PBSTRACT OF THE THESIS OF
AIJ
Thomas Leroy Grlshn
for the
Dge)
tNe)
Date thesis is presented
Title
An
3.
September 28,
rror Bound for rn Iterative
in
iathemtics
ti
1953
ethod of So1vinT
Fredhoim Integral Ecuations
_
a
___________________
Abstract approved
Various problems in physics and engineering lead to
integral equations of the Fredholm type and second kind.
Generally speaking, Fredholin's solution of such equations
is given in terms of the ratio of two infinite series.
This method is not customarily thought of as useful for
computation because the direct calculation of the tenns
of these series is formidably difficult. However, the
successive terms of the two series can be calculated
from recursion formulas as suggested by G. C. Evans.
The principal result contained in this thesis is the
th step of this process.
bound for the error at the
Techniques of F. Tricorni are adapted to this problem.
An illustrative example is worked, and tables are given
which w111 facilitate use of the error bound by others.
AN EBROR
EOITND
FOR AN ITtA!IVE METHOD OP
OLV ING- PREDHOLM INTEGRAL E4TAT I ONS
by
THOMAS LEROY GLAHN
A
THESIS
submitted to
OREGON STATE COLLEGE
in partial fulfillment of
the requirernents for the
degree of
MASTER OF
June
C lERCE
19511.
APPROVED:
?rofessor of Mathematics
In Charge of Major
Head, of
Department of Mathematics
airman of
choo1 G'rad.uate Committee
Dean of Gi,adnate
Date thesis is presented
TyDed by Margaret Schnjdt
choo1
!ePtemb0r28, 193
TABLE OF OONTL1TS
Page
INTRODUCTION
.
. . . . . ,
.
PREDHOLii1S SOLUTION
,
.
. . . . . . . . .
. , . . . . . .
,
. . . . . .
,
THEAPPROXIMATEOLUTION.,.,.,.,,..........
CO}4JTATIONOP
A
n
AND
THEBOU1DPORTHEEEROR
,,,...........
.,...,.........
B(x,y)
TABULATIONS AND A NUIitgRICAL EXAPLE
. , , . . . . . . . . . .
i
14.
6
7
il
17
AGIGOWLEDGET
The author wjshes to express his gratitude
to Dr. A, T.
Lonseth for his guidance
during the preDaration of this thesis,
and.
patience
AN NRROI BUlLID FOP. AN IERAT IvE METHOD OP
SOLV ING FPtEDHOLM INTEGRAL EQUATIONS
INTRODUOT I ON
±hys:ica1 irob1erns are
inteal
equations or
more
familiar,
equations.
many
The
usually formulated as differential
equations. Älthouh differential ecjuations are
problems can be conveniently expressed as integral
calculus of variations is a rich source of such
equations and the use of the Green1s function for the differential
eouations of a boundary value problem results in an integral ecluation.
Ccnsidering everything, there is reason to suspect that the extensive
use of differential equations is more habitual than expedient.
boundary value problem) when exDressed. as an
integral equat ion,
A
may
appear rather cumbersome as compared with the corresponding differen-
tial
ecjuation.
it
However,
must be remembered
equation expresses not only the
that the integral
Localu condition of the Droblem,
but also the boundary conditions which must be
satisfied. Por
instance, a linear boundary value problem in one variable can usually
be exoressed as
u(x)
where
Iniown
of the second kind and
here.
The
f(x)
The
K(x,t) u(t) dt
as a Fredhoim type linear integral ecjuation
it
represents the class which will
function 1(x) and the
a loiown parameter and
+
a
contain the boundary conditions,
K(x,t) and f(x)
This is
=
u(x)
is the
'kernel'1
un1mown
solution of this equation,
K(x,t)
are
be
considered
Irnown,
X
is
function,
if it exists,
can be
cressed
2
as (6, pp.19-20),
b
u(x)
with
H(':,t;X)
as the
=
f(x)
'reso1vent
+J
H(x,t;X)
(t) ct
kerne1.
There is a well-known expansion ±or the resolvent kernel
which was derived by Predholin
pp.365-390).
(3,
This expansion cives
the resolvent kernel as the quotient of two integral power series and,
generally speaking, both series contain an infinite nuther of tarm$.
If the
series contain only a finite number of terms,
compute the exact solution.
in the general case, however, actual
computation of a solution by Fredhoim'
approximation.
It
it is Dossible to
s
method
mtist
result in an
is our purpose here to determine a bound for the
error contained in this approximate solution.
One of the major objections to the use of
sion for the resolvent kernel
The coefficient of the
th
s
expan
has been the amount of labor involved.
term of each series involves
successive integrations of a determinant of order
n.
n
Beyond the
second or third term, the direct compitation of these coefficients is
far too laborious to be practical.
lead. to
owever, relationships
two relatively simple recursion formulas.
outing the series is not usually
it seemed that
it
est
which
This method of
found. in the literature
corn-
and, therefore,
should appear here.
