The Numerical Solution of Fredholm Integral Equations of the Second Kind
Author(s): Kendall E. Atkinson
Source: SIAM Journal on Numerical Analysis, Vol. 4, No. 3 (Sep., 1967), pp. 337-348
Published by: Society for Industrial and Applied Mathematics
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SIAM J. NUMER. ANAL.
Vol. 4, No. 3, 1967
Printed in U.S.A.
THE NUMERICAL SOLUTION OF FREDHOLM INTEGRAL
EQUATIONS OF THE SECOND KIND*
KENDALL E. ATKINSONt
1. Introduction. A general method is presented for the numerical solution
of the Fredholm integral equation,
b
-
(Xx(s)
K(s, t)x(t) dt
=
y(s),
a _ s ? b.
a
In the equation, X is a nonzero complex inumber, [a, b] is a finite iinterval,
y( s) is complex-valued and contiiluous on [a, b], and the integral operator 3C,
rb
(aCx)(s)
(2)
K(s, t)x(t) dt,
= f
a < s < b,
is assumed to be a compact (completely continuous) operator oin [a, b]
inito C[a, b]. The set C[a, b] consists of all complex-valued continuous functions on [a, b], and with the maximum norm,
lxii = a?smax Ix(s)!,
_
it is a Bainach space.
Although quite general, the method preseinted is iinteindedto treat (1)
when the kernel K(s, t) has singularities, e.g.,
log Is -
t/,
Is -
for a > -1,
tI'
log Icos s -
cos t.
When the kernel has several continuous derivatives, the method reduces to
replacing the integral with a numerical integral and then to solving a finite
linear system; see [1], [2], [3], [5], [11], [13].
In the followiing section, a generalized form of numerical inltegration is
given for functioins of oine variable. It is applied to (1) in ?3, aild convergence of the resulting method is showii in ?4. Section 5 conitains computational notes and a numerical example.
2. Generalized quadrature.Assume f E C[a, b] and sp(t) is Lebesgue
b
integrableon [a, b];denote f
(For most practicalprobIp(t) Idt by I(p111f.
* Received by the editors June 13, 1966, and in revised form January 18, 1967.
Contributed at the Symposium on Numerical Solution of Differential Equations,
SIAM 1966 National Meeting at the University of Iowa, sponsored by the United
States Air Force Office of Scientific Research, May 11-14, 1966.
t Department of Mathematics, Indiana University, Bloomington, Indiana 47401.
337
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338
KENDALL
E. ATKINSON
b
ft so(t) Idtbe an ordinary singularintegral defined
lems, it is sufficient that
as a limit of Riemann integrals.)
Consider the problem of numerically integrating f(t)(p(t) over [a, b].
Most procedures amount to finding an approximating sequence {f,,} to f,
and then to using
Jb
rb
f f(t)(p(t) dt
(t) dt
Jfn(t)s(
with the error decreasing as n increases. Usually fn has been an interpolating
polynomial of degree n for f or an nth partial sum of a series E axjrj(t)for
f, [41,[6], [71, [121, [14], [16], [17]. The maini practical drawback to either of
these approaches is the problem of evaluating integrals of the form
rb
Jrj(t)So(t) dt,
where the integrationls become increasingly complex with increasing n.
The approach given here is to use piecewise polynomial interpolation and
to develop the natural generalizations of the trapezoidal rule, Simpsonl's
rule, etc. The error for such quadrature is given by
En(f)
=
rb
[f(t) - fn(t)]p(t)
f
I En (f) I -< 15 Ill1| f
dt,
fn 11I
-
2.1. The generalized trapezoidal rule. Let h = (b - a)/n, anid definie
ti = a + jh, j = 0,1, ***, n. Let fn(t) be the piecewise linear interpolation
function of f(t) at the nodal poilts to, ti, * , , i.e.,
fn(t) -h [(t-
t)f(t4-1) + (t
tp-_)AWL
ti-i < t < tijX
The proofthat En(f)
0 as n -oo
|| f
(5)
1 *** , n.
followsfrom (3) and the inequality
-fnl 11 c(f ;h))
where co(f; h), the modulus of continuity, is defined by
w(f; h)
max
=
lf(si)
-f(82)|.
Is 1-8 21 C h
Since f is continuous, w(f; h)
-*
0 as h -O 0.
rb
To find an explicit expression forJ
fn(t)(
t) dt, substitute from (4)
to obtain
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FREDHOLM
INTEGRAL
339
EQUATIONS
n
rb
fn(t)Xo(t)dt = E [ajf(tj5-) + fjf(tj)]
(6)
with
(7)
ai
h
(t -tj 4-0s(t)
i=
dt,
(tj -t)(P(t)
i
h
ti-i
dt.
