Solving singular boundary value problems for

ordinary di↵erential equations

Isom H. Herron⇤

Abstract

This work seeks to clarify the derivation of the Green’s matrix for the

boundary value problem with a regular singularity, based on a theorem

of Peter Philip. Singular Sturm-Liouvile problems are illustrated by the

Bessel di↵erential equation. Several other examples, including applica-

tions to physical problems, serve as illustrations.

Key words. Green’s matrix, singular, boundary-value

AMS subject classifications. 34B27, 76D55, 76E06

1 Introduction

Among the important applied problems in analytical methods is that of solvingsingular boundary value problems for di↵erential equations. They arise natu-rally and repeatedly in physical models, often because of the coordinate systeminvolved or because of an impulsive source or sink term ([4], [16], [18] [26]).

In this paper we will discuss the resolution of the following problems:(a) Under what conditions the Green’s matrix exists for a system of ordinarydi↵erential equations with a singularity of the first kind. The contention of thisarticle is that writing a single equation as a system of first order equations canbe greatly illuminating. This approach also automatically finds the connectioncoe�cients needed for a single Green’s function. (b) An introduction to naturalsingular Sturm-Liouville problems. These eigenvalue-eigenfunction problems lieat the heart of many formulations. Two introductory examples serve to illustratesome of the issues.

1

1.1 Conduction of heat in a spherical shell with sources

Consider the conduction of heat in a spherical chamber with sources. The earlytime behavior of the heat di↵usion process is of great interest in nuclear scienceand engineering. The ability to be able to predict the early stages of a nuclearreactor transient is of great importance. Here we consider a simple relevantmodel of heat conduction in a spherical chamber, with a spherically symmetricsource ([29], [37]). It might represent the heating of the pressure shell in a watermoderated and cooled reactor. We consider the heat conduction problem, asstated below. Owing to the symmetry the problem, it is described by the partialdi↵erential equation for the temperature T (r, t) as

@T

@t=

@2T

@r2+

2r

@T

@r+

f(r)r

, t > 0, 0 < r < r2, (1)

with the initial conditionT (r, 0) = 0. (2)

Here the thermal di↵usivity is scaled to unity. To start, we suppose that theouter surface is kept at some zero reference temperature implying the boundarycondition T (r2, t) = 0. We also suppose that in this model the temperature re-mains finite at the origin so lim

r!0+T (r, t) < 1. This is a singular initial-boundary

value problem.If we make the transformation

u(r, t) = rT (r, t), (3)

the one-dimensional heat conduction problem results:

@u

@t=

@2u

@r2+ f(r); (4)

the initial condition remainsu(r, 0) = 0,

and the boundary condition is are still

u(r2, t) = 0. (5)

However, the condition at the origin is now

limr!0+

u(r, t)r

finite. (6)

We consider the possibility that a steady-state is reached so that @u/@t = 0.Then the governing equation (4) reduces to

d2us

dr2+ f(r) = 0. (7)

2

The transformation (3) which led to (6) is typical. Sometimes, a change of vari-ables will render a problem more tractable, or the transformation will illuminatethe approach to a solution. In this case, in textbook fashion [33], (7) may bedirectly integrated (twice) to give

us

=Z

r2

0g(r, ⇢)f(⇢)d⇢,

where the Green’s function is

g(r, ⇢) =⇢

r(r2 � ⇢)/r2, r < ⇢⇢(r2 � r)/r2, r > ⇢

= (min(r, ⇢)r2 � r⇢)/r2. (8)

1.2 Unsteady fluid flow through a micro-tube

The following initial-boundary value problem occurs in the theory of nano-scaleflow in a circular tube for the time dependent flow W (r, t) in the axial direction([25]):

@W

@t=

@2W

@r2+

1r

@W

@r, t > 0, 0 < r < 1, (9)

We suppose that the velocity initially is parabolic, W (r, 0) = 1 � r2 + 2`, andremains finite at the origin so lim

r!0+W (r, t) < 1, with the boundary condition

W = �`@W

@r, on r = 1.

Here ` � 0 is the dimensionless slip-length parameter, while the Reynolds num-ber is scaled to unity. Note: This same initial-boundary value problem couldarise in describing radial heat conduction in a long circular cylinder where onthe boundary the rate of heat flow is proportional to the temperature [29]. Thisis another example of a singular initial-boundary value problem. We make thecommon separation of variables assumption

W (r, t) = '(r)f(t)

to find that'00(r) +

1r'0(r) + �'(r) = 0, (10)

whilef 0(t) = f0e

��t.

The boundary conditions for the ordinary boundary value problem are then

limr!0+

' finite, '(1) + `'0(1) = 0. (11)

The equation (10) is the Bessel equation of order zero. See [24], an onlinereference, and successor to the classical handbook [1]. We find both to be usefulin analytical treatments. Together (10),(11) form a singular Sturm-Liouvilleproblem.

3

1.3 Singular Sturm-Liouville problems

An eigenvalue problem is called singular if the interval (a, b) on which it isdefined is infinite or if one or more of the coe�cients of the equation havesingular behavior at x = a or x = b. For instance, in

d

dx

✓p(x)

dy

dx

◆+ (�w(x)� q(x)) y = 0, (12)

if p(x) ! 0 as x ! a or x ! b, the problem is in general singular. This wouldhappen for the Bessel di↵erential equation of order ⌫:

(xy0)0 + (�x� ⌫2

x)y = 0, 0 < x < b (13)

at x = 0. Here the coe�cient p(x) = x and q(x) = �⌫2/x are both singular atx = 0.

1.3.1 Boundary Conditions

One issue which arises for singular problems is: What are the correct bound-ary conditions to apply at a singular point? Suppose u(x) and v(x) are bothoperated on by the di↵erential operator L such that

Lu = (pu0)0 � qu,

and similarly for v. In general the boundary conditions for (12) at the singularpoint x = a, say, depend on the behavior of

limx#a

{p(x) [u0(x)v(x)� u(x)v0(x)]} .

