Kelly J.J. Graduate Mathematical Physics, With MATHEMATICA Supplements
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9 Boundary-Value Problems
Abstract. Several methods are developed for constructing solutions to partial dif-
ferential equations that match boundary conditions on a surface. We focus on equa-
tions based upon the Laplacian operator, such as the Helmholtz equation, that are
expressed in separable orthogonal coordinate systems. Eigenfunction methods can
then be used to develop series expansions or integral representations. Green func-
tions provide general solutions for inhomogeneous problems. Examples include elec-
trostatics, magnetostatics, and scattering theory. Exercises at the end of the chapter
explore a wider variety of problems.
9.1 Introduction
We are often faced with the problem of solving a partial differential equation within a vol-
ume V bounded by a surface S subject to boundary conditions that specify the solution
and/or some of its derivatives on S. Since neither time nor space nor patience nor expertise
permits an exhaustive survey of this extremely broad subject, we shall be content to study
equations based upon the Laplacian operator expressed in separable orthogonal coordi-
nate systems. We take as a prototype the inhomogeneous Helmholtz equation
+
2
k
2
Ψ
r,t4ΠΡ
r,t (9.1)
which includes among its special cases Laplace’s equation (k 0, Ρ0), Poisson’s equa-
tion (k 0), the wave equation (k Ω/c), the diffusion equation (k
2
!Α), and the
Schrödinger equation (k
2
2mE, 4ΠΡ 2mV Ψ). In the happy circumstance that the
boundary consists of surfaces on which one of the coordinates is constant, such problems
are often amenable to the method of separation of variables.
Rather than plunging directly into the general theory, we illustrate the basic strategy
by solving Poisson’s equation
+
2
Ψ
r4ΠΡ
r (9.2)
within a rectangular box 0 x a,0 y b,0 z c with specified potentials
upon its surfaces. We can divide the problem into several parts. First, we determine the
potentials Ψ
i
r,t that satisfy Laplace’s equation
+
2
Ψ
i
r0 ,
r S
i
Ψ
i
V
i
,
r S
ji
Ψ
i
0 (9.3)
where Ψ
i
is specified by the two-dimensional function V
i
on surface S
i
and vanishes on the
other five faces of the box. Then we determine the Green function that satisfies
+
2
G
r,
r
'
4Π∆
r
r
'
(9.4)
Graduate Mathematical Physics. James J. Kelly
Copyright © 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ISBN: 3-527-40637-9
328 9 Boundary-Value Problems
with boundary conditions
Ψ0,y,zΨa, y, zΨx, 0,zΨx, b, zΨx, y, 0Ψx, y, c0 (9.5)
on the surface of a grounded box. Finally, recognizing that Poisson’s equation is linear, we
can superimpose these solutions to obtain the general solution
Ψ
r
V
G
r,
r
'
Ρ
r
'
V
'
6
i1
Ψ
i
r (9.6)
for an arbitrary internal charge distribution and arbitrary potentials on each of the six sides
of the box.
9.1.1 Laplace’s Equation in Box with Specified Potential on one Side
Suppose that we wish to solve Laplace’s equation +
2
Ψ0 in a rectangular volume 0
x a,0 y b,0 z c with Ψ0 on all faces except
z c Ψx, y, cV x, y (9.7)
This problem is ideally suited for the method of separation of variables, wherein we pro-
pose a factorized solution of the form
Ψx, y, zXxYyZz (9.8)
for which
+
2
Ψ0
1
X
2
X
x
2
1
Y
2
Y
y
2
1
Z
2
Z
z
2
0 (9.9)
Recognizing that each of the three terms is a function of a different variable, each must
separately be constant, such that
1
X
2
X
x
2
Α
2
,
1
Y
2
Y
y
2
Β
2
,
1
Z
2
Z
z
2
Γ
2
(9.10)
where the separation constants Α
2
, Β
2
, Γ
2
must satisfy
Α
2
Β
2
Γ
2
(9.11)
Note that the separation constants were chosen with the foresight of experience (disingen-
uous hindsight), but lacking such foresight we would have used simply A, B,C and then
discovered later their most natural interpretations. The boundary conditions with respect
to x, y are satisfied using
X0Xa0 X
n
xSinΑ
n
x , Α
n
nΠ
a
(9.12)
Y0Y b0 Y
m
ySinΒ
m
y , Β
m
mΠ
b
(9.13)
9.1 Introduction 329
with integer m, n, while the lower boundary condition on z requires
Z00 ZzSinh
Γ
nm
z
, Γ
nm
Α
2
n
Β
2
m
(9.14)
A linear superposition of solutions of this form then gives
Ψx, y, z
n,m
A
nm
SinΑ
n
x SinΒ
m
y SinhΓ
nm
z (9.15)
where the upper boundary condition on z requires
Ψx, y, cVx, yA
nm
4
ab SinhΓ
nm
c
a
0
x
b
0
y SinΑ
n
x SinΒ
m
yVx, y
(9.16)
Solutions to more general problems in which nonzero potentials are specified on more than
one face can then be constructed simply by adding several solutions of this type.
