AN ABSTRACT OF THE THESIS OF
for the
SERGEI KALVIN AALTO
(Name)
Date thesis is presented
Title
May
M.A.
(Degree)
in
Mathematics
(Major)
3, 1966
REDUCTION OF FREDHOLM INTEGRAL EQUATIONS WITH
GREEN'S FUNCTION KERNELS TO VOLTERRA EQUATIONS
Redacted for Privacy
Abstract approved
(Major professor)
G. F.
Drukarev has given a method for solving the Fredholm
equations which arise in the study of collisions between electrons
and atoms. He transforms the Fredholm equations into Volterra
equations plus finite algebraic systems. H. Brysk observes that
Drukarev's method applies generally to a Fredholm integral equation
(I -> G)u
In
=
h
with a Green's function kernel.
this thesis connections between the Drukarev transforma-
tion and boundary value problems for ordinary differential equations
are investigated.
In
particular, it is shown that the induced Volterra
operator is independent
of the
boundary conditions.
operator can be expressed in terms
regular
X
.
of
The resolvent
the Volterra operator for
The characteristic values of
G
satisfy a certain
transcendental equation. The Neumann expansion provides a means
for approximating this resolvent and the characteristic values. To
illustrate the theory several classical boundary value problems are
solved by this method. Also included is an appendix which relates
the resolvent operator mentioned above and the Fredholm resolvent
operator.
REDUCTION OF FREDHOLM INTEGRAL EQUATIONS
WITH GREEN'S FUNCTION KERNELS TO
VOLTERRA EQUATIONS
by
SERGEI KALVIN AALTO
A THESIS
submitted to
OREGON STATE UNIVERSITY
partial fulfillment of
the requirements for the
in
degree
of
MASTER OF ARTS
June 1966
APPROVED:
Redacted for Privacy
Professor
of Mathematics
In
Charge
of
Major
Redacted for Privacy
Chairman of Department of Mathematics
Redacted for Privacy
Dean of Graduate School
Date thesis is presented
Typed by Carol Baker
May
3,
1.966
TABLE OF CONTENTS
Page
Chapter
I.
INTRODUCTION
1
II.
GREEN'S FUNCTION FOR A SECOND ORDER
DIFFERENTIAL EQUATION
7
III.
IV.
GREEN'S FUNCTION FOR AN NTH ORDER
DIFFERENTIAL EQUATION
SOLUTION OF THE INTEGRAL EQUATION
(I -X K)u
V.
VI.
13
=
h +
X
fCu)
APPROXIMATE SOLUTION OF THE INTEGRAL
EQUATION (I -X K)u = h + X fcD (u)
18
31
SOLUTION OF THE INTEGRAL EQUATION
(I-X K)u
=
h
+
X
lrf.(D.(u)
37
BIBLIOGRAPHY
46
APPENDIX
48
REDUCTION OF T'REDHOLM INTEGRAL EQUATIONS
WITH GREEN'S FUNCTION KERNELS TO
VOLTERRA EQUATIONS
CHAPTER I
INTRODUCTION
We sha1l be concerned with a certain rnethod of solving
particular Fredholrn
(r. 1)
'W'e
inte
gral equations of the second kind,
u(x) -
assurne
^ t,
G(x, s)u(s)ds = h(x).
that G, h and u are continuous cornplex valued
functions on their closed domains of definition.
Physicists are interested in obtaining solutions to integral
equations of form (1. 1) which often arise in the study of collisions
between electrons and atorns
IZ,S] .
One method physicists use to
obtain approxirnate solutions to (1. I) is to calculate one of the Born
approximations which are truncations of the Neurnann series solu-
tions lZ, p. 1536i L2, p. 1073] . The Born approxirnatisns are use-
ful only when the Neurnann series converges and, generally, this
occurs only for sufficiently small values of the parameter ). in
equation (1.I).
It is of physical interest to obtain solutions to (I.1)
for larger values of }.
Thus there is motivation to study other
rnethods of solving Fredholrn equations.
2
Integral equations
(1.
1)
often come from ordinary differential
equations with two point boundary conditions, e. g. a one dimensic,_nal
scattering problem.
In this
case
is the Green's function associ-
G
ated with the given boundary value problem. Thus we are led to con-
sider
(1.
with a Green's function type kernel,
1)
G(x, s)
(1. 2)
In (1. 2) let
V(s)f(s)g(x),
0 < s <
V(s)f(x)g(s),
0 <
x
<
1
< s <
1
,
=
V, f and g
x
.
be continuous complex valued functions
defined on the closed unit interval and further suppose that
g
*0
and
kernel
of
*
0,
*0.
f
G. F.
V
Drukarev gave a novel method
form
(1. 2)
[
solving (1.
of
2, p. 1536; 5, pp. 309 -320]
transform the Fredholm equation into
a
.
1)
with the
He was able to
Volterra equation and a finite
algebraic system for certain constants.
H. Brysk observes that
Drukarev's transformation
of the
Fredholm equation into a Volterra equation is possible because the
kernel is
of
Green's function type
attempts to show that the solution
[
2, p. 1536]
of the
.
Brysk further
Volterra equation leads to
the solution of the Fredholm equation obtained by using the Fredholm
resolvent
[
2, pp. 1537 -1538]
.
Briefly, the transformation
of a
Fredholm equation with
3
kernel
(1. 2) depends on
('1
J G(x, s)u(s)ds
(1.3)
x
=
0
J
V(s)f(s)g(x)u(s)ds
0
1
V(s)f(x)g(s)u(s)ds
+
x
=
J
V(s)[f(s)g(x)-f(x)g(s)] u(s)ds
0
1
+
V(s)g(s)u(s)ds.
f(x)
0
Let
(1.4)
K(x,
Then equation (1.
1)
s) =
V(s)[f(s)g(x)- f(x)g(s)]
1
V(s)g(s)u(s)ds
.
.
1
K(x,$)u(s)ds
u(x)-X
x
can be rewritten
x
(1. 5)
0 < s <
<
,
=
h(x)+Xf(x)
0
J
0
The right member of (1. 5) is of the form
h(x)
where
h
and
f
are known and
+
cf(x)
c
is a constant depending on
4
u
and
X
.
function u.
Note that (1.5) is a Volterra equation for the unknown
If it is solved for
stitution of the solution into
u
with
arbitrary, then sub-
c
yields an equation for
(1. 5)
technique we shall develop to solve (1.
5)
The
c.
is somewhat analogous to
the "shooting method" discussed by Henrici [ 7, pp. 345 -346]
Brysk deals with the special case h
(cf. Chapter VI).