If the di↵erential operator with the boundary conditions is to be well defined,this limit must vanish [19, p.198] It is helpful to know from simply appealingto the coe�cients, whether this is the case. The simplest such conditions arethose derived by Kaper, Kwong and Zettl [21]. (See also [38].) Consider (12);they suppose that (i) on (a, b), p(x) > 0, but p(x) # 0 as x # a so that

Zb

x

d⇠

p(⇠)= O

�(x� a)��

�, as x # a, 0 < � <

12.

(ii) q(x) is bounded on (a, b).With these assumptions the authors prove that there are many equivalentboundary conditions at x = a which lead to a well defined operator. We men-tion two of these(H1) lim

x#ay(x) exists and is finite, or

(H2) limx#a

(py0)(x) = 0.

4

1.3.2 Orthogonality

Still it is necessary to derive certain orthogonality conditions for (12) whichwe illustrate on Bessel’s equation (13). Its basic solution is y = J

⌫

(p

�x). Wemay take the interval to be [0, 1] after any necessary re-scaling. It is desirableto investigate conditions under which one can be sure that

Z 1

0J

⌫

(p

�k

x)J⌫

(p

�j

x)xdx = 0. (14)

For simplicity, we assume that ⌫ is real and that ⌫ > �1 in order that theintegral will converge. Relevant boundary conditions are

�0y (1) + �1y0 (1) = 0, (15)

while at x = 0 we allow a condition such as (H1) or (H2). Write (13) as

Ly + �xy = 0

and suppose that (z(x), µ) is another eigenfunction-eigenvalue pair, satisfyingthe same boundary conditions as (y(x), �). Evaluate

Z 1

0+[z(Ly + �xy)� y(Lz + µxz)] dx = 0,

which after integration reduces to

x(y0z � z0y)|x=1 � lim

x#0[x(y0z � z0y)] + (�� µ)

Z 1

0+xyzdx = 0. (16)

By either of the assumed forms:(H1) lim

x#0y(x), lim

x#0z(x) exist and are finite, or

(H2) limx#0

(xy0)(x) = limx#0

(xz0)(x) = 0,

it follows quite readily that the limits in (16) are 0 as x # 0. The conditionsat x = 1 are symmetric and consequently since besides (15) we have �0z (1) +�1z

0 (1) = 0, no boundary terms remain. The desired relation (14) follows aslong as � 6= µ.

1.3.3 Expansions in eigenfunctions

A very important question about a sequence of eigenfunctions yn

of a regularSturm-Liouville problem, orthogonal and square integrable with respect to aweight function w(x), is the following: Can every square-integrable function f beexpanded into an infinite series f =

Pcn

yn

of the yn

? With a suitable meaningattached to sense of convergence of the series, the answer to this question isa�rmative, and the sequence of eigenfunctions y

n

is said to be complete [5].A very important example of this is the Fourier series. In the singular case,

5

we would like to rely on an analogous expansion, an example of which is theFourier-Bessel series.

The expansion coe�cients for an expansion in the eigenfunctions of the typ-ical problem involving (12) are found by assuming for a suitable function f(x)

f =X

cn

yn

. (17)

Then by orthogonalityR

b

a

w(x)yn

(x)ym

(x)dx = 0, n 6= m, soZ

b

a

f(x)w(x)ym

(x)dx =X

cn

Zb

a

w(x)yn

(x)ym

(x)dx

= cm

Zb

a

w(x)y2m

(x)dx.

So mean convergence is defined as limN!1

���f �P

N

n=1 cn

yn

���2

= 0.

These calculations are illustrated on the example of section 1.2. The solutionof (10) finite at r = 0 is given by

'(r) = J0

⇣p�r

⌘.

Applying the boundary conditions (11), because of the identity

d

drJ 00(hr) = �hJ1(hr),

find that the locations of the eigenvalues are given by the roots of the transcen-dental equation

J0

⇣p�

n

⌘� `

p�

n

J1

⇣p�

n

⌘= 0, n = 1, . . . , (18)

Graphs of the first four eigenfunctions are shown, drawn in Maple.The expansion coe�cients may be evaluated explicitly using certain other

properties of the Bessel functions. For instance, an expansion

f(r) = c1J0

⇣p�1r

⌘+ c2J0

⇣p�2r

⌘+ · · ·

is called a Fourier-Bessel series of the second type [35, Chapter 8]. (We examinean expansion of the first type later in section 2.2.) Its coe�cients are given by

cn

=R 10 rf(r)J0

�p�

n

r�dr

R 10 rJ2

0

�p�

n

r�dr

=2

J21

�p�

n

�+ J2

0

�p�

n

�Z 1

0rf(r)J0

⇣p�

n

r⌘

dr

=2

J21

�p�

n

�(1 + `2�

n

)

Z 1

0rf(r)J0

⇣p�

n

r⌘

dr,

by (18). The following theorem is applicable.

6

Figure 1: J0(p

�1x)

Figure 2: J0(p

�2x)

7

Figure 3: J0(p

�3x)

Figure 4: J0(p

�4x)

8

Theorem 1 [35, Chapter 8] Suppose the eigenvalues are determined by (18).Let f(r) be a piecewise smooth (continuous or discontinuous) function defined on[0, 1]. Then the Fourier-Bessel series of the second type converges for 0 < r < 1.Moreover, its sum equals f(r) at every point of continuity of f(r) and

12

[f(r + 0) + f(r � 0)]

at every point of discontinuity of f(r).

For our calculation W (r, 0) = f(r) = 1 � r2 + 2`. The coe�cients may befurther computed using [1]:

Zz

0tJ0(t)dt = zJ1(z)

and Zz

0t3J0(t)dt = (z3 � 4z)J1(z) + 2z2J0(z).

(In each integral make the change of variables t =p

�n

r.) Hence, again with(18), the solution to the original problem may be given as

W (r, t) = 81X

n=1

J0

�p�

n

r�e��nt

�3/2n

(1 + `2�n

) J1(p

�n

).

Graphs of the complete time-dependent solution is given in [25]. Because theinitial function W (r, 0) satisfies the boundary conditions (11) it may be shownthat the series is uniformly convergent [35], [36].