9.1.2 Green Function for Grounded Box
We can reduce the three-dimensional equation
+
2
G
r,
r
'
4Π∆
r
r
'
(9.17)
with boundary conditions
Ψ0,y,zΨa, y, zΨx, 0,zΨx, b, zΨx, y, 0Ψx, y, c0 (9.18)
to a one-dimensional equation by using the expansions
∆
r
r
'
∆z z
'
3
4
4
4
4
4
5
2
a
i1
SinΑ
i
x SinΑ
i
x
'
6
7
7
7
7
7
8
3
4
4
4
4
4
5
2
b
j1
SinΒ
j
y SinΒ
j
y
'
6
7
7
7
7
7
8
(9.19)
G
r,
r
'
i, j1
g
i, j
z, z
'
SinΑ
i
x SinΑ
i
x
'
SinΒ
j
y SinΒ
j
y
'
(9.20)
to obtain
Γ
2
i, j
(
2
(z
2
g
i, j
z, z
'
16Π
ab
∆z z
'
,g
i, j
0,z
'
g
i, j
c, z
'
(9.21)
where
Γ
2
i, j
Α
2
i
Β
2
j
(9.22)
is the separation constant formed from two eigenvalues for the degrees of freedom that
we chose to eliminate. The sine expansions satisfy the boundary conditions on four of the
six sides automatically, but we must use the resulting one-dimensional inhomogeneous
330 9 Boundary-Value Problems
Helmholtz equation to reconstruct the delta-function source. We chose to retain the z vari-
able because the nontrivial boundary condition occurs on a side with constant z.
Using solutions to the homogeneous Helmholtz equation
0 z z
'
g
i, j
z, z
'
A
i, j
SinhΓ
i, j
z (9.23)
z
'
z c g
i, j
z, z
'
B
i, j
SinhΓ
i, j
c z (9.24)
that satisfy the boundary conditions one either side of the source and applying the matching
conditions
$g
i, j
z
'
,z
'
0 B
i, j
SinhΓ
i, j
c z
'
A
i, j
SinhΓ
i, j
z
'
0 (9.25)
$g
'
i, j
z
'
,z
'
16Π
ab
B
i, j
CoshΓ
i, j
c z
'
A
i, j
CoshΓ
i, j
z
'
16Π
abΓ
i, j
(9.26)
across the z z
'
interface, we find
A
i, j
16Π
abΓ
i, j
SinhΓ
i, j
c z
'
SinhΓ
i, j
c
,B
i, j
16Π
abΓ
i, j
SinhΓ
i, j
z
'
SinhΓ
i, j
c
(9.27)
Thus, we find
g
i, j
z, z
'
16Π
abΓ
i, j
SinhΓ
i, j
z
<
SinhΓ
i, j
c z
>
SinhΓ
i, j
c
(9.28)
where z
<
is the smaller and z
>
is the larger of z and z
'
. This form should be very familiar,
by now, from our work with Sturm–Liouville systems. Therefore, we can assemble the
entire result
G
r,
r
'
16Π
ab
i, j1
SinΑ
i
x SinΑ
i
x
'
SinΒ
j
y SinΒ
j
y
'
,
SinhΓ
i, j
z
<
SinhΓ
i, j
c z
>
Γ
i, j
SinhΓ
i, j
c
(9.29)
in the form of a piecewise continuous, two-dimensional Fourier sine expansion. The expres-
sions may be lengthy, but the analysis is hardly more complicated than that for one-
dimensional Sturm–Liouville systems. Two other similar representations can be obtained
by simply permuting variables and their associated boundary conditions; the most conve-
nient choice may depend upon the representation of the source density, Ρ
r, in the Poisson
equation. Notice that the reciprocity condition
G
r
'
,
rG
r,
r
'
(9.30)
is satisfied automatically.