1537]
=
f
,
[
2, p.
and thus his Volterra equation has the form
u(x) -
('x
K(x, s)u(s)ds
f(x)[ 1+X
=
('
V(s)g(s)u(s)ds]
1
J
0
1
.
0
This is a brief summary of the results of Drukarev and Brysk
dealing with the mathematical aspects of the problem. The author
intends to set the problem in a more abstract setting and to extend
the results obtained by Drukarev and Brysk.
As we will be dealing with integral equations with continuous
kernels, it will be convenient to work in the complex Banach space
C
of continuous
complex valued functions defined on the closed unit
Ifli
interval with the norm
=
max {If(x)I:xe[0,
letters will denote continuous linear mappings
For example, define
G
and
(Gu)(x)
G(x, s)u(s)ds,
=
0
and
C
by the equations
K
1
(1. 6)
of
1] }.
Capital
into itself.
5
(Ku)(x)
(1. 7)
=
('x
J K(x, s)u(s)ds
0
where
and
G
are given by
K
(1. 2) and (1. 4).
Now (1.
1)
may be
expressed by
( 1.
=h
(I - XG)u
8)
where
is the identity operator on
I
C.
Capital Greek letters will denote continuous linear functionals;
continuous linear mappings of
i. e.
The set of all linear functionals on
In
particular, define
(u)
(1.9)
where
SEC*
V(s)
and
g(s)
C
C
into the scalar field
forms a Banach space
J
C*.
by the equation
1
=
V(s)g(s)u(s)ds
1
J
0
are as above. Note that
1,
f
0
.
As a final notational convention, the symbols for elements of
C
will also be used to indicate mappings from the scalars into
C.
This convention is adopted because of the obvious isomorphism be-
tween
f:
y -
(1. 10)
and these mappings: for each f EC
C
C
by
(fy)(x)
=
yf(x)
define the mapping
6
where
C
ye
into
.
C
With this convention
DI.
with the one dimensional range
operator ft. has rank one where the rank
dimension
of
is a linear operator on
{yf:
of an
NE
J
}
.
Thus the
operator is the
its range.
Now (1. 3) and (1. 4) may be
(1. 11)
G
=
expressed by
K + RD,
and
(I-X K)u = h
(1. 12)
Thus, the Fredholm operator
of the
Volterra operator
K
G
+ X fsl
(u)
.
has a decomposition into the sum
and the operator
f.T.
of
finite rank.
In Chapters II and III of this thesis the above decomposition
of a
Fredholm operator with a Green's function kernel arising from
an ordinary differential equation is investigated.
and VI, the solution of (1.
ters
IV and VI
12)
In
Chapters
IV
is developed. Also included in Chap-
are examples worked out using the techniques inspired
by Drukarev. Approximate solutions of (1. 12) are discussed and
error estimates given
in Chapter V.
Finally the solution
of (1. 12)
is related to the Fredholm resolvent operator in the Appendix.
7
CHAPTER II
GREEN'S FUNCTION FOR A SECOND ORDER
DIFFERENTIAL EQUATION
In this
chapter a brief outline
construction
of the
of a
Green's
function for a boundary value problem arising from a second order
ordinary differential equation is given. Then the integral operator
arising from this construction is decomposed as in Chapter
I.
A
close examination of the Volterra operator shows that it is independent of the boundary values. Further discussion is given to show the
relation
of this decomposition to more
classical results
of
ordinary
differential equations.
The Green's function
will be constructed for the
G(x, s)
second order differential operator
Lu=u"
where
u
is defined on
[
+
plu'
p2u
+
pi, p2
and
0, 1]
C
E
with boundary
conditions
(2. 1)
a
a2u'(0)
=
0,
(2. 2)
ß1u(1)+ 132u'(1)
=
o,
lu(0)
+
I
al
Ißl
I
+
+
I
a2I > 0,
IR2I > o
.
8
We wish to solve the equation
(2.3)
Lu
(heC)
h
=
subject to boundary conditions
(2. 1) and (2. 2).
We
assume that this
boundary value problem has a unique solution.
Since
it follows from the theory of ordinary dif-
pl, p2 e C,
ferential equations that there exist two linearly independent functions
ul
and
satisfying the homogeneous equation
u2
Lu
(2. 4)
Thus the Wronskian of
=
u1
0
[
and
u2
xe [ 0,
1]
u2(x)
t
_
ui (x)
for
The assumption that the solution to the boundary
.
chosen such that
.
0
u2(x)
value problem is unique assures us that
378]
.
is nonzero; that is
ul(x)
W[ul(x),u2(x)]
3, p. 106]
u1
satisfies
and
u1
(2. 1) and
can be
u2
satisfies
u2
(2. 2)
[
9, p.
The method of variation of parameters yields a solution of
(2. 1) -(2. 3) in the
form
çx u
u(x)
=
0
(s)u (x)
W[u (s), u ()]
s
2
1
1
h(s)d(s)
+
x
u(x)u
(s)
W[u (s), u (s)]h(s)ds
2
1
9
[
9, pp. 378-379]
.
Let
u1(s)u2(x)
W[ul(s),u2(s)]
G(x, s)
x <
1
< s <
1
0 < s <
'
,
=
u1(x)u2(s)
w[ul(s), u2(s)]
0 <
'
x
.
Then
1
u(x)
(2. 5)
G(x, s)h(s)ds
=
.
0
The function
is the Green's function associated with the
G(x, s)
differential operator
Now using the
L
with boundary conditions (2.
and (2. 2).
1)
decomposition developed in Chapter I the right
side of (2. 5) may be rewritten
(s)u2(x) - ul(x)u2(s)]
1
G(x, s)h(s)ds
0
=
0
1
u2(s)
1
+
u1(x)
S-'0
Let
h (s)ds
W[ul(s), u2(s)]
W[u
1
(s), u 2 (s)]
h(s)ds
.
10
K(x,$) -
u1(s)u2(x)
ul(x)u2(s)
-
,
W[ul(s), uz
0 < s <
x<
1
1
Then
1 u2(s)h(s)ds
('x
r
J K(x, s)h(s)ds + u 1 (x) J W[u (s) ,u (s)]
0
0
2
1
(2. 6)
G(x, s)h(s)ds
SI
=
0
1
The kernel of the Volterra operator has the property that it is in-
variant under linear combinations
of the
functions
u1
and
u2,
that is, if
vi(x)
=
y1u1(x)
+
Y2u2(x)
v2(x)
=
61u1(x)
+
62u2(x)
(2. 7)
and
Ni62 - N261 f 0,
then
vi(s)v2(x)
K(x,$) -
-
v1(x)v2(s)
W[v1(s),v2(s)]
.