1.3.4 Other issues and the plan to follow

Another issue that arises is: What happens to the spectrum? That is, when isthe spectrum real? When will there still be an infinite sequence of eigenfunctionsand eigenvalues? If the nature of the spectrum is di↵erent it is necessary tocharacterize the change. In the case of singular Sturm-Liouville problems, theother possible spectral points are said to be continuous as opposed to discrete.We will not pursue this issue further here. However, there are many goodreferences where this is done ([34], [3], [38]).

Another important factor for many readers is the need to calculate the spec-trum numerically and determine the eigenfunctions. This is not our objectivehere. However, in the last few years several freely available programs and codeshave been developed. Indeed this is an area requiring theoretical sophisticationin both di↵erential equations and in numerical analysis. A useful place to beginthis search would be at the website http://www.math.niu.edu/SL2/.

In the next section, we will review the underlying theory of the solution ofa regular system Ly = f in the formulation of the Green’s matrix for regular

9

systems. In the succeeding section, we and state and prove a theorem for theexistence of the Green’s matrix for a system which has a singularity of the firstkind in just the first order case and we look at the tube flow problem as itillustrates the singular Sturm-Liouville problem and its perturbation. In thelast section we will give worked examples and applications of the theory.

1.4 Regular Green’s matrix

For systems of di↵erential equations the desired representation of the solutionmay be expressed in terms of a Green’s matrix [31]. The functions derived hereare sometimes referred to as Green’s “tensor” ([11]) or Green’s “dyadic” ([27,Chap. 13]) by other authors. Consider the operator defined by

Lu = P0(x)u(m) + · · ·+ Pm�1(x)u0 + P

m

(x)u, a < x < b,

where Pi

(x), i = 1, . . . m, are n⇥ n matrices, P0(x) is non-singular and

u(x) =

2

64u1(x)

...u

n

(x)

3

75 .

To be solved is the non-homogeneous problem

Lu = f

with boundary conditions

Bi

(u) = 0, i = 1, . . . ,m.

The boundary conditions involve linearly independent combinations of u,u0, . . . ,u(m�1),at x = a and x = b. The solution is expressible as

u =Z

b

a

G(x, ⇠)f(⇠)d⇠,

where G is an n ⇥ n matrix. To construct G there are four conditions whichapply to it as a function of x:

(i) G is continuous on a x b, and has continuous derivatives up to order(m� 2).

(ii) Its (m� 1)st derivative has an upward jump at x = ⇠, that is,

@(m�1)

@x(m�1)G

�x=⇠

+

x=⇠

�= P�1

0 (⇠).

(iii) G satisfies L(G) = 0, except at x = ⇠. In particular, we use the operationaldefinition based on the Dirac function, which is developed in standard courses[15] as

L(G) = �(x� ⇠)I, (19)

10

where I is the n⇥ n identity matrix.(iv) G satisfies the boundary conditions.

Some special representations of the Green’s function may be found if the oper-ator L is symmetric or skew-symmetric [17].

Example 2 Determine the Green’s matrix for the system:

�u001 � u2 = f1(x)�u002 = f2(x) , 0 < x < 1,

u1(0) = u2(0) = u1(1) = u2(1) = 0.

Solution. The Green’s matrix satisfies (19) as"� @

2

@x

2 �10 � @

2

@x

2

# G11 G12

G21 G22

�= �(x� ⇠)

1 00 1

�,

and column-wise the boundary conditions so that

G11(0) = G21(0) = G11(1) = G21(1) = 0,

andG12(0) = G22(0) = G12(1) = G22(1) = 0.

The sub-problems to be solved are:

� @2

@x2G21 = 0 for all x, ⇠,

so that with the boundary conditions G21 ⌘ 0. Thus since

� @2

@x2G11 �G21 = �(x� ⇠),

G11 is a simple Green’s function, from (8)

G11(x, ⇠) = min(x, ⇠)� x⇠.

Likewise we have for G22,

� @2

@x2G22 = �(x� ⇠),

with the same boundary conditions as G11, so that G22 = G11. The last entryG12 is found from

� @2

@x2G12 �G22 = 0. (20)

This may be solved in several ways. It may reduced to a fourth order problemby applying �@2/@x2 to obtain

✓� @2

@x2

◆2

G12 = � @2

@x2G22 = �(x� ⇠).

11

This may also be solved for the Green’s function G12 with the boundary condi-tions

G22(0, ⇠) =@2

@x2G22(0, ⇠) = G22(1, ⇠) =

@2

@x2G22(1, ⇠) = 0.

Alternatively, from (20) we may write

G12(x, ⇠) =Z 1

0G11(x, ⇠1)G22(⇠1, ⇠)d⇠1.

A first order exampleWe specialize to the system of n equations in vector form,

Ly = y0 + A(x)y = f(x), (21)

with boundary conditions

B0y(0) + B1y(1) = 0. (22)

where y, f are n�vectors and A = (aij

(t)) is an n⇥n matrix of functions. LetD : = B0Y(0) + B1Y(1) be non-singular, where Y(x) is a fundamental matrixof the homogeneous form of (21). Then the Green’s function is given explicitlyby [10]:

G(x, t) =⇢�Y(x)D�1B1Y(1)Y�1(t), x < tY(x)D�1B0Y(0)Y�1(t), x > t

. (23)

We further conclude that we have found the operator inverse of L giving

y(x) = L�1f =Z 1

0G(x, t)f (t) dt.

An example of the usefulness of this approach may be found by consult-ing Wikipedia: http://en.wikipedia.org/wiki/Green%27s matrix. The bound-ary conditions on which that reference focuses are initial conditions for whichB0 = I, and B1 = O, the zero matrix. The interested reader will find thereworked out in detail the example where (in our notation) in problem (21)

y =

y1

y2

�and A =

0 �11 0

�.

So, a fundamental matrix is

Y(x) =

cos x sin x� sin x cos x

�, and Y�1(t) =

cos t � sin tsin t cos t

�.