9.1 Introduction 331
Alternatively, we can employ Fourier sine expansions in all three dimensions
∆
r
r
'
8
abc
i, j,k1
SinΑ
i
x SinΑ
i
x
'
SinΒ
j
y SinΒ
j
y
'
SinΓ
k
z SinΓ
k
z
'
(9.31)
G
r,
r
'
i, j,k1
g
i, j,k
SinΑ
i
x SinΑ
i
x
'
SinΒ
j
y SinΒ
j
y
'
SinΓ
k
z SinΓ
k
z
'
(9.32)
where now we define Γ
k
kΠ/c with a single index. Substituting into Poisson’s equation
and using orthogonality to equate coefficients
Α
2
i
Β
2
j
Γ
2
k
g
i, j,k
32Π
abc
SinΑ
i
x
'
SinΒ
j
y
'
SinΓ
k
z
'
(9.33)
we obtain
G
r,
r
'
32Π
abc
i, j,k1
SinΑ
i
x SinΑ
i
x
'
SinΒ
j
y SinΒ
j
y
'
SinΓ
k
z SinΓ
k
z
'
Α
2
i
Β
2
j
Γ
2
k
(9.34)
Equivalence between Eq. (9.29) and Eq. (9.34) may be verified by performing a Fourier
sine expansion of g
i, j
z, z
'
, whereby
g
i, j,k
2
c
c
0
g
i, j
z, z
'
SinΓ
k
zz
32Π
abcΓ
i, j
SinhΓ
i, j
c z
'
SinhΓ
i, j
c
z
'
0
SinhΓ
i, j
z SinΓ
k
zz
SinhΓ
i, j
z
'
SinhΓ
i, j
c
c
z
'
SinhΓ
i, j
c z SinΓ
k
zz
(9.35)
Although it is not difficult to evaluate these integrals by hand and simplify the result using
standard trigonometric identities, we prefer to let perform the ‘grunt’ work
to obtain
Simpli fy
32Π
abcΓ
i, j
Sinh
Γ
i, j
c z
!
z
!
0
Sinh
Γ
i, j
z
Sin
Γ
k
z
z
Sinh
Γ
i, j
c
Sinh
Γ
i, j
z
!
c
z
!
Sinh
Γ
i, j
c z
Sin
Γ
k
z
z
Sinh
Γ
i, j
c
$
%
%
%
%
%
%
%
%
%
%
%
%
%
&
/.
Γ
2
i, j
Α
2
i
Β
2
j
, Sin
Γ
k
c
0
/ / Simpli fy
32Π Sin
Γ
k
z
#
abc
Α
2
i
Β
2
j
abcΓ
2
k
in agreement with our analysis based upon Poisson’s equation.
332 9 Boundary-Value Problems
A point charge q located at position
r
'
within a grounded box produces an electrostatic
potential
Ρ
rq∆
r
r
'
Ψ
rqG
r,
r
'
(9.36)
while a charge density distributed within the box produces
Ψ
r
V
G
r,
r
'
Ρ
r
'
V
'
(9.37)
where V denotes the enclosed volume. The integration can be performed using either form
of the Green function, whichever seems simplest for the actual interior charge distribution.
Of course, in real life, such integrations usually must be performed numerically outside
the classroom.
9.2 Green’s Theorem for Electrostatics
In electrostatics or magnetostatics we are often given or seek to define either the potential
or the field on some closed surface and then need to determine those quantities everywhere
within the volume enclosed by the bounding surface. Although one could, in principle,
compute the potentials by adding the contributions of all charges and currents, we often do
not know the detailed distributions of charge or current outside the volume of interest. For
example, if we use a battery to maintain a constant potential on an electrode, we probably
cannot guess the distribution of charge on its surface except in the simplest of geometries;
hence, we are faced with a boundary-value problem. Once we know the Green function for
a point charge within the volume of interest subject to the specified boundary conditions,
we could compute the distribution of surface charge induced upon the electrode by the
interior charge density. Dirichlet boundary conditions specify the value of a scalar poten-
tial Ψ on the surface S while Neumann boundary conditions specify its normal derivative.