This follows since
vi(s)v2(x)-vi(x)v2(s)
=
(Y162-Y261)(ul(s)u2(x)-u1(x)u2(s) )
and
w[vl(s),v2(s)]
_
(v152-Y251)w[ul(s), u2(s)]
1 1
If
boundary conditions (2.
and (2. 2) are changed to
1)
(2. 1)'
alu(0)
+
a2u'(0)
=
0,
I
(2. 2)'
plum
+
R2u'(1)
=
0,
I
then the linearly independent functions
the homogeneous equation (2.
and
+
I
u2.
(2. 7) for some
> 0,
and
v2
I
v1
satisfying
and the boundary conditions (2. 1)'
In other words they can be
Y2,
Y1,
51
and 52.
Ylul(x)+Y2u2(x)
('x
K(x, s)h(s)ds
=
a2I > 0,
ßl I+ ß2I
expressed as in equations
The solution to the new
boundary value problem written in terms of
u(x)
I
respectively can be expressed as linear combinations of
and (2. 2)'
u1
4)
al
+
S
0
1
Therefore the kernel
2
-
K(x, s)
Y
2
S
1
,
u1
and
u2
is
1[S1u1(s)+S2u2(s)]
h(s)ds
W[u (s), u(s))
0
1
2
is independent of the boundary con-
ditions and the second term on the right varies with the boundary
conditions.
It is easy to
u (x)
p
satisfies equation
In fact
u (x)
verify that
=
xK(x, s)h(s)ds
1
0
(2. 3) and the
initial conditions
u(0)
=
u'(0)
= O.
is the "particular" solution to (2. 3) which can be
12
found by the method of variation of parameters if any two linearly
independent solutions to equation (2.
4)
are given.
other hand
On the
it is well known from the theory of ordinary differential equations
that any solution to equation (2. 3) can be written
u
In our
=
u
+
+ 13u2
[
9, p. 356]
.
case we have
1
a
and
aul
ß = O.
- S' 0
u2(s)
h(s)ds
W[u1(s),u2(s)]
Thus the solution obtained via the Green's function
and the decomposition gives the parameters
a
and
ß
as
functionals operating on the function h.
As a final remark it should be noted that boundary value
problems with inhomogeneous boundary conditions can be transformed into boundary value problems with homogeneous boundary
conditions. The term on the right side of (2.
3)
is modified by this
transformation, but the discussion is simpler for homogeneous
boundary value problems. Thus the techniques used here apply with
greater generality than indicated above.
13
CHAPTER III
GREEN'S FUNCTION FOR AN NTH ORDER DIFFERENTIAL
EQUATION
results
In this chapter the
of the
previous chapter are
generalized to a boundary value problem arising from an nth order
ordinary differential equation. The more general results do not
appear in as simple a form as the second order case considered in
the previous chapter. In the second order case the Fredholm opera-
tor
G
admitted the decomposition
G
In the
nth
K + f(1).
=
order case we obtain
a
decomposition
n
(3. 1)
G
=
K+
f.(D.
i i
.
i=1
However the Volterra operator is still independent of the boundary
conditions.
The Green's function will be constructed for the
differential operator
n
(3. 2)
Lu
dn-iu
=
pj
j=0
dxnj-
nth
order
14
where
u
p0(x)
0
>
is defined on
for X [ 0,
[
0, 1]
pi EC,
,
j =
,n and
0, 1,
with boundary conditions
1]
n -1
(3. 3)
Ui(u)
ßlJu(j)(1)]
alJu(j)(0)
=
0
=
j =0
i
=
1, 2,
,
We wish to solve the
n.
boundary value problem
given by
(3.4)
Lu
=
(hEC)
h
As before we assume that this
and boundary conditions (3.3).
boundary value problem has a unique solution.
The usual definition of the Green's function for the operator
L
given in (3.
2)
with boundary conditions (3.
and its derivatives up to and including the
G(x, s)
(a)
derivative are continuous for
(n -2)
(b) ElimO+
(c)
-
axn -1
0+
for each fixed
Ui(G)
=
In the region
tion
G(x, s)
0,
i
x, s
0 <
< 1,
an -1
an -1
E
is
3)
=
G(s +E, s) -axn
-1 G(s
s
E
[
1, 2,
0 < s <
and all
0, 1]
,
x < 1,
has the representation
n
we
[
-E
x
,
#
s)} - p0(s)
L(G(x, s))
s
8, p. 254]
,
=
0,
.
assume the Green's func-
15
n
G (x, s)
ai(s)ui()
x
=
i=1
and in the region
0 <
x
< s <
1
n
G (x, s)
b. (s )u. (x)
i
i
=
i=1
{u.(x): i
where
i
=
,n}
1, 2,
is a linearly independent set of
solutions to the equation
Lu
=
0.
Using conditions (a) and (b) unique solutions for the quantities
c.(s)
i
i
=
1, 2,
. .
.
,n are obtained
K(x, s)
a.(s)
i
=
[
-
b.(s)
i
8, pp. 254 -255]
ci(S)ui(x) ,
_
Let
.
0 < s <
x
<
1
.
i=1
Using condition (c) unique solutions for the
terms
of the
a.(s)
c.(s)
i
=
i
ci( s)
+
b.(s),
i
and the boundary terms
i
=
1, 2,
,n and thus
b.(s)
i
[
are obtained in
8, p. 255]
a.(s)
i
and
.
But
bi(s)
can be found such that the assumed representation of G(x, s)
in
16
the appropriate regions are satisfied. Therefore
n
n
c.(s)u.(x)
i=1
G(x, s)
b.(s)u.(x),
i
i
+
0 < s < x <
=
_
_
1 ,
i=1
=
n
Ib.(s)u.(x),
0 <
x
< s <
1
i=1
n
K(x, s)
bi(s)ui(x),
+
i
0 < s <
x
<
1
,
i
i=1
n
0 <
bi(s)ui(x),
x< s <
1
.
i=1
Thus the solution to (3.
with boundary values (3. 3) can be repre-
4)
sented by
x
1
(3. 5)
u(x)
G(x, s)h(s)ds
_
bi(s)ui(x))h(s)ds
K(x,$)h(s)ds+
=
0
0
0
n
('
+
J
bi(s)ui(x))h(s)ds
1(
x
i=1
n
K(x, s)h(s)ds
=
0
1
u.(x)
+
i=1
b.(s)h(s)ds.