The Green’s matrix is simply

G0(x, t) =⇢

O, x < tY(x)Y�1(t), x > t

=

8<

:

O, x < tcos(x� t) sin(x� t)� sin(x� t) cos(x� t)

�, x > t

.

12

2 Singular systems

2.1 Structure of Green’s matrix

We specialize to the case of first order systems. Consider a system much like(21) on the interval 0 < x < 1, except that the system has a singularity of thefirst kind at x = 0 so that it may be written as

Ly = y0 +1xSy + Q(x)y = f(x), (24)

where S is a constant matrix and Q(x) is regular on [0, 1]. An existence theoryhas been developed for the solutions near the singular point x = 0 ([9], [10]).In response to the question, what are reasonably simple boundary conditions toappend to solutions of (24) [28], Brabston ([6],[7]) proved that the condition

limx#0

B0y(x) + B1y(1) = 0 (25)

is adequate, even though y(x) may not be defined at x = 0.We can define the Green’s matrix verbatim as we did in properties (i)-(iv),

except that the boundary conditions of (22) are replaced by (25). We are thenable to conclude that

y(x) = L�1f =Z 1

0G(x, t)f (t) dt

is a true solution of (24)-(25). The verification of this is next.From the theory of Brabston we have the following:

Theorem 3 A necessary and su�cient condition for a solution of (24)-(25) toexist is that

limx#0

B0Y(x) and limx#0

B0yp

(x) exist

where Y(x) is a fundamental matrix with yp

a particular solution of (24), andthat

D := limx#0

B0Y(x) + B1Y(1)

be non-singular.

Proof. See [6], [7].We now make the following theorem:

Theorem 4 [30] A necessary and su�cient condition for the Green’s matrix toexist for (24)-(25) is that

limx#0

B0Y(x) + B1Y(1)

is non-singular.

13

Proof. The su�ciency condition is readily given by the theorem of Brabston.To show necessity, let us assume that the Green’s function exists. Since

G(x, t) is a formal solution we can write

G(x, t) =⇢

G1(x, t), x < tG2(x, t), x > t

,

where G1(x, t) = Y(x)H1(t) and G2(x, t) = Y(x)H2(t), and G(x, t) satisfiesthe boundary condition.

Let us consider

limx#0

B0G(x, t) + B1G(1, t)

= limx#0

B0G1(x, t) + B1G2(1, t)

= limx#0

B0Y(x)H1(t) + B1Y(1)H2(t)

= limx#0

B0Y(x)H1(t) + B1Y(1)⇥Y�1(t) + H1(t)

⇤

=limx#0

B0Y(x) + B1Y(1)�H1(t) + B1Y(1)Y�1(t)

= P, some matrix.

Further limx#0

B0Y(x) + B1Y(1)�H1(t) = P�B1Y(1)Y�1(t).

So, H1(t) is unique if and only if limx#0

B0Y(x) + B1Y(1) is non-singular. Hence

the theorem.

The expression for the Green’s function in the singular case is thus:

G(x, t) =⇢�Y(x)D�1B1Y(1)Y�1(t), x < t

Y(x)D�1B00Y�1(t), x > t, (26)

where B00 = limx#0

B0Y(x).

The material presented in this section has also been ably developed morerecently in [38].

2.2 Stability of flow through a tube: a singular boundary

value problem

The stability of the axial steady flow considered earlier in section 1.2 is ex-amined in more detail as was done by [25]. The time-dependent calculationsperformed earlier showed that disturbances which are independent of z and ✓are unlikely to produce an instability. So, a complete study would be to considerfully non-axisymmetric perturbations. But because of the nature of this articlewe will consider only the axisymmetric perturbation equations for the velocity

14

components (vr

, vz

) and refer the interested reader to other more comprehensivereferences ([14], [26]):

✓d2

dr2+

1r

d

dr� k2 � 1

r2� ikRW0(r)

◆v

r

= Rdp

dr� �v

r

, 0 < r < 1, (27)

✓d2

dr2+

1r

d

dr� k2 � ikRW0(r)

◆v

z

� 2r

1 + 2`v

r

= ikRp� �vz

, (28)

1r

d(rvr

)dr

+ ikvz

= 0. (29)

The boundary conditions on this singular boundary value problem are

vr

, vz

, p finite at r = 0, vr

(1) = 0, vz

(1) + `dv

z

dr(1) = 0.

Here R is the Reynolds number, and

W0(r) =1� r2 + 2`

1 + 2`

is the basic axial velocity, k is the axial wave number, p(r) is the pressureperturbation and the eigenvalue � = �ikRc, for wave speed c.

We take a more abstract approach. First consider the operator counterpartto (13) as

M⌫

' :=✓� d2

dr2� 1

r

d

dr+ k2 +

⌫2

r2

◆' = �', 0 < r < 1, (30)

where the domain of M⌫

⇢ H is determined by the boundary conditions. Wewill principally be interested in the cases where ⌫ = 0 and ⌫ = 1. For M1 weconsider functions such that lim

r#0' finite and '(1) = 0, while for M0 we consider

limr#0

' finite and '(1) + `'0(1) = 0. The inner product is

h', �i =Z 1

0+r'(r)�(r)dr, ',� 2 H, (31)

where

H =⇢

' |Z 1

0+r |�|2 dr

�< 1,

and normk ' k= h', 'i1/2

.

Remark 5 The operators M⌫

are selfadjoint, that is M⌫

= M⇤⌫

and positivedefinite so that hM

⌫

', 'i � µ⌫

k ' k2, µ⌫

> 0 [15].

Several other things can be shown and are embodied in the following.

15

Lemma 6 The eigenvalues of M⌫

are real, simple and satisfy 0 < �(⌫)1 <

�(⌫)2 < · · · , where �

(⌫)n+1 � �

(⌫)n

= O(n) as n ! 1. The problem (30) has acomplete set of eigenfunctions in H.