The necessary relationships for this method appear in the
works of Predhoim (3, o.37l), as part of his derivation of the solution.
ia1esco
(5,
pp.25-33), used the same relationships to derive Predhoim's
resolvent kernel from the itòrated or Neuman resolvent kernel.
They
ii
also appear in several books which present the $'redholm theory but
the suggestion that they be used for computation appears very seldom
in the literature.
vans,
in a veview of a book written by Lovitt, apparently
was the first to suggest the use of the relationships for computation
(1, pD.l1-i)--5),
(Lt,
p,)432),
Prank and von Mises
(2,
p.516),
and.
Hildebrand
included them as computational aids in their presentation
of the Predholm theory.
In the chapters that follow,
we shall consider first the
true solution as given by Predholm and then the approximate solution.
Following this will be the method of computation and the bound for the
error in the approximate solution.
The final chapter will contain a
table for use in evaluating the bound and a numerical example.
The error bound is developed along the lines of Tricomi's
derivation of an error bound for a similar problem (7, pp.l-83-486),
and.
(8,
pp.26-30).
His approach to the problem is used.
bound. is given in terms of
the functions that he used.
difference is that Tricorni used an approximate kernel
and.
the error
e
principal
K(x,y), while
here an approximation is made directly to the true resolvent kernel.
To be
elicit,
where
K*(x,y)
Tricorni replaced the kernel
by
K*(x,y)
satisfied. the condition
¡
uniformly in
K(x,y)
x
and.
y
solution obtained by using
K(x,y) - K*(xy)J <
He then developed. an error bound for the
K*(x,y)
as the kernel.
11.
OLUT ION
ThEDHOLM' S
The integral equations under consideration are of the type
f(x) + x
u(x)
(2.1)
K(x,t) u(t) dt
J
"a
Where
K(x,t)
meter, and
and
f(x)
a-re
X
imown functions,
is a laiown para-
is the unknown function.
u(x)
Pred.holm solved this equation subject to the following
conditions:
a.
K(x,y)
b.
f(x)
c.
D(X)
is real and continuous on
I:a
is real and continuous on
O.
R:a = x,y = b.
x
b.
(See equation 2.5)
Under these conditions, the unique, continuous
solution is given by
('b
u(x)
(2.2)
=
¿(x t;X)
f(x)
The functions
i(x,y;X)
and
are the integral power series
D(X)
L(x,y;X)
referred to in the introduction and
,_,\flXn+l
(x,y;X)
(2.3)
with
B(x,y)
=
K(x,y)
(2.)L)
=
K(x,y)
b
n0
and for
f(t) dt.
D(X)
+ J
n
fl
is defined by
B (x,y)
n
1,
K(x,t1) ... K(x,t)
b
B(x,y)
dt ... dt
=5a
¿
K(t,y) K(t,ti)... K(t,t)
n
The series in the denominator is defined by
f-i)x
co
(2.)
-
D(X)
=
n=O
with
A
=
3.
and for
b
(2.6)
n
b
n
1,
IK(t1,t1)
I...
A
Ja
Ja
Both series converge for
imiformly with resDect to
A
nt
..,
..*.....s.
K(t,t1)
...
- co< X < +
x
and
y
dt1 ... clt.
I
K(tn,tn)I
co
and
on R: a
A(x,y;X)
x,y
converges
b.
TRE JPPROXIMATE SOLUTION
Ou.r
taking only a finite number
and.
D(X).
u(x)
approximate solution
will be obtained by
of the terms in the series for A(x,y;X)
m
That is
A (x,t;X)
u
(3.1)
m
(x)
=
m
f(t) dt
(l)nXfl+l
=
A (x,y;X)
(3.2)
m
f(x) +
fl0
In
B(x,y)
fl
Ill
D(X)
(3.3)
with
=
nO
B(x,y)
and.
A
defined by
n.
(2.1k-)
and.
(2.6).
where
and.
1
A
COMPUTATION OP
AND
B
(x, y)
As mentioned in the introductIon, the direct comDutation of
A
and
B(x,y)
for
n > 2
is quite laborious.
Therefore, we will
present here two recursion fornmi.as which are relatively easy to use.
The recursion f oriailas are
(Li,i)
B (t, t) dt
and
b
(.2)
=
Ecjuations
and
A1K(x,y)
- (n+l)j
(.i) and (.2) are valid for all
B(x,y) = K(x,y), the desired
n
B(x,t) K(t,y)
and since
O
nujnber of terms
dt.
A0
= 1
may be computed.