1
0
i
i
i
17
1
Let
(Gh)(x)
_
G(x, s)h(s)ds
1
0
(Kh)(x)
J
=
K(x, s)h(s)ds
0
and
1
i(h)
Then (3.
5)
=
J
bi(s)h(s)ds
i
=
,n
1, 2,
.
0
can be rewritten as
u=Gh=Kh+
ui1)i(h)
i =1
and we see that the Fredholm operator
(3. 1)
where
K
(b)
admits the decomposition
is a Volterra operator and the
functionals. Further, we note that
c.(s)
G
K(x, s)
,i
is determined by the
which were given by conditions (a) and (b).
are independent
of the
are linear
But (a) and
boundary conditions (3. 3). Hence
K(x, s)
is independent of the boundary conditions.
The same remarks made about the inhomogeneous boundary
conditions in Chapter II can be repeated here, so that there is no
need to consider inhomogeneous boundary conditions separately.
18
CHAPTER IV
SOLUTION OF THE INTEGRAL EQUATION (I -XK)u
In this
(4.
=
h +Xf.1)(u)
chapter the equation
(I-A G)u
1)
=
h
(h
E
C)
is solved assuming that the Fredholm operator
G
has the de-
composition
(4. 2)
where
G
K
=
K +
f
is a Volterra operator,
f E C,
(DE
C *, f tO
and
1, # O.
This is the decomposition considered in Chapter II.
The solution obtained for (4.
and an entire function in
comprise all
of the
X
.
1)
is a quotient of an operator
The zeros of this entire function
characteristic values
of
G
(characteristic
values are inverses of eigenvalues). The resolvent operator ob-
tained by solving (4.
1)
exists for all noncharacteristic values
Thus this resolvent and the Fredholm resolvent are equal
[
X.
11, p. 15]
.
Also to be discussed are solutions of characteristic value problems
for integral equations. Finally several examples
of
characteristic
value problems are solved using techniques developed in this chapter.
At this point it is convenient to note that the operator
G
is
19
an operator such that the Fredholm alternative holds for (4.
follows since
asserts that
G(x, s)
(4. 1)
1).
This
The Fredholm alternative
is continuous.
has a unique solution for arbitrary h
E
C
iff the
homogeneous equation
(I-X G)u
(4. 3)
= 0
has only the zero solution [ 11, p. 46]
are called eigenfunctions
From
(I-AK)u
is equivalent to (4.
exists for all
X
(4. 5)
u
for
1).
(I-X K) - lh
h
K
and the corresponding
G.
that
Aft(u)
+
is a Volterra operator,
+ X
(I-X K) -
lft(u).
it
and transposing yields the following equation
4(u);
(u) [1-X t(I-A K)
(4. 6)
Equation (4.
6)
(I -X K)- 1
such that (4. 5) holds, then operating on both
u
5) by
Since
=
G
3)
Thus equation (4. 4) is equivalent to
.
=
there exists
sides of (4.
of
(4. 1) and (4. 2) it follows
(4. 4)
If
operator
of the
are called characteristic values
X
Nonzero solutions of (4.
.
lf]
=
has a unique solution for
(I-X K) -
4(u)
lh.
iff
20
d(X)=
Assuming
1-X
(I-
K) -
we can solve for
d(X) f 0,
if f
0.
and we obtain
Vu)
equation
(4. 7)
u
=
That is to say, if
(I-K)-1 h+d() (I-XK)-lf(I-XK)-lh.
d(X)
which implies that
contrapositive
of
value then d(X)
value of
0
then (4.
is not a
X
7)
= O.
if
Now suppose
1)
that
6)
since
(I -X K)
E
-1
we assumed that
If
is not a characteristic
X
d(X)
is a one to one mapping of
g f 0,
hence
d(X) f
The
G.
of
1)
is a characteristic
X
holds for arbitrary h
holds for arbitrary h C.
that (4.
is the unique solution to (4.
characteristic value
this statement is,
Then (4.
G.
t
O.
=
E
C
0,
which implies
then
c
= 0
onto itself.
C
But
To summarize, the
following two theorems are recorded.
Theorem
4.
1
Theorem 4. 2
X
is a characteristic value of
If
X
1)
d(X)
= 0
7)
gives
1).
has a unique solution iff the one dimensional system
(4. 6) has a unique solution.
operator
iff
is not a characteristic value, then (4.
the unique solution to (4.
Note that (4.
G
(I -XG)
Thus the Fredholm alternative for the
reduces to the Fredholm alternative for the one
21
dimensional system (4. 6).
From Theorems
4.
and 4. 2,
1
(I -X G) -1
exists iff
d(),.)
t
in which case
(I-AG)-1= (I-XK)-1+d() (I-XK)-lf(I-aK)-1
(4.8)
As
K
.
is a Volterra operator
00
(I-XK)-1
=
X
nKn
n=0
where
K
=
I
and
Kn
operator norm for all
+1
X
.
KKn
=
Letting
fx
(4. 8) may be
The series converges in the
.
(I-X K) - if
=
rewritten
oo
1
n 0
n[ d(X)Kn+l
X.
(4. 9)
(I-X G)-1 =I+
X
d(%)
The Fredholm resolvent operator
(I -AG)
-1
of
F'X
(I-xG)-1
whenever
+ fX (DKn]
exists. Thus by
=
I+
(4. 9)
G
xF'x
is defined by
0
22
oo
X
rk
)Kn+
d(k
1
+ fX
Kn]
n=0
-
It might be
n[
d(X
)
remarked at this point that Brysk attempts to prove
similar result by showing that the numerator and denominator
a
of
his solution are the same as the numerator and denominator of the
solution obtained via the Fredholm resolvent [ 2, pp. 1537- 1538]
.
His proof is faulty, but a proof can be established using techniques
developed by Manning
In
[
10]
,
(cf. appendix).
general d(X) is an entire function in
since
X
00
d(X)
=
1
-
X
(I-X K)
-
if
= 1
n+1
-
n f.
n=
Thus there is some difficulty in attempting to use the equation
d(X)
= 0
to calculate the characteristic values of
it is easier to calculate
determinant
[11, p. 56]
d(X)
.
G.
However
than to calculate the Fredholm
More specifically, in making approximate
calculations of characteristic values it may be easier to use a truncation of
d(X)
than to use a truncation -of the Fredholm determinant.
Consider the characteristic value problem,
(4.10)
(I-X G)u
= 0
.