For this form of quadrature, it is necessary to evaluate the integrals of
p(t) and tbp(t) over arbitrary intervals. These integrations are generally
not as difficult as they first might seem siinceusually the singularity of an
integrand can be isolated as a simple function; this is illustrated in ?5.
For the case p(t) = I- -c c for a > -1, and log I t-c
1, the necessary
1, we obtain the ordinary
integrations are quite easy. When p(t)
trapezoidal rule since a( = 3,j= h/2. Also, s may be continuous, and it still
might be advantageous to integrate it exactly because of a singularity in
a low order derivative, e.g., (p(t) = NIt on [0, 1].
When f" is conltinluous,a better error bound can be found by using the
error formula for Lagrange iilterpolation [8, p. 56]. Using it on each subinterval [tj_,, tj], aind combining it with (3), we obtain the following.
THEOREM 1. Let f" E C[a, b]. Then the generalized trapezoidal rule has
the errorbound
I < Wh2lf"
jEn(f)
(8)
.
The order of convergence is the same as that of the regular trapezoidal
rule, but this will not be true for the generalization of all quadrature rules;
the generalized Simpson's rule will only have an h3 order of convergence.
2.2. Other generalized quadrature rules. For the generalized Simpson's
rule, let n > 1, h = (b - a)/(2n), and tj = a + jh, j = 0, 1, * *, 2n.
Define fn as the piecewise quadratic interpolation function to f on to, ti,
t2nXfn being quadratic on each subinterval [t2j2 , t2j], j - 1 ** n.
The quadrature formula becomes
b
n
dt =
jfn(t)>p(t)
a
E
[aj f(t2j-2)
j=1
+ ,j3f(t2j-1) + 'yjf(t2j)]
with
2k2
a=
(t
ft
t2j-2
Fi-2h2 Jt
-12
t3=
1
(t
-
t2j)(t
-t2j--1)(t
-
t2j-1)i(t)
dty
-t2i-2)(P(t)
dt,
r2j2
ft2i
(t
- t2j-2)(t
-
t25)(P(t)
dt.
t2j-2
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340
KENDALL
E. ATKINSON
The analogue of (5) is
c(f ; 2h).
lif - fn II< 5w
For f"' E C[a, b], the analogue of (8) is
IEn(f)
f
27 h3|I
I<
||! || so
p Ill.
The reason for this being of a lower order than the regular Simpson's rule
is that the interpolation polynomial for (t - t2j1)3 is odd about t2j_1 on
[t2j-2, t2j]. Whenp =_ const., this oddness will imply that both the integral
and the numerical integral of (t- t2j_1)3 over [t2j_2, t2j] will be zero, and this
implies that the numerical integration of cubics is exact. But for so 4 const.,
this behavior will not hold.
The generalized quadrature rules of a higher order can be defined in an
analogous manner to that used above. For piecewise mth degree polynomial
interpolation with n subintervals, the analogues of (5) and (8) are, respectively,
11f-fn || - Cm w(f; mh),
9
En,(f)
J
_
Dm
IIf(m+l) 1P 1 1l
(m + 1)
hm+?
with Cm,the norm of the Lagrangian interpolation operator of degree m
and
Dm =
max jIh(/u-1)
(,u-m)
j.
o < A:! l
See [3, p. 31] for more details.
3. The numerical solution of (1). In order to show that the integral
operator of (2) defines a compact operator on C[a, b] into C[a, b], it is sufficient to show that the kernel satisfies the following properties.
Al. For each s, a < s < b, K(s, t) is Lebesgue integrable on [a, b], and
1I5z11=
rb
sup
a?8?b
f
a
K(s, t) I dt
is finite.
b
A2.
f
IK(si, t)
as Is-
s2I
-K(s2,
t) Idt O uniformly for a < si ,
S2
0.
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<
b
FREDHOLM
INTEGRAL
341
EQUATIONS
These hypotheses seem to be as general as possible, and the important
kernels
log I s-tt
and
a > -1,
s-tja,
satisfy them easily. In order to show that these hypotheses are satisfied
for a more complicated kernel, it is sufficient to be able to split it into a
finite sum
n
(10)
K(s, t)
=
ZHk(s,
k=l
t)Lk(s, t),
where each Lk is continuous and each Hk is known to satisfy Al and A2.
To apply the generalized quadrature to the operator 3C of (2), it is
necessary to identify so and f. The choice f(t) = x(t) and so(t) = K(s, t)
will generally not be adequate since most kernels K(s, t) cannot be integrated exactly with respect to t. Rather it will be assumed that K(s, t)
can be split as in (10) with each Hk capable of being integrated exactly.