The detailed proof of the completeness will not be presented. That theeigenfunctions of M

⌫

are complete follows from the fact that (30) may be trans-formed into the Bessel equation so that the boundary-value problem producesthe Fourier-Bessel series, whose expansion theory is standard ([2, p.214], [35]).However, what makes the proof of the rest of the lemma particularly a↵ect-ing is that it depends on the asymptotic behavior of the eigenfunctions of theSturm-Liouville problem.

For M0 we may refer to the results of sections 1.2 and 1.3.3. The equations

developed were (10)-(11). The eigenfunctions are therefore 'n

= J0(q

�(0)n

r)with the eigenvalues satisfying

J0

✓q�

(0)n

◆� `

q�

(0)n

J1

✓q�

(0)n

◆= 0, n = 1, . . . . (32)

We have by standard asymptotic methods [24] that

J0

✓q�

(0)n

◆⇠

vuut2

⇡

q�

(0)n

sin✓q

�(0)n

� ⇡/4◆

, (33)

and

J1

✓q�

(0)n

◆⇠

vuut2

⇡

q�

(0)n

cos(q

�(0)n

� ⇡/4), (34)

as �(0)n

! 1, so the eigenvalues of large index may be estimated. Put the

estimates (33),(34) into the transcendental equation (32) to findq

�(0)n

⇠ (n +3/4)⇡ as n !1. Hence �

(0)n+1 � �

(0)n

= O(n) as n !1.For M1 we are able to identify directly the locations of the eigenvalues

because setting�2 = �� k2, (35)

the eigenfunctions are' = J1(�r). (36)

Explicitly one finds that the numbers �(1)n

(k) are given by the roots J1(�n

) =0, n = 1, 2, . . . . This leads to what is called a Fourier-Bessel series of the firsttype [35, Chapter 8]. A common notation for these zeros is j1,n

, so

�(1)n

= j21,n

+ k2.

From our references or by a computer algebra system, such as Maple we maydetermine these numbers to six decimal places, if desired. The figures shown

16

Zeros of the Bessel function J1(x)n j1,n

= �n

1 3.8317059702 7.0155866703 10.173468144 13.32369194

Figure 5: J1(�1r)

were drawn in Maple and show the behavior of the first four eigenfunctions.One notices that the number of zeros on the basic interval [0, 1] exactly matchesthe index of the eigenfunctions, as is the case for Fourier series. However, thespacing of the zeros is unlike Fourier series, whose periods are uniform.

Now by (35) as � ! 1, the eigenvalues of large index may be estimated.Fix k, then using (35), the conclusion is � ⇠

p�. Related to (34),

J1(�) ⇠r

2⇡�

sin(� � ⇡/4) as � !1. (37)

Put the estimate (37) into (36) and find thatq

�(1)n

⇠ (n + 1/2)⇡ as n ! 1.

Hence �(1)n+1 � �

(1)n

= O(n) as n !1.

2.3 Completeness for the tube flow stability equation

The verification of the completeness of the eigenfunctions of the tube flow stabil-ity problem (27)-(29) may be analyzed as an operator perturbation. So considerthe equation

A� + P� + B� = ��, (38)

17

Figure 6: J1(�2r)

Figure 7: J1(�3r)

18

Figure 8: J1(�4r)

whereA =

M1 00 M0

�, B =

ikRW0(r) 0

2r

1+2`

ikRW0(r)

�(39)

� =

vr

vz

�

and an operator P implicitly by

P� =

R dp

dr

ikRp

�.

It is understood that the axisymmetric continuity equation (29) applies to (38).In the space of 2-vectors H⇥ H. it can thereby shown that A+P is selfadjoint.From that point of view we can rely on one of the theorems of [22, p.293], [13],or [12], which ensure that (38) also possesses a complete set of eigenfunctionsin H. For example, based on the theorem of [13] we can assert the following.

Remark 7 Given A as above, with compact resolvent and simple eigenvaluessatisfying �

n+1 � �n

= O(n) as n !1.Let S = A+ P + B, where B denotes the bounded operator of multiplication.Then (i) S has compact resolvent and its eigenvalues lie within circles of radiik B k about those of M+ P and the eigenfunctions of S span H⇥ H.

3 Worked examples and applications

3.1 An axisymmetric potential equation

An example which many authors (see [7]) have used to test the accuracy of theirnumerical methods is the following.

19

Example 8 Consider the boundary value problem

u00 +�

xu0 = f(x), 0 < x < 1, (40)

where 0 < � 1, withu(0) finite and u(1) = 0.

The example in the introduction considered the case where � = 2, which is typicalof the situation where � > 1. However the range 0 < � 1 is more subtle. Inorder to illustrate the situation which might occur for a larger system,let

y =

uxu0

�. (41)

Using (41), (40) is transformed into the system

y0 � 1x

0 10 1� �

�y =

0

x�f

�. (42)

When 0 < � < 1, a compatible boundary condition is of the form

limx#0

B0y(x) + B1y(1) = 0, (43)

where the boundary matrices are

B0 =

1 00 0

�, B1 =

0 01 0

�.

A basis to (40) is {1, x1��}, � 6= 1, and with (41) a fundamental matrix givenby

Y(x) =

u1 u2

xu01 xu02

�=

1 x1��

0 (1� �) x1��

�,

andY�1(t) =

1 �1/(1� �)0 t��1/(1� �)

�.

We have

B00 = limx#0

1 00 0

� 1 x1��

0 (1� �) x1��

�= lim

x#0

1 x1��

0 0

�=

1 00 0

�.

Consequently

D = B00 + B1Y(1) =

1 01 1

�, D�1 =

1 0�1 1

�

and combining the matrices according to (26) we obtain desired Green’s matrix:

G(x, t) =

8>>>><

>>>>:

"�x1��

x

1��(1�t

��1)1��

[�(1� �) x1�� x1��

�1� t��1

�

#, x < t

"1� x1��

x

1���11��

� (1� �) x1�� x1��

#, x > t

.

20

Finally

y(x) = L�1f =Z 1

0G(x, t)f (t) dt

is the solution to the problem. The expression for u(x) depends directly onG12(x, t) as

u(x) =Z 1

0G12(x, t)t�f(t)dt.