23
By (4. 2) this may be
rewritten
(I-X K)u
= X
f0u)
or
u
= 1. (u)X
(I-X K)
-1
f
Thus the general form of the eigenfunctions of
.
G
will be
co
(4. 11)
uX =
aX(I-XK)-1f
=
XnKnf,
ax
n=0
and
uX
will satisfy (4.
10)
only if
d(X)
=
0.
To conclude this chapter two examples of
classical differen-
tial eigenvalue problems are solved using the integral equation
generated by the Green's function for the given eigenvalue problem.
The
first is the eigenvalue problem for the vibrating string problem.
The second example is the heat equation in cylindrical coordinates:
Bessel's equation with two boundary conditions. This boundary value
problem does not have an ordinary Green's function since the coefficient
of the
highest derivative vanishes. However in this special
case an integral equation for the eigenfunctions can be derived.
Furthermore this integral equation has a kernel
sidered in Chapter
I.
of the type con-
Thus we can solve this problem by methods
developed in this chapter.
24
Example 4.
where
Consider the vibrating string problem;
1
a2v
a2
ax2
a
is defined for
v(x, t)
t2
x<
_
0 <
_
1
and
t
> 0
with boundary
conditions
v(0, t)
=
v(1, t)
= 0
and the initial condition
v(x, 0)
=
g(x).
Separating variables, the following boundary value problem is obtained;
u" (x)
(4. 12)
_ -X
u(0)'= u(1)
(4. 13)
0<x<
u(x),
= 0
1,
.
The Green's function associated with the differential operator
L
=
d2
with boundary conditions (4. 13) is
dx
G(x, s)
s(x-1),
0 < s <
x(s-1),
0 <
x <
1
,
< s <
1
.
=
x
25
Thus the following characteristic value problem arises;
u(x)
(4. 14)
=
('1
J G(x, s)u(s)ds
-X
.
0
Using the decomposition outlined in Chapter I we obtain
x
u(x)
= X
J
1
(s-x)u(s)ds
+
(1-s)u(s)ds
Xx
0
0
or in symbolic form
u
where
(Ku)(x)
=
=
XKu
+ Xf(1)(u)
('x
j (s- x)u(s)ds,
f(x)
=
and
x
(u) =5.
0
In
order that there exist nontrivial
d(X)
u
satisfying (4.
=
1-X
K)
-
if
= O.
Now
00
(I-X K)
l f]
(x)
n(Knf)(x)
=
1
=
siAx
'
n=u
iD(I-XK)-lf
(1- s)u(s)ds.
o
sufficient that
[
1
sinNFX
3/2
X
12) it
is
26
and thus
d (A
Hence
values
d(A)
of
iff
= 0
(4.12) are
TA
X
n
=
=
Also we have
a sin(nTrx)
=
n
a 2v
is defined for
av
s as
1
at - as2
v(s, t)
Thus the eigen-
n2Tr2.
=
Consider the heat equation in cylindrical coordinates;
av
where
X
Thus known results are obtained.
as eigenfunctions.
2
s inNFA
_
or
±nTr
n2Tr2.
u (x)
n
Example 4.
)
+
0 < s <
1
and
t
> 0
with the
boundary conditions
v( l, t)
=
v(0, t)
0,
<
co
and the initial condition
v(s,
0)
=
g(s).
Separating variables the following boundary value problem is obtained;
(4. 15)
[
su'(s)]
'
=
-X
su(s),
0 < s < 1,
27
u(1)
(4. 16)
=
0,
u(0) <
oo
.
Although no ordinary Green's function exists for the operator
(Lu)(s)
s
=
0),
=
[su'(s)]
'
a function
(since the coefficient of
right member of (4.
15)
µ
vanishes at
can be found which is integrable with
G(s, r)
respect to the measure
u"
(dr)
=
rdr.
Since
s
it may be reasonable to
appears in the
try working in this
measure space.
Proceeding formally, we notice that
u1(s)
=
satisfies the boundary condition at
u2(s)
satisfies the boundary condition at
u2
1
s
=
=
and
0
log s
Furthermore
s = 1.
satisfy the homogeneous equation associated with
u1
(4. 15).
and
Car-
rying out the calculations in the same spirit as suggested in Chapter
II we find a function
log s,
H(s,r)
0 <
r
< s <
1,
=
log
r,
0 < s <
r
<
1
.
Thus formally we expect that a solution to the equation
28
(4. 17)
[
su'(s)]
satisfying boundary conditions
'
=
sh(s)
(4. 16) would be
1
u(s)
(4. 18)
Let
G(s, r)
=
rH(s, r).
J
H(s, r)h(r)rdr.
0
Then (4. 18) can be rewritten
u(s)
(4. 19)
=
('1
=
\
G(s, r)h(r)dr
.
0
The kernel
G(s, r)
shows that
u(s)
Further
is continuous.
as given by
(4. 19)
satisfies
a simple calculation
(4. 17)
as well as the
Thus we expect that solutions to the
boundary conditions (4. 16).
characteristic value problem
1
u(s)
(4. 20)
G(s, r)u(r)dr
_ -X 5"
J
0
will give eigenfunctions for (4. 15). Clearly
G(s, r) is the same
type of kernel as was encountered in Chapter
I (cf.
equation (1.
Thus we find that (4. 20) can be written
s
u(s)
= X
1
(r log r -r log s)u(r)dr
1
0
or symbolically,
+ X
Ç
J
(
0
-r log r)u(r)dr
2))..
29
u
XKu
=
+
X flu)
s
where
(Ku)(s)
=
J
(r log r - r log s)u(r)dr,
f(s) =
and
1
0
1
(10(u) =
J
(
-r log r)u(r)dr.
0
In
order that there exist
f
uX
0
satisfying
(4. 20) it is
sufficient that
d(X)
= 1
-A.(I-XK)-1f
=
0
.
As before
co
[(I-XK)-lf] (s)
Xn(Knf)(s)
+
= 1
n=1
By induction it can be
(Knf)(s) -
verified that
(-1)
n 2n
s
...
22- 42.
n
=
(2n)2
1,2,
and therefore
oo
[(I-XK)-lf](s)=
1
n=1
Furthermore
-ln(s)2n
+
2
2
4
2
(2n)2
=
JO( s)
.
30
00
AflI-XK)- lf
=
(- 1)n(NFX )2n
2. 42.
(2n)2
-
...
n=
and thus
d(X)
=
JO(NFX ).