In addition, each Lkshould have some differentiability in order to guarantee
a good order of convergence in solving (1). A trivial example of such a
split is
Yo(j s -t
1) = A(j s -t
1) log s
-
t j + B(j s
-
t 1),
where YOis a Bessel function of the second kind of zero order and both
A(u) and B(u) are analytic functions of u. With each product in (10),
identify f( t) = Lk(s, t)x( t) and so(t) = Hk(s, t).
For developmental purposes, we will assume that
K(s, t) = H(s, t)L(s, t);
all derivations and proofs will easily generalize. Now denote by
[L(s, t)x(t)]. the mth degree piecewise polynomial interpolation function
on n subintervals with respect to the variable t; it is the analogue of fn(t)
in ?2. To approximate the operator 3C,define the numerical integration
n
operators 3C, n>
1, by
b
(11)
(3cnx)(s) =
f
H(s, t)[L(s, t)x(t)]. dt,
a < s <b.
For numerically solving (1), (X - 3C)x = y, replace 3Cby 3C, some n > 1,
and solve the approximating equation, (X - 3C) xn = y. The details for
the trapezoidal rule are given in the next subsection.
3.1. Application of the generalized trapezoidal rule. For the trapezoidal
rule, (11) will reduce to the following by using (6) and (7):
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342
E. ATXINSON
XENDALL
n
=
(3C,nX) (S)
E
i=l
(12)
aj(s)
=
,l(s)
=
[aj(s)L(s, tj_-)x(tj-1) + /3(s)L(s, tj)x(tj)],
(tj(t
h ft
-
t)H(s, t) dt,
t) dt.
tj-,)H(s,
The functions cajand fj are continuous by assumption A2 on H. Also 3Cn
is a finite rank operator on C[a, b] into C[a, b], and is therefore a compact
operator. For simplicity, (12) can be written as
n
w
Wk(s)L(s,
=
(3Cnx)(S)
k-O
The approximating equation (X
= y is
3Cn)xn
-
tk)X(tk).
n
(13)
xXn(S)
-
k-O
wk(s)L(s,
A system is produced from (13) by setting s
=
a
- y(s),
tk)xn(tk)
*
ti, i = 0,1,
s
?
< b.
, n, to obtain
n
(14)
Xxn(ti)
,
,wk(ti)L(ti
-
=
tk)xn(tk)
k==o
Y(ti),
i
=
0, 1,
n.
,
This system is easy to solve, but more important, every solution to the
system
writing
to a solution
corresponds
(13) as
of (13).
This
can be partially
seen
by
_n
=
xn(s)
-y(S)
X\
and noting
L
the right-hand
that
+
Es Wk(s)L(s,
tk)Xn(tk)
k-O
side depends
on xn at just the quadrature
nodes.
Only
xn(to),
*
it turns
analysis
rather than for
will be wanted
to be more natural
, xn(tn)
out
max
0!, k:!Sn
For a more extensive
discussion,
I x(tk)
-
in most cases, but for the error
to find a bound for || x -Xn
xn(tk)
see [1], [2], [3].
4. An error analysis. The error analysis is based on the general theory of
and Moore [1], [2]. Their theory assumes that a sequence of linear
operators {3Cn satisfies the following hypotheses:
Bi. 3Cnx -* 3Cx for all x E C[a, b].
x 11< 1} has compact closure in
= {3Cnx n ? 1 and 11
B2. The set
Anselone
C [a, b].
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FREDHOLM
INTEGRAL
343
EQUATIONS
From these hypotheses, it is proven that (X - C)-' exists if and only if
for all sufficiently large n, (X - 3Cn' exists and is bounded uniformly with
respect to n, [1, Theorem 7.3]. Their general theory gives computable error
bounds for 1 x- xn 11,but those bounds cannot be used here because of
11,a quantity used in their esthe difficulty in evaluating 11(,W-3C)3C
timates.
LEMMA. The sequence {J3X,}of (11) satisfies Bi and B2.
Proof. (i) For x E C[a, b], apply (9) to I (3Cnx)(s) - (Cx)(s) I to obtain
3CX -
3K,X
? Cm.l3C
max
max
aCsCb
IL(s, rl)x(r1) -
L(s,
r2)x(r2)
|
1r1-r21?5mh
where Ie 11 is obtainied by applying Al to H(s, t). In the inequality, the
right-hand side goes to zero by the uniform continuity of L(s, t), thus
proving Bl.