When � = 1, a compatible boundary condition is is also of the form (43),where

B0 =

0 10 0

�, B1 =

0 01 0

�.

A basis of solutions to (40) is {1, log x} when � = 1. With (41), a fundamentalmatrix is

Y(x) =

u1 u2

xu01 xu02

�=

1 log x0 1

�,

andB00 = lim

x#0

0 10 0

� 1 log x0 1

�= lim

x#0

0 10 0

�=

0 10 0

�.

And we haveD = B00 + B1Y(1) =

0 11 0

�= D�1,

andY �1(t) =

1 � log t0 1

�.

Then combining all of the matrices according to (26),

G(x, t) =

8>><

>>:

�1 log t0 0

�, x < t

0 log x0 1

�, x > t

.

Again, the solution has the integral representation

y(x) = L�1f =Z 1

0G(x, t)f (t) dt.

The expression for u(x) depending on G12(x, t) is

u(x) =Z 1

0G12(x, t)tf(t)dt.

21

3.2 Green’s matrix for the stability of tube flow

We derive the Green’s matrix suitable for use in the tube flow stability problem.The Green’s function was first used years ago in proving completeness in theaxisymmetric case [32]. There, the system was reduced to one fourth-orderdi↵erential equation. Our approach will be such that it may be used in thenon-axisymmetric case [26] and others in which the order is higher.

Example 9 Green’s matrix involving the Bessel operator We wish toanalyze (38) and invert part of the equation, the operator A�� by obtaining theGreen’s matrix. This will permit a further analysis by converting the problem toan integral equation.This example continues the material of section 2.2. We make use of the ideasfrom Example 2. The two operators that must be analyzed are the diagonalentries of the operator A� �, for (M1 � �)�1 and (M0 � �)�1

First for

(M1 � �) ' =✓� d2

dr2� 1

r

d

dr+ k2 � � +

1r2

◆' = f,

For this to be true, we must have from limr#0

' finite and '(1) = 0.

We will need the functions I1(�r) and K1(�r), the modified Bessel functions oforder 1 [24] and �2 = � � k2. Then by standard methods its Green’s functionmay be written

g(r, t;�) =

(u1(r)u2(t)/(p(t)W (t)), r < t

u2(r)u1(t)/ (p(r)W (t)) , r > t. (44)

for the canonical di↵erential operator

(p(r)u0)0 + q(r)u = f(r), a < r < b, (45)

So, here p(r) = �r where u1 and u2 satisfy the left and right boundary conditionsrespectively. We take

u1 = I1(�r), u2 = [K1(�)I1(�t)� I1(�)K1(�t)] .

The Wronskian W is thereby

W (t) = det

I1(�t) K1(�)I1(�t)� I1(�)K1(�t)d

dt

I1(�t) d

dt

(K1(�)I1(�t)� I1(�)K1(�t))

�. (46)

It is known from [24] that the Wronskian

W{K1(z), I1(z)} = I0(z)K1(z) + I1(z)K0(z) = 1/z,

involving the modified Bessel functions of order zero. Hence in (46) we have

W (t) =�I1(�)

t.

22

We transform back to the original operator (M1 � �)�1. Its Green’s function is

therefore

G11(r, t;�) =⇢

I1(�r)[K1(�)I1(�t)� I1(�)K(�t)]/I1(�), r < t[K1(�)I1(�r)� I1(�)K1(�r)]I1(�t)/I1(�), r > t

.

In order to make the calculations of this section more instructive we constructthe Green’s function for (M0 � �)�1 by use of the Green’s matrix as was donein section 3.1 when � = 1. That is we consider

(M0 � �)' =✓� d2

dr2� 1

r

d

dr+ k2 � �

◆' = f

where the boundary conditions are (11)

limr!0+

' finite, '(1) + `'0(1) = 0. (47)

We sety =

'

r'0

�.

The system becomes

y0 +1r

0 �10 0

�y+r

0 0�

�� k2�

0

�y =

0�rf

�.

The boundary condition is

limr#0

B0y(r) + B1y(1) = 0

whereB0 =

0 10 0

�, B1 =

0 01 `

�.

A suitable fundamental matrix is given by

Y(r) =

'1 '2

r'01 r'02

�=

I0(�r) K0(�r)

�rI1(�r) ��rK1(�r)

�.

We may then determine

D = limr#0

B0Y(r) + B1Y(1)

= limr#0

�rI1(�r) ��rK1(�r)

0 0

�+

0 0

I0(�) + `�I1(�) K0(�)� `�K1(�)

�.

Make use of the limiting forms [24]:

I0(�r) ⇠ 1, I1(�r) ⇠ 12�r and

K0(�r) ⇠ � log(�r), K1(�r) ⇠ 1�r

as r ! 0, � fixed.

23

Then

D =

0 �1I0(�) + `�I1(�) K0(�)� `�K1(�)

�,

D�1 =1

I0(�) + `�I1(�)

K0(�)� `�K1(�) 1� (I0(�) + `�I1(�)) 0

�.

soY�1(t) =

�tK1(�t) K0(�t)�tI1(�t) �I0(�t)

�.

We are able to supply the required Green’s matrix as

G(r, t;�) =⇢�Y(r)D�1B1Y(1)Y�1(t), r < tY(r)D�1B0Y(0+)Y�1(t), r > t

, (48)

where B0Y(0+) = limr#0

B0Y(r). The original problem may be converted to

y(r) = (L� �)�1 f =Z 1

0G(r, t;�)f (t) dt,

The expression for '(r) depends directly on G12(r, t) as

'(r) = ��

Z 1

0G12(r, t;�)'(t)dt,

The usual expansion theory for the Fourier-Bessel series G12 is known from [2]:

G12(r, t;�) =X

n

yn

(r)yn

(t)�� �

n

. (49)

We use the operational definition based on the Dirac function, which is developedin standard courses [15] as

MG12 + �G12 = �(r � t),

where M is the same operator as in section 2.2. Hence applying this to (49)obtain formally

�(r � t) = (M+ �)X

n

yn

(r)yn

(t)�� �

n

=X

n

(M+ �)yn

(r)yn

(t)�� �

n

.