As expected the squares of the eigenvalues of (4. 15) are the zeros
of the
Bessel function
uk (s)
=
J0
and the eigenfunctions are
ak JO(Nrik s),
k
=
1, 2,
31
CHAPTER V
APPROXIMATE SOLUTION OF THE INTEGRAL EQUATION
(I -X K)u
In this chapter
of the
=
h
+ X f,D(u)
is approximated by truncating all
(I -X G) -1
series which appear in the expression on the right side
of
equation (4. 9). Error estimates are calculated giving an error
bound for the approximate solution. From this calculation an
error
bound for the approximate calculation of the characteristic values
arises.
Equation (4.
9)
is
/
co
(I-x G)-1
=
I +
X
Xn[d(X)Kn+l+fX
.T.Kn]
n=0
d(X
)
where
/
co
(5. 1)
d(X)
=
1
-
X
X
nKnf
n=0
and
00
(5. 2)
XnKnf
fX =
.
n=0
Truncatingthe series which appear on the right side
obtain
of (4. 9) we
32
Xn[ dm(X
(I-X G)-1
m
=
I +
)Kn+ 1+fX mCKn]
0
Xn
d
m
(A)
where
dm(A )
(5. 3)
=
1
-
X
XnKnf
(1.
n=0
and
m
fXm
(5. 4)
IAnKnf.
=
n=0
Let
00
o(X)=
)
X
X
n[
)Kn+ l
d(X.
fx dKn]
+
n=IO
and
m(X) the analogous truncated expression.
X
r'
II
(I-XG)1-(I-XG)mll=
-
o(X
)
d()
Then
and
II
d-
II
dm
a°
I
d-ami
I
=
-
Ild
I
of
Idi
I
I
I
+[d1
m
I
m11
'
I
a ml
l
I
33
To find an
and
dl
I
I
K(x, s)
error bound
I
I
A
pm
K(x,
n
I
It is
and a lower bound for
I
Mn(x-s)n- 1
I<-
s)
Id!.
Id-dm I,
HAIL
Let
Then
(0 < s < x < 1).
< M,
I
I
we need upper bounds for
[11, p.
(n-1)!
16]
.
easily verified that
MnI,IfII
IIKnfII <
(5. 5)
for any
Thus
f EC.
n
(5. 6)
From
IIKnII
CX
=
I
(5.7)
X
I
'
I
I
[d(X)
(I
d(X
I"
I
I
)
f
I< 1+
I
I
e
m(X)
m
=
1
n m+
that
I
e
-d m (X)I
co
where
Mn,
(5. 1) and (5. 5) we have
I
where
<
(IX I!M)
n.
1
<
=
I
CX
M
1x1I'
and from (5. 1), (5.3) and (5.
IMI'
IIfII"ETTl(X)
n
From
(5. 2) and (5 5) it
follows
5)
34
IIfx
Again using (5.
H
<
IIfII.eIxIM
easy to show that
5) it is
IIfX
-fXmll
<_
m()
IIfIIE
.
Now
00
I
I
A
I
I
00
i
= II
X
n[ dKn+
l+fx Kn]
I
n+1IIKn+1ll
< Idl
n=-1
I
n=0
00
Ix I. IIfx II'
+
nllKnll
IMI'
<
[
1+2Cx] eIx IM
n=0
A
short calculation shows that
,
00
IIA-AmII<_ IdI
n=m+ 1
CO
In+1IIKn+IIl
+ IX
I' IIfx II' IMI
IX InIIKnII
n=m+l
m
+
I
d-d
IX
In+lIlKn+lll
n=0
m
+
'XI'
IIf
f
nll IMI' i,
n=0
Then using (5.
6)
and the estimates
I InIIKnII'
.
35
oo
Ix
2,
In+lliKn+lll
<Em(),
n=m+ 1
m
(Ix IM)n+l
ek IM
<
(n+1)!
n=0
\
m
(1XIM)n
LL
<eIXIM
n!
n=0
we obtain
II- mIl
< (1 +
4Cx.)Em(X)
.
Then
2
I
I
G)
1
-(I-X G)
1+ 6CX +6CX
1
I
I<
I
To complete the
for
Idl
From
dm
error analysis
I
I
d
E
m(A ).
i
a lower bound
(5. 7)
Idmi
-
Idi <
I'
IMI'
IIfIIEm(x)
lx I'
IIiI.
IIfIIEm(X)
IX
or
Idi
>
IdmI
-
must be found
36
For fixed
X
assuming
,
E
and further
I
d(X)
I
m
(X
> O.
for sufficiently large
m
is not a characteristic value, we have
X
)
Since
that
Thus for sufficiently large
m,
dm(X)
dm
I
I
---
-
I
X
+
6C
II <
X
+
_ la ml(Idml-IX I'
I'
HO'
O'
on the
error. Thus
I
I
f
II0.
scalar field, then
we have that the
I
---
I
E
00
we have
m(X)
> 0.
error is
6C 2
If (5. 3) is used to compute approximate
on any compact subset of the
as m
d(X)
the final form of the
1
II(I-XG)1-(I-XG)m
m -00
as
0
IIfII'Em(X))
Em(X).
characteristic values
(5. 7)
gives a bound
characteristic values can be
uniformly approximated on compact sets.
Due to
results obtained in the appendix an error analysis
developed by Glahn[6, pp.
7
two methods of analyzing the
-16] also applies to this problem.
error are
The
not directly comparable
since different parameters appear in the two methods. Thus more
work could be done here.
37
CHAPTER VI
SOLUTION OF THE INTEGRAL EQUATIONS
(I-X K)u
In this
h
/
+ X
f.(D.(u)
chapter we shall solve the equation
(I-X G)u
(6. 1)
where
=
=
h
(h E C)
has the decomposition
G
n
(6.2)
G
=K+
i=1
As before
.
i
E
C*,
i
K
=
is a Volterra operator,
1, 2,
,n and the
f.
i
f.
and
E
C,
'.
i
i
=
1, 2,
,n,
are linearly inde-
pendent. Results similar to those in Chapter IV are obtained.
However, instead of the Fredholm alternative reducing to the
alternative for a one dimensional system, it reduces to the alternative for an n- dimensional algebraic system. Again the solution is
expressible as a quotient
of an
operator and an entire function
of
X
Also similar to the case dealt with in Chapter IV, the zeros of this
entire function comprise all
operator
G.
of the
characteristic values
of the
Finally an example is worked using the techniques
.
38
of
this chapter.
In
order to obtain the solution in a form comparable to the
solution in Chapter IV, it is necessary to introduce certain notation.