Sometimes one encounters mixed boundary conditions for which the value is specified on
some portions and the normal derivative on others. Cauchy boundary conditions specify-
ing both the potential and its normal derivative are too restrictive for Poisson’s equation,
generally precluding existence of a solution, but can be useful for other subjects.
In this section we use Green’s theorem to construct formal solutions to either Dirichlet
or Neumann boundary value problems for Poisson’s equation. First we review the deriva-
tion of Green’s identities based upon the Gauss divergence theorem. Let
A represent a vec-
tor field within a volume V bounded by a surface S. The divergence theorem then states
+,
A V
A ,
S (9.38)
where
S ˆn S is the directed element of surface area with ˆn being the outward normal.
If we choose
A Ψ
+Φ where Φ and Ψ are differentiable scalar fields, substitution of
A Ψ
+Φ
+,
A Ψ+
2
Φ
+Ψ ,
+Φ (9.39)
9.2 Green’s Theorem for Electrostatics 333
into Gauss’ theorem immediately yields Green’s first identity
Ψ+
2
Φ
+Ψ ,
+Φ V
Ψ
+Φ ,
S
Ψ
(Φ
(n
S (9.40)
where
(Φ
(n
ˆn ,
+Φ (9.41)
is a convenient shorthand notation for the normal derivative, the component of the gradient
in the direction of the outward normal to a surface. Interchanging Φ and Ψ and subtracting
the new form of the first identity then provides the more symmetric Green’s second identity
Ψ+
2
ΦΦ+
2
Ψ V
Ψ
+Φ Φ
+Ψ ,
S
Ψ
(Φ
(n
Φ
(Ψ
(n
S (9.42)
Poisson’s equation for the electrostatic potential Ψ takes the form
+
2
Ψ
r4ΠΡ
r (9.43)
where Ρ is the charge density found within volume V . The contributions of charges that
might be found outside the volume of interest are represented by the boundary conditions,
either the potential or its normal derivative on the bounding surface. It is useful to define
the Green function G
r,
r
'
to be the potential at
r produced by a unit point charge at
r
'
as
the solution to
+
2
G
r,
r
'
4Π∆
r
r
'
(9.44)
subject to appropriate boundary conditions to be specified later. Next we apply Green’s
second identity by choosing Ψ to be the electrostatic potential and Φ to be the Green func-
tion, such that
Ψ
r
'
+
'2
G
r,
r
'
G
r,
r
'
+
'2
Ψ
r
'
V
'
Ψ
r
'
(G
r,
r
'
(n
'
G
r,
r
'
(Ψ
r
'
(n
'
S
'
(9.45)
Using Poisson’s equation and the definition of the Green function this becomes
4Π
Ψ
r
'
∆
r
r
'
G
r,
r
'
Ρ
r
'
V
'
Ψ
r
'
(G
r,
r
'
(n
'
G
r,
r
'
(Ψ
r
'
(n
'
S
'
(9.46)
Thus, we obtain a formal solution to Poisson’s equation in the form of Green’s electrostatic
theorem
Ψ
r
G
r,
r
'
Ρ
r
'
V
'
1
4Π
G
r,
r
'
(Ψ
r
'
(n
'
Ψ
r
'
(G
r,
r
'
(n
'
S
'
(9.47)
If there were no bounding surfaces, the first term would represent the familiar electrostatic
potential produced by a specified charge distribution. The second term then represents
334 9 Boundary-Value Problems
the contribution made by external charges that determine the conditions on the bounding
surface.
For Dirichlet boundary conditions specifying Ψ on S, it is convenient to require that
the Dirichlet Green function, G
D
, vanish on S such that
r
'
S
'
G
D
r,
r
'
0
Ψ
r
G
D
r,
r
'
Ρ
r
'
V
'
1
4Π
Ψ
r
'
(G
D
r,
r
'
(n
'
S
'
(9.48)
The corresponding result for Neumann boundary conditions, specifying (Ψ/ (n
'
on S
'
,is
slightly more complicated because we cannot simply require (G
N
/ (n
'
to vanish on S
'
.