EQuation (.2) is essentially an exoression of Fredholmts
first findamental relationship which is (3, p.373),
rb
xJ
L(x,y;X) = XD(X) K(x,y)
A(x,t;X) K(t,y) dt
+
a
and may be derived from this relationship by associating like powers of
the parameter
X
We will derive both equations directly from the
definition in the following manner.
In view of (2)4), we may write
3(t,t) dt
,f
.. K(t,t)
K(t,t)
K(t,t1)
K(t1,t)
K(t1,t1) ... K(t11t11)
.. el...... .................
K(t,t)
K(t,t1)
...
dt1
... dt
nj
dt.
Replacing
by
t
by
t1, t1
,,
t2,
by
t
this may be
t141,
written
fb
b
b
B (t,t)dt
K(t,t1)
IK(t1,t)
I
n
.Ja
=5a
a
dt1 ...
K(t1,t1) ,. K(t+1,t+1)I
and considering (2.6) this becomes
b
3(t,t) dt
4n1
=
which is the desired res'ilt.
In order to derive equation (11.2) consider (2.11) which is
b
b
B(x,y)
=J...f
K(x,t)
K(x,y)
K(x,t1)
K(t,y)
K(t1,t1) ... K(t1,t)
.
e
..........
K(t,y)
..
s..
5
K(t,t1)
e ..
...
dt.
dt1 ...
.. s.
K(t,t)
Developing the determinant in the integranö. in terms of the elements
of the first column, it is found that
b
B(x,y)
J
a.
b
1K(t,t)
...
IK(t ,t
...
dt1 ...
K(x,y)
..ja
n
i
)
K(t ,t
n
)I
ni
K(x,t)..........K(x,t)
n
i
.sse.esessIeess
irb
n
+
K(t
i-b
(-i)
,t
a
1(t ,t
nl
Ir
i
)....... K(t
...
1=1
a
i-1
)
t.
by
and we obtain for that summation
t,
t1
n)
K(t.41t)
s..
s....
........ K(t
view of (2.6), the first term reduces to
of the summation, replace
,t
i-1
n.
t,
.dt.
e.
,t)
AK(x,y).
by
e
dti5
...
In the terms
,
tn
by
tn_i,
1=1
dt1.
J
t2t
a
a
'
(X, t1, ...
t t.
.
..
litt tjt
t I
.
.
etn_i
t I
I
tn_i
where
=
(
IK(x,t1) t,..... K(x,t11)
K(x,t.)..... K(x,t_1)
K(x,t)
:i.
I
I
. . . . . . . . . . .
I
t
t a t I I h t I t I I
t I
I t
t I I t I I t t t t I t I I I t .
I S I t I t
IK(t_1t1) IIS.S...........I......................I
IK(t1,t1) SIP...I,,...s,.,.,II
(
t1 )
t
. I I S t I
I
S I I
I I I
K(t1,t)
I II t t
I
I
S
........,.. K(t,t_1)
I S I I t t
I I t
I I I I
K( t
,
tn_i )
IItIIIIIS*SItStS.I....I....I*.S.....S.....I....I....II..I..SIt
L(t
1,t1)
...., K(t
,t1)
n-].
K(x,t)
into the first column,
the summation becomes
t,y)
K(x,t)
K(x,t1) S.... K(x,t_1)
K(t1,t)
Kt1,t1).,,.. K(t1,t_1
IS.It,S,..I,. ..... ..I.........
K(t_1,t) K(t_1t1)...
K(tn_i,tfl_l)I
dt dt1 ... d.t_1
This forni of the integrand shows that the terms of the
summation are all
respect to
t1
integrations.
and we have
.
ecjual.
ta_1
K(t,y)
I
K(tn_itt) K(t_1,t1).. K(tn_l,t_lu1
Now bringing the co1uim containing
n2i_1f
t
In addition, we may integrate first with
and consider
K(t,y)
constant for these
may be taken before the (n-1)-fold. integral
b
K(x,t)
{I
J
.
(x,t1)
..... ...s.*s.s
dt1 ...
(2.)4-),
.
e...
...*...
K(t_1t1) ..
(t_1t)
Conidering
K(xt_1)
K(t1t1) ..... K(t1,t_1)
K(t1,t)
s.
..
dt.
this may be written
_nj
K(t,y) B
(x,t) dt
and the final relationship is
ß(x,y) = AK(x,y) -
nf
_1(x,t) K(ty) dt.
ib!
THE BOUED FOR THE ERROR
We will now develop a bound for the error
u(x)
as the
Sibtractin
(5.1)
solution and
tru.e
um(x)
ju(x)
j
with
as the approximate solution.
u(x)
(3.1) from (2.1) we obtain
u(x)
-
f(t)
j L;t;
Um(x)
- ______
Omitting the argu.ments for the present,
=1b
uu
u-u
or
In
(5.1) may be written
D -AD
m7
rm
aLm J
In
b
=1
[(D
D)+ D(&-A
In
I
J
fdt
)I
fdt.