Let
(6. 3)
G
where
fn:
n-
=
K
fnen
is defined by
C
r-
al
a
2
n
fn
LLLL
,
a.f.
i i
i=1
an
n
and
,n: C
-n
is defined by
1(u)
2(u)
tn(u)
1)n (u)
where the
C.
f.
and
For any norm in
(D.
are as above. Thus
cn
g
e. g.
fn1.n
maps
C
into
39
a
1
a
2
max
=
{
la
i
:
+
i
=
1, 2,
,
n}
an
n
n
fn
J
bounded linear mapping of
From
(6. 3) we see
C
=
h
1)
into
n
into
that (6.
(I-X K)u
(6. 4)
n
C 11
is a bounded linear mapping of
C
Hence
and
e
fne
is bounded.
is equivalent to
+ X
fne(u)
and hence to
(6. 5)
Let
u
fn
:
g- n
C
=
(I-k K) ih
+ k
(I-A K)
- l fne(u).
be defined by
MN.
a
1
a2
n
f
n
I-XK)- lf i
=
k
i=1
an
Then (6.
5)
may be rewritten
.
is a
40
(6. 6)
u
(I-X K) ih
=
Assuming there exists a
en(u)
The operator
=
nf
fn
such that (6.
u
e
on both sides of (6. 6) with
(6. 7)
+ X
en(I-X K)
(u)
(1)11
6)
.
holds, we may operat
to obtain
lh
+ X enfxn
en(u)
.
is a linear mapping of
may therefore be characterized by a matrix Ax
n
7
the identity mapping of
onto itself.
n
into itself and
.
Let
Then (6.
7)
be
In
may be re-
written
(In-Axn )en(u)
(6. 8)
The matrix equation (6.
(In-An )-
B
of
=
l
exists iff
adj(In -An
cofactors
of
)
8)
(I- K)
lh
.
has a unique solution for
dn(X)
where
=
=
det(In -Axn) f
adj(In -Axn)
(In -Axn ).
If
O.
Let
is the transpose of the matrix
dn(X) f 0,
the solution for
is given by
.1.n(u)
B
-
dn(X
If
dn(X) f 0,
then
n(I- K) - lh
X
)
n(u) iff
.
.1.n(u)
41
(6. 9)
u
(I-ñ K)
=
-
lh
+
dn(X
is equivalent to (6.
- lh
fn BnX tn(I-X K)
X
)
By exactly the same reasoning as in Chapter
1).
IV we have the following two
theorems.
Theorem
6.
iff
Theorem
6. 2
tion to (6.
1).
dn(X)
1
If
= 0
dn(X)
0,
#
is a characteristic value of
X
then equation (6.
As in Chapter IV we have if
(I-XG)-1
(6. 10)
=
(I-XK)-1
dn(X)
+
X
dn(X
It is of
fact
interest to note that
B
(6. 10) is
Finally as in Chapter
value of
G
iff
dn(X)
=
IV we
0.
gives the solu-
then
0,
fnnn(I-XK)-1
similar in form to
see that
dn(X)
Now if
.
)
the adjoint matrix, is equal to (1) if
,
9)
=
1.
0,
then
a2
=(I-XK)1fn
an
are the eigenfunctions
of
G
where the
a.
In
is a characteristic
X
=
n
(4. 8).
al
ux
G.
are appropriate
42
constants.
As an example of a
we shall solve the
of
G
the form considered in this chapter
characteristic value problem which arises from
the transverse oscillations of a homogeneous bar clamped at one
end and free at the other.
Example 6.
The oscillations are determined by the equation
1
a4z
a22
+
where
-
at 2
ax
0
{(x,t):
is defined on the strip
z(x,t)
0 <
x < 1,
t > 0}
and the boundary conditions are
z(0,t)
[
13, pp. 26 -29 ]
.
=
z
x
(0,t)
=
z
xx
(1,t) = z
Assuming z(x, t)
=
xxx
(1,t) = 0
u(x)ei Wt,
we
are led to the
ordinary differential equation
(6. 11)
d
4
ux)
A
4u(x),
0< x<
1
,
dx
with the boundary conditions
(6. 12)
u(0)
=
u'(0)
=
u"(1)
=
u"'(1)
The Green's function for the operator
L
=
=
0.
d
4
dx
4
with boundary
43
conditions (6.
12)
is
2x
3s x
-
s3
G(x, s)
- - x <-
1
,
x < s
1
.
0 < s <
3!
=
3 sx2 -
x3
0 <
3!
We have the following
integral equation for
<
u;
1
u(x)
= X
4
G(x, s)u(s)ds
11
.
0
Decomposing the integral operator, we obtain
X
(6. 13)
u(x) -
X
4
0
(x-s)3 u(s)ds
3!
=
x4
1
{
su(s)ds
J
0
He re it may be of some
treatment
of
J
2
2!
3
1
+ [ -
x
u(s)ds]
}
.
o
interest to briefly mention Tricomi's
this problem so that similarities in both methods can be
compared . Initially, Tricorni develops a method by which the solution to an initial value problem can be written as a Volterra equation
[
13, pp. 18 -19]
that
u" (0)
=
c
.
In the
and
case
u "'(0)
of equation (6. 11),
=
c3.
Tricorni assumes
Then by the above mentioned
method for solving initial value problems, he arrives at the equation
44
u(x)-X
(6. 14)
[
13, p.
2 8]
('x
4
j0
3S)
3
u(s)ds
= X
4(
c2x2 2!
+
c3x3/ 3!
)
These two methods are connected by the "shooting"
.
method mentioned by Henrici [7, pp. 345 -346]
.
Briefly the
"shooting" method assumes that every boundary value problem is
equivalent to an initial value problem. The solution to the boundary
value problem is represented as a parameterized solution to the
initial value problem (e.
are
c
and
Now
equation (6.
14)
where the parameters
Then the parameters are determined.
c 2).
method used to obtain (6.
functionals of
g.
13)
The
gives these parameters as linear
u.
let us proceed to the problem of finding the character-
rewritten as
istic values. Let (6.
13) be
(6. 15)
(I -X4K)u
=
A4f2cb 2(u)
where
a
f2
a2l
=
al x2,2!
+
a2 x3/3!
and
1
su(s)ds
(u)
1,
1
2(u)
=
('0
2(u)
-J u(s)ds
0
45
There exists a solution
-
1
2
d (X)
=
A
if
u
d2(X)
=
41(I-X 4 K) -1 x2/2!
where
0,
_
-
A
41(I-X 4K) -1 x3/ 3!
det
-X 41,2(I -x K)
-1
x2/2!
