for is then

the graph of which can be seen in Figure 8.

Figure 8 Graph of partial sum of the Fourier-Legendre series for

To estimate the accuracy of this approximation, the difference is plotted in Figure 9.


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Figure 9 Graph of as an estimate of theaccuracy of the partial sum

Legal Notice: The copyright for this application is owned by Maplesoft. The application is intended todemonstrate the use of Maple to solve a particular problem. It has been made available for productevaluation purposes only and may not be used in any other context without the express permission of

Maplesoft.

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