The problem is that the average value of the normal derivative on the bounding surface is
constrained by Poisson’s equation and Gauss’ theorem such that
+
'
,
+
'
G
r,
r
'
V
'
4Π∆
r
r
'
V
'
+
'
G
r,
r
'
,
S
'
(9.49)
requires
(G
r,
r
'
(n
'
S
'
4Π (9.50)
Hence, the simplest boundary condition we can impose upon G
N
is
(G
N
r,
r
'
(n
'
4Π
S
(9.51)
where here S is the area of the bounding surface. Thus, a formal solution to the Neumann
boundary-value problem can be expressed as
(G
N
r,
r
'
(n
'
4Π
S
Ψ
rΨ
S
G
N
r,
r
'
Ρ
r
'
V
'
1
4Π
G
N
r,
r
'
(Ψ
r
'
(n
'
S
'
(9.52)
where Ψ
S
denotes the average potential on S. Often Neumann problems appear in exterior
form where the region of interest is outside an inner boundary and can be interpreted as
within an outer boundary that is taken to , such that
S !Ψ
S
! 0 ,
(G
N
r,
r
'
(n
'
! 0 (9.53)
Ψ
r
G
N
r,
r
'
Ρ
r
'
V
'
1
4Π
G
N
r,
r
'
(Ψ
r
'
(n
'
S
'
(9.54)
where only the inner surface provides a finite contribution.
9.3 Separable Coordinate Systems 335
9.3 Separable Coordinate Systems
The Laplacian operator can be expressed in a curvilinear coordinate system with
r
Ξ
1
, Ξ
2
, Ξ
3
as
+
2
Ψ
1
h
1
h
2
h
3
3
i1
(
(Ξ
i
h
1
h
2
h
3
h
2
i
(Ψ
(Ξ
i
(9.55)
where
h
2
i
3
j1
(x
j
(Ξ
i
2
(9.56)
are diagonal elements of the metric tensor
g
i, j
3
k1
(x
i
(Ξ
k
(x
j
(Ξ
k
(9.57)
relating the cartesian coordinates x
1
,x
2
,x
3
x, y, z to the curvilinear coordinates (Ξ
1
, Ξ
2
,
Ξ
3
). We consider a coordinate system
r Ξ
1
, Ξ
2
, Ξ
3
separable when a product function of
the form
ΨΞ
1
, Ξ
2
, Ξ
3
X
1
Ξ
1
X
2
Ξ
2
X
3
Ξ
3
(9.58)
permits the Laplacian to be expressed in the form
+
2
Ψ
Ψ
f
1
Ξ
1
1
X
1
Ξ
1
(
(Ξ
1
gΞ
1
(X
1
Ξ
1
(Ξ
1
f
2
Ξ
2
X
2
Ξ
2
(
(Ξ
2
gΞ
2
(X
2
Ξ
2
(Ξ
2
f
3,2
Ξ
2
f
3,3
Ξ
3
X
3
Ξ
3
(
(Ξ
3
gΞ
3
(X
3
Ξ
3
(Ξ
3
(9.59)
One can then separate the homogeneous Helmholtz equation
+
2
Ψk
2
Ψ0 (9.60)
in a sequence of steps
f
3,3
Ξ
3
X
3
Ξ
3
Ξ
3
gΞ
3
X
3
Ξ
3
Ξ
3
Λ
3
(9.61)
f
2
Ξ
2
X
2
Ξ
2
Ξ
2
gΞ
2
X
2
Ξ
2
Ξ
2
Λ
3
f
3,2
Ξ
2
Λ
2
(9.62)
f
1
Ξ
1
1
X
1
Ξ
1
Ξ
1
gΞ
1
X
1
Ξ
1
Ξ
1
Λ
2
k
2
(9.63)
that produces three second-order ordinary differential equations coupled through the two
separation constants Λ
3
and Λ
2
.
It turns out that there are 11 orthogonal coordinate systems that are separable in this
manner. A comprehensive analysis can be found in the treatise by Morse and Feshbach.
Here we will be content to illustrate the general method using just the most common of
these systems: rectangular, spherical polar, and cylindrical. We have already discussed the
rectangular case and now proceed to spherical and cylindrical coordinates.