1
-X
412(I -X 4K)
-1
1
1-X 2/2
=
1
l
1
s(coshX s-cosX.$)ds (-X)/2
Ç
J0
s(sinhXs-sinT.$)ds
det
1
X
1
2/2 J (coshXs-cosXs)ds 1+X/2
0
=
x3/3!
(sinhXs-sinXs)ds
0
2(1 +coshX cos X).
Hence we obtain the anticipated result that the characteristic values
are the zeros
of the
1
+
transcendental equation
coshX cos X
= 0
[
13, p. 29]
.
46
BIBLIOGRAPHY
1.
Allis, W. P. and Philip M. Morse. The effect of exchange on the
scattering of slow neutrons from atoms. Physical Review
44: 269-27 6.
1933.
2.
Brysk, Henry. Determinantal solution of the Fredholm equation
with Green's function kernel. Journal of Mathematical
Physics 4: 153 6 -1538. 1963.
3.
Coddington, Earl A. An introduction to ordinary differential
equations. Englewood Cliffs, N. J. , Prentice Hall, 1961.
292 p.
4.
Davis, H.T. The theory of linear operators. Bloomington,
Indiana, Principia Press, 1936. 628 p.
5.
Drukarev,
6.
Glahn, Thomas Leroy. An error bound for an iterative method
of solving Fredholm integral equations. Master's thesis.
Corvallis, Oregon State University, 1954. 22 numb. leaves.
7.
Henrici, Peter. Discrete variable methods in ordinary differential equations. New York, Wiley, 1962. 407 p.
8.
Ince, E. L. Ordinary differential equations. New York, Longmans,
Green and Company, 1927. 558 p.
9.
Indritz, Jack. Methods in analysis. New York, Macmillan,
G. F. The theory of collisions of electrons and atoms.
Soviet Physics. JETP 4: 309 -320. 1957.
1963. 481 p.
10.
Manning, Irwin. A theorem on the determinantal solution of
the Fredholm equation. Journal of Mathematical Physics
5: 1223 -1225. 1964.
11.
Miklin, S. G.
12.
Morse, Philip M. and Herman Feshbach. Methods of theoretical
physics. 2 vol. New York, McGraw -Hill, 1953. 1978 p.
Integral equations and their applications to certain
problems in mechanics, mathematical physics and technology.
New York, Pergamon Press, 1957. 338 p.
47
13.
Tricorni, Francesco. Integral equations. New York,
Interscience, 1957. 238 p.
APPENDICES
48
APPENDIX
COMPARISON OF THE FREDHOLM RESOLVENT OPERATOR AND
EQUATION (4. 9)
As remarked in Chapter IV, Brysk's
assertion can be proved
using tools developed by Manning. This result can be used to show
that the numerator and denominator of
co
(1)
X
n[
d(X
)Kn+ l+fX
IKn]
/ d(X
)
nLL=LO
n=0
are equal to the numerator and denominator respectively
Fredholm resolvent operator. Theorems
immediately from the equality
4.
of (1) and the
1
and 4.
2
Fredholm resolvent
The Fredholm resolvent operator is given by
r
(2)
_
nDn)
d0
=
D0
1,
=
/
(
X
ndn
)
n=0
n=0
where
/
oo
X
(
G,
1
dn
and the
D
n
=
the
then follow
operator.
00
of
(-1)n-1 J Dn-1(s, s)ds, n
0
=
1, 2,
are given by the recursion relation
,
Dn
(3)
[
dnG
10, pp. 1223 - 1224; 11, p. 54]
+
GDn-
n
l'
=
49
,
1,2,
Brysk's assertion was that
.
co
(4)
X
nGKnf /d(X
)
n=0
and
r
were identical
f
[
2, pp. 1537 -1538]
obtained from (1) by operating on
f
,
(note that (4) is
and simplifying).
developed by Manning are stated below in Lemmas
1
proof.
Lemma
{Gh}
1
where
{h}
=
for
Ch)
lim h(x) /f(x)
x
Lemma
=
hEC
.
0+
dn= - {Dn- if },
2
n
=
1,2,
These tools yield
Theorem
1
do +l=- cIKnf,
n
=
n
=
0,1,,
and
GKnf
=
D f,
n
0,1,
.
..
The tools
and
2
without
50
Proof: (by induction)
For
GKof
=
Gf
=
n
D0f
dn+2
{Dn+lf}
=
{dn
=
{- 13(Knf)Gf
=
GDnf}
+
+1f
- {Gf}
13(Knf)Cf)
+
=
- c1,K0f,
and
1
f
)
(by the inductive hypothesis)
{G(GKnf)}
+ .13.(GKnf)
- (D(Knf)(1) (f) +
Kn+
(by (3)
G(GKnf)}
+
-''(Knf) {Gf}
= -
=
=
2
=
=
Also by
- {D0f}
=
.
By Lemma
-
dl
0,
=
[
(by Lemma
1)
(by equation (4. 2)
(K +f (D)Knf]
)
.
(3)
Dn+ 1f
+
GDnf
=
dn+ 1Gf
=
-(1)(Knf)Gf
=
G(G- f')Knf
=
GK Knf
=
GKn+
1
+
G(GKnf)
(by the inductive hypothesis)
(by equation (4. 2)
)
51
Hence the conclusion follows by induction.
An interesting conclusion which may be drawn from Theorem
is
1
00
d(A
dnX n
) =
.
n
n=0
Using this we can write
co
00
X
n[d(X)Kn+ l+fX
Kn]
p
P[
=
Then defining
D'
D'
mKp-m+1
+
Kmf CKP-m)]
m=0
p=0
n=0
d
by
d
=
P
m
Kp-m+l
elf
(KP-m),
m=0
it is easy to verify that
D0
=
G
and that
D'
P
satisfies the recur -
sion relation (3). Hence (1) is identical to (2) and thus Theorems
4.
1
and 4.
2
follow immediately since these properties hold for the
Fredholm resolvent operator.
This approach while being more tedious (the proof of Lemma
2
as proved by Manning is largely bookkeeping) yields the more
satisfying result that
do
o +1
(1.Knf
and
D
=
P
D'
P
.
This conclusion
cannot be drawn from the remarks in Chapter IV leading up to the
52
equality
of
the expression given in (1) and
rx
.
The method given in this thesis of solving equation (4.
does
1)
give greater insight into the nature of the decomposition and how it
is used to solve the problem. Also it generalizes to operators with
the type of decomposition considered in Chapter VI with greater
ease (once the notation was developed the proof of Theorems
and 6.
2
were identical to the proofs of Theorems 4.
1
6.
and 4. 2).
1