(ii) The proof of B2 is the stanidard one of showing 0 to be uniformly
bounded and equicontinuous, and of then citing the Arzela-Ascoli theorem.
* with I L 11*
A uniform bound can be proven directly to be Cm.i
I lI II L 11
the maximum of I L(s, t) |, a < s, t < b, or the principle of uniform boundedness [15] can be cited oni the basis of Bi.
s$ , s2 < t and 11x 11< 1. Then use
To prove the equicontinuity, let a <
(9) and the definition of Cm to obtain
|(3nX))(Sl)
-
(WnX)(82)
I<
Cm||
3
i|
max IL(si, t)
-
L(s2, t)|
atb
+ Cm||LLI*
IH(s', t) - H(s2,
t) I dt.
The equicontiniuity of the right-hanid side follows from the conitinuity of
L(s, t) and the assumption A2 on H(s, t).
THEOREM 2. Let K(s, t) = H(s, t)L(s, t) with L(s, t) continuous and
H(s, t) satisfying Al and A2. Define Wnc n _ 1, by (11), and assume that
(X - 3C)1 exists. (Note that 3Ccompact invokes the Fredholm alternative
for X - X, and therefore it is necessary to show only that X - 3 is one-toas a bounded operator
one in order to infer the existence of (X
defined on the whole space C[a, b].) Then
(i) for all sufficientlylarge n ! N, (X - cn)f-' exists with a boundB < oo
which is uniform for n > N;
(ii) for (X - 3)x = y and (X - 3n)Xn = y, we have the bound
(15)
ix - xn11 < BIICx- anxn
n > N,
thus showing Xn -- x as n -s o.
Proof. Part (i) follows from the lemma and the remarks preceding it.
The proof of (ii) follows immediately from taking norms and bounds in
the identity
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344
E. ATKINSON
KENDALL
(16)
= (X
X-Xn
To prove this, subtract (X
as follows:
X(X
-
Xn)
=
Wn)(X
n ? N.
-
= y from (X
Xn)Xn
3 X
(X -
3n)->(3C
-
nXn =
(3C -
-
=
Xn)
3Cn)x
(3C -
-
3C)x
+
3C(x
=
y and proceed
-Xn)
Wn)X2
to obtain (16).
and multiply by (X -3C)-1
The above lemma and theorem can be generalized to the case of a coiitinuous [L(s, t)x(t)]n obtained in some other manner than interpolation.
All that is necessary is that
[L(s, t)X(t)]n -- L(s, t)x(t)
uniformly in s and t. I would like to thank the referee for pointing this out.
COROLLARY. Let 3Cnbe defined using the generalized trapezoidal rule of
= y and
(12), and assume a2L(s, t)/at2 is continuous. Then for (X -3C)x
x" E C[a, b], we have the bound
I1BfhI
(17)
h
-8
?B-11c11
a2L(s, t)x(t)
Proof. Apply Theorem 1 to I (3Cx)(s)
Theorem 2 to complete the proof.
at2
-
(3Cnx)(s)
, and
thein use
5. A numerical example. Practical computational notes will be given in
the course of developing the example. Let
K(s, t) = log Icos s-cos
(18)
O < si t < r.
t,
As noted at the end of ?3, the answer is usually desired at only the quadrature points, and that will be the case for this example. For most of the
development, the generalized trapezoidal rule will be used.
The first thing to do in using generalized quadrature is to split the kernel
into a sum of the form (10), and the initial split that came to mind was
(19)
H(s, t)
=
I
t -/2
L(s, t) = I-
t |"2 log I cos s -
cos t|.
Its main disadvantage is that aL(s, t)/at is unbounded at s = t, and therefore there is no guarantee of even an h order of convergence.
Producing the matrix of coefficients of (14) can be considerably simplified, and a general idea of how this is done will be given below for a better
split of K(s, t). Using the split (19), the following equation was solved
numerically:
(20)
x(s)
x(t) log I cos s-cos
Its exact answer is 1/(1 + ir log 2)
t I dt = 1,
0 < s < r.
0.31470429802. At the most accurate
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FREDHOLM
INTEGRAL
345
EQUATIONS
point, s = ir/2, the answers converged with only an h order of convergence.
Moreover, the approximations for low order n were quite poor, and for
n = 32, x,Qir/2) = .35347, in error by .038.