From this, make use of the fact that yn

is an eigenfunction with eigenvalue �n

,to obtain

�(r � t) =X

n

yn

(r)yn

(t) =X

n

yn

(r)yn

(t), (50)

24

since �(r � t) = �(t� r). It may also be obtained by examining the poles of theGreen’s function G12(r, t;�) in the complex ��plane [15].

We can view G12(r, t;�) as an analytic function of �, except for poles at theeigenvalues (here simple), where � = �

n

. If C is a closed contour in the complex��plane, containing all the poles of G12 then from (49)

12⇡i

Z

C

G12(r, t;�)d� =1

2⇡i

Z

C

X

n

yn

(r)yn

(t)�� �

n

d�

=X

n

yn

(r)yn

(t), sinceZ

C

d�

�� �n

= 2⇡i.

We thus have the expansion of � :

�(r � t) =1

2⇡i

Z

C

G12(r, t;�)d�. (51)

3.3 Convection in a fluid sphere

The concluding example is of a higher order boundary value problem whichmay be treated by Green’s matrix methods. There are several geophysical andastrophysical applications in which this model is important [23, Section 8.4].We will consider a homogeneous fluid sphere in which the surface is rigid, asat the interface with the mantle of the liquid core of the earth. Another pos-sible application is with a free surface boundary condition of an object in freespace. At the onset of convection there will arise perturbations in temperatureand velocity within the fluid [8, Chapter VI]. The same system arises in study-ing nonlinear convection, using energy theory, and by bifurcation methods [20,Chapter X].

Example 10 A problem considered some time ago by Chandrasekhar [8] in-volves solving a system of higher order boundary value problems, one part ofwhich is

M2`

w = f(r), 0 < r < 1,

where M`

:=d2

dr2+

2r

d

dr� `(` + 1)

r2, ` = 1, 2, . . . . The rigid surface boundary

conditions are

w,dw

drfinite at r = 0, w =

dw

dr= 0 at r = 1.

The free surface boundary conditions have d

2w

dr

2 = 0 instead of dw

dr

= 0 at r = 1.Spherical harmonics were used and contribute to the form of the operator M

`

.For simplicity and clarity of exposition we specialize to the case ` = 1. Numericalevidence suggests that the easiest modes to excite are those belonging to ` = 1[8, p.234]. Because the boundary conditions involve both w and dw

dr

at r = 1, themethod of the last section, splitting the operator into two second order factors

25

is not simple. Instead put the system in the form treated in section 2.1. We setu = r2w. (In the general case set u = r`+1w.) Then after some manipulationfind that

M1w =d

dr

✓1r2

du

dr

◆.

Now set v = M1w and the original equation becomes

M21w = M1v =

d

dr

✓1r2

dv

dr

◆= f(r),

with r2w, r2v finite at r = 0. Letting primes denote di↵erentiation with respectto r, we may transform this to a first order system of the form of (24) by setting

y =

2

664

r2w�r2w

�0

r2v�r2v

�0

3

775 , (52)

obtaining

y0 +1rSy + Q(r)y = f(r)

with

S =

2

664

0 0 0 00 �2 0 00 0 0 00 0 0 �2

3

775 , Q =

2

664

0 �1 0 00 0 �1 00 0 0 �10 0 0 0

3

775 , f(r) =

2

664

000

r2f(r)

3

775 .

Though other matrix decompositions may be possible, the boundary conditionsare determined by the particular form of (52). We may take

B0 =

2

664

1 0 0 00 0 1 00 0 0 00 0 0 0

3

775 , B1 =

3

775

0 0 0 00 0 0 01 0 0 00 1 0 0

3

775 .

A suitable fundamental matrix may be found from the elements of M21w =

0. Setting w = rm in this 4th order Euler-Cauchy equation we find that itscharacteristic equation is

m (m� 1) (m + 2) (m� 3) = 0.

So, by (52), we may take

Y(r) = [y1 y2 y3 y4] =

2

664

r2 r3 1 r5

2r 3r2 0 5r4

�2 0 0 10r3

0 0 0 30r2

3

775 .

26

The next step is to calculate D = limr#0

B0Y(r) + B1Y(1) which here is

D =

2

664

1 0 0 00 0 1 00 0 0 00 0 0 0

3

775

2

664

0 0 1 00 0 0 0�2 0 0 00 0 0 0

3

775 +

2

664

0 0 0 00 0 0 01 0 0 00 1 0 0

3

775

2

664

1 1 1 12 3 0 5�2 0 0 100 0 0 30

3

775

=

2

664

0 0 1 0�2 0 0 01 1 1 12 3 0 5

3

775 ,

so

D�1 =14

2

664

0 �2 0 0�10 3 10 �24 0 0 06 �1 �6 2

3

775 .

The form (26) is employed. The desired Green’s matrix has the form

G(r, t) =⇢�Y(r)D�1B1Y(1)Y�1(t), r < tY(r)D�1B0Y(0+)Y�1(t), r > t

, (53)

where

Y�1(t) =

2

664

0 0 � 12

16 t

0 13 t�2 1

3 t�1 � 16

1 � 13 t 1

6 t2 � 130 t3

0 0 0 130 t�2

3

775 .

The result, of course, is G(r, t) when all of the factors are evaluated. A com-puter algebra system such as MAPLE R� is invaluable in these calculations. Ofparticular interest is G14(r, t) which figures prominently in the evaluation ofw(r). We are able to write

w(r) =Z 1

0r�2G14(r, t)t2f(t)dt,

where

G14(r, t) =

(� 1

12

�3r3 � r5

�t + 1

6r3 + 160

�5r3 � 3r5

�t3 � 1

30r

5

t

2 , r < t112 (2r2 � 3r3 + r5)t� 1

30 (1� 52r3 + 3

2r5)t3, r > t.

3.4 An exercise

The interested reader may find it illuminating to solve the following variant tothe last example.