In contrast to this, another split of K(s, t) gives excellent results. Using
simple trigonometric identities, it is easy to show that
[sin
_l
_
log Icos s
-
cost
log
=
2s
t
+ log Is
+
tI +
-
log [(t
1
t)j
+ s (+2r -s
log (s + t) + log (2r
-s
-
t)
for 0 < s, t < r. Define
t
sin
Li(s, t)
(21)
=
log
2s)
2-s
L
L
2
H2(s, t) =log I s-t
1,
h(s, t) = log(s + 0,
10~
2 1/s
sin
+ log
s)(2
-
J
H3(s, t) =log (2ir-s-t),
H1i L2-L3
L4_1.
Then all of the Li's are analytic on [0, r], and each of the Hi's can be integrated exactly. Using the corollary to Theorem 2, we can expect an h'
order of convergence in solving (20).
Before giving results, a simplification will be given for computing the
weights aj( t) and fj( t ) which occur in the system (14). The effort required
to calculate these weights can be made negligible. A table with 200 entries
can be created, and with it the weights can be calculated trivially for n ? 50
and for each of the functions He, H3, and H4.
The table will be developed for log Is - t 1,and it can be shown to easily
apply to the other two singular functions. The definitions of a,j(ti) and
tj1(ti) are
aj(ti)
=
h
MO(t) =-J
(tj-
t) log ti
(t
tj_-) log ti
-
t Idt,
-
-t
i dt
for i = 0, 1, 2, ... , n and j = 1, 2, * , n. Make the change of variables,
t = tj_l + Vh, 0 < V ? 1, dt = h dV, and recall that t, = a + ih. Then,
aj(ti)
=
MO(t) =
h log h + h j(1-V)
hlogh+
h
log I i-j
V log i i-j
1-V
+ 1-V
I dV,
I dV.
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346
KENDALL
E. ATKINSON
TABLE
2
4
8
16
32
1
xn(O)
Error
Ratio
.3051691
.3122181
.3140722
.3145453
.3146644
.0095
.00249
.000632
.000159
.0000398
3.8
3 9
4.0
4.0
Define
Wo(k) =f
W1(k)
(1-V)loglk
=f
-VIdV
1
VlogIk-
VIdV.
By producing Wo(k) and Wi(k) for -49 _ k < 50 we can calculate aj(ti)
and fj( ti) for log Is - t j on any interval [a, b] and for any n ? 50. To evaluate W0(kl) and W1(k), first define
1
*Jo(k)
1
If(k)=
=log I k-V I dV,
V log Ik-V I dV.
These are more natural to work with than Wo and W1, particularly in
methods of a higher order than the trapezoidal rule. For calculating 'o
and 41',
to(k) = k log Ik -(k
*1(k) =
2 [(k-
I2O
-
1) log k
k
k-1
-
1
-1,
1 -klol k lk] + l[k2
(k -)
+
kfo(k) .
Then W0 = to1, W1 = 1
Using this way of finding the weights, (20) was solved with the split (21),
and the answers at the worst point, s = 0, are given in Table 1. Note the
h2 order of convergence.
The same equation was also solved with the generalized Simpson's rule
and the same split (21). Table 2 gives the answers at the worst point,
s = 0, and the number of quadrature points corresponding to n is n + 1.
The accuracy is quite good, and the order of convergence is h4 rather than
the usual h3. This can be explained on the basis that (a) the true answer
is a constant and the generalized Simpson's rule is exact for constants when
the function L(s, t) is of degree <2 in t, and (b) the h4 order holds for the
1 results in the regular Simpson's rule.
kernel Li(s, t)Hl(s, t) since HI
Results consistent with these were also obtained for (20) with the righthand side replaced by other functions. These examples showed that Simpson's rule would generally be needed for practical problems.
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FREDHOLM
INTEGRAL
TABLE
347
EQUATIONS
2
n
xn()
Error
Ratio
2
4
8
16
32
.31449048976
.31468781770
.31470316978
.31470422550
.31470429345
.000214
.0000165
.00000113
.0000000725
.00000000456
13
1436
14.6
15.6
.
It should be emphasized that the programming of generalized quadrature is not particularly difficult once the mathematics accompanying it is
fully understood. Once the tables referred to earlier are calculated, the
speed of solving the integral equation is comparable to the speed in using
regular quadrature to solve integral equations with smooth differentiable
kernels.
Acknowledgments. This paper was part of my doctoral thesis, and I wish
to express my sincere appreciation to Dr. Ben Noble, my major professor,
for his guidance in my research. The computation of ?5 was done on a CDC
1604, and I acknowledge the financial support of the Graduate Research
Committee of the University of Wisconsin, together with the Wisconsin
Alumni Research Foundation and the National Science Foundation. I
also acknowledge the assistance of a U.S. Army Mathematics Research
Center Fellowship.
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All use subject to JSTOR Terms and Conditions
348
KENDALL
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