Find the Green’s matrix in the free surface case:

M21w = f(r), 0 < r < 1,

27

w,d2w

dr2finite at r = 0, w =

d2w

dr2= 0 at r = 1.

This problem may be solved in the following way. Reduce it to a first ordersystem. Note that the same B0 is applicable but owing to the assumed form(52), the appropriate B1 has to be found. Because

r2v = M1w = y3(r),

w(1) = w00(1) = 0 ) the relevant boundary condition is y3(1) = y2(1). Thisleads to the choice

B1 =

2

664

0 0 0 00 0 0 01 0 0 00 1 �1 0

3

775 .

Acknowledgement The author would like to thank Ms. Emaan Abdul-Majidwho read the manuscript and made valuable suggestions from a learner’s pointof view.

References

[1] M. Abramowitz and I. A. Stegun, eds. Handbook of MathematicalFunctions, Dover, New York, 1972

[2] N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators inHilbert Space, vol II, Ungar, New York, 1963

[3] M. Ya. Antimirov, A.A. Kolyshkin and R. Vaillancourt, AppliedIntegral Transforms, American Mathematical Society, Providence, 1993

[4] G. K. Batchelor and A. E. Gill, Analysis of the stability of axisym-metric jets, J. Fluid Mech. 14 (1962), pp.529-551

[5] G. Birkhoff and G. C. Rota, On the completeness of Sturm-Liouvilleexpansions, Am. Math. Month., 67, 835-841 (1960).

[6] D. C. Brabston, Numerical Solution of Singular Endpoint BoundaryValue Problems, Ph.D. dissertation, California Institute of Technology,Pasadena, 1974

[7] D. C. Brabston and H. B. Keller, A numerical method for singulartwo point boundary value problems, SIAM J. Numer. Anal. 14 (1977), pp.779-791.

[8] S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability, OxfordUniversity Press, London, 1961.

[9] E. Coddington and N. Levinson, Ordinary Di↵erential Equations,McGraw-Hill, New York, 1955.

28

[10] R. H. Cole, Theory of Ordinary Di↵erential Equations, Appleton-Century-Crofts, New York, 1968.

[11] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. I,Wiley-Interscience, New York, 1953

[12] S. H. Davis, On the principle of exchange of stabilities, Proc. Roy. Soc.A 310, 341-358 (1969).

[13] R. C. DiPrima, R. C. and G. J. Habetler, A completeness theory fornon-selfadjoint eigenvalue problems in hydrodynamic stability, Arch. Rat.Mech.. Anal. 34, 218-227 (1969).

[14] P. G. Drazin, Introduction to Hydrodynamic Stability, Cambridge Uni-versity Press, 2002.

[15] B. Friedman, Principles and Techniques of Applied Mathematics, ,Dover,New York, 1990

[16] M. A. Heckl and M. S. Howe, Stability analysis of the Rijke tube witha Green 0s function approach, Journal of Sound and Vibration, 305 (2007),pp. 672-688.

[17] I. H. Herron, A method of constructing Green 0s functions for ordinarydi↵erential systems, Bull. Inst. Math. and its Appl. (1989) 25, 233-237.

[18] I. H. Herron and M. R. Foster, Partial Di↵erential Equations in FluidDynamics, Cambridge University Press, 2008

[19] S. S. Holland, Jr., Applied Analysis by the Hilbert Space Method, Dekker,New York, 1990.

[20] D. D. Joseph, Stability of Fluid Motions II, Springer-Verlag, Berlin, 1976

[21] H. G. Kaper, M. K., Kwong. and A. Zettl, Characterizations ofthe Friedrichs extensions of singular Sturm-Liouville expressions, SIAM J.Math. Anal. 17 (1986), pp. 772-777.

[22] T. Kato, Perturbation Theory for Linear Operators (2nd ed.), Springer-Verlag, New York , 1976.

[23] E. L. Koschmieder, Benard Cells and Taylor Vortices, Cambridge Uni-versity Press, 1993.

[24] D. Lozier, F. Olver, C. Clark and R. Boisvert, eds. , Digital Li-brary of Mathematical Functions, National Institute of Standards and Tech-nology, 2010, http://dlmf.nist.gov/.

[25] M. T. Matthews and J. M. Hill, E↵ect of slip on the linear stabilityof flow through a tube, Z. angew. Math. Phys. 59 360–379 (2008).

29

[26] A. Meseguer and L. N. Trefethen, Linearized pipe flow to Reynoldsnumber 10 7, J. Comp. Phys. 186, 178-197 (2003)

[27] P. M. Morse & H. Feshbach, Methods of Theoretical Physics, McGraw-Hill, New York, 1953.

[28] F. Natterer, A generalized spline method for singular boundary valueproblems of ordinary di↵erential equations, Linear Algebra Appl. 7 (1973),pp.189-216

[29] M. N. Ozisik, Boundary Value Problems in Heat Conduction, Dover, NewYork, 1968

[30] P. Philip, The Green 0s Function for a Singular Boundary Value Problem,M.S. Thesis, Department of Mathematics, Howard University, WashingtonD. C., 1975

[31] W. T. Reid, Ordinary Di↵erential Equations, Wiley, New York, 1971

[32] I. V. Schensted, Contributions to the theory of hydrodynamic stability,Ph.D. Dissertation, Univ. of Michigan, 1960.

[33] I. Stakgold, Green 0s Functions and Boundary Value Problems, SecondEdition, Wiley-Interscience, New York, 1998.

[34] E. C. Titchmarsh, Eigenfunction Expansions Associated with Second-order Di↵erential Equations, Clarendon, Oxford, 1946.

[35] G. P. Tolstov, Fourier Series, Dover, NY 1976

[36] H. F. Weinberger, A First Course in Partial Di↵erential Equations,Dover, NY 1995

[37] J. E. Wilkins, Jr., Conduction of heat in an insulated spherical shell witharbitrary spherically symmetric sources, SIAM Rev.1, 149-153 (1959).

[38] A. Zettl, Sturm-Liouville Theory, American Mathematical Society, Prov-idence, 2005

30