DD
I
Ja
t.
m
I
Taking absolute values, we arrive at the inequality
fb[Al!Dm_Di+lDIlA_AJ1
(5.2)
fu-taj
Ifj
In order to get (5.2) into a usable form, upper
be found for
¡DI,
¡Aj,
and
¡Da_DI,
bounds in terms of the functions;
(5.3)
(I)
__Ç_
=
jA_Aj.
2
E
x
n=O n
co
__n._ '(z)
=
n
2
_
n]
n
co
(x)
In
e
E
n=ni+1
n.
dt.
bnds nst
will get these
12
n-1
co
fl(x)
m
and
=
X
nm1
(n-l),
In deriving the bounds, freqaent use will be made of
Hadamardts theorem concerning determinants.
RLDAMARD1 S THEQREM.
It
If the elements
is stated here briefly.
ajj
of the determinant
a11 ... a
a1
...
are real and satisfy the inecjuality
1<=
M
la1jI
then
n2
A1
tiPPER
BOU1D FOR
Restating (2.5), we write
1D(X).
D(X)
n0
with
A0 = 1
n
and for
n,
1,
(b
K(t1,t1) ... K(t1,t)
r'°
An
=J
a
since
K(x,y)
.)
a
.................e
K(t,t1)
...
.
dt
K(t,t)
R,
is continuous in the closed rectangle
also bounded there and we may assume that
fore,
cit1
Hadaardts theorem may
K(x,y)j
M
in
There-
be applied to the absolute value of the
determinant in the integrand. and the resulting inegumlity is
IAl
R.
it is
<Sb
dt1 ,,, dt
13
(5.U)
LAi
Mow
D(X)j
M
n
(b-a)
ooNn
<
=
n=o
and. if
we substitute
,
jAl
nt
we will have the
into this exoression,
(5)1W)
inequality
lxE
2
ID(X)I
fl:O
If we let
N =
N
(b-a)
ni,
fi
D(x)I
AN
U?P. BOUND
POE.
(ba)L
consider (5.3), the resait is
and.
(5.5)
n
(N)
jL(x,y;X)1.
e
will start with equation
(2.3) which is
(l)nxfl+1
co
(x,1;X)
=
where
B(x,y) = K(x,y)
>
n = i
and for
K(x,t1) .... K(x,t)
K(x,y)
1b
.(x,y) =
çb
JJ
...
a
,)
K(t
1(x,y)
nl
n=o
t1).... K(t1,t
K(t
)
...................
i
dt,
J.
n
a
K(tt)
K(t,y) K(t,t1)
Again we may apply Hadanard'
s
theorem to the absolute value
of the determinant and it is f oimd that
<C
b
b
i±2.
2
C
(n+l)
t]311(x,y)1
a
n+1
M
dt1 ... dt
'a
or
2
(5.6)
... dt
B(x,y)j
=
(n+i)
+1
Mn
(b-a)
l4
i
<
Now
¡(x,y;X)1
IB(x,y)I
=
=O
and. if
n.
I
we substitute (5.6) into this inequ.ality, it will become
n
1n+l
(b-a)
(n+1)
Letting
N =
¡XI
M (b-a)
changing the index of summation this may
and.
be written
¡A(x,yX)j
and.
considering (5.3),
¡x
i
n1
the final form is
¡(x,y;X)j
(5.7)
AN
'
-1)'
¼!'
UPPi BOUND FQR
¡XI
Mfl...*(N).
- D(X)l.
¡Dm(A)
Subtracting equation
(2.5) from (3.3) we get the difference
Dm(A) - D(X)
=
-E
n=m+].
Por the absolute value, we get the inequality
¡xv'
- D(A)I
Using
(5.ui.)
with
N =
lxi
nm+l
M (b-a)
and.
¡A
n.
n
in view of (5.3), this may be
written as
(5.8)
ID(A) - D(X)j
AN UPPER BOUIW POR
as previously defined,
£(x,y;X) -
')
n+i
j(x,y;X) - &(x,y;X)
j
.
With B(x,y)
the difference between (2.3) and (3.2) is
Am(x,y;x)
=
E
nm+1
(-l)"X"1
fl.
B(x,y)
15
Using (5.6), we get the ineauality
in+1
co
IA(x,y;X) - Am(x,y;X)j
With
N
(n+1)
nnrfi
as 2reviously defined,
t+1
2
n.
n
(b-a)
this may be written
w
-
-fl(Y?l
X
M
n_-m+2 (n-i)'.
Considering (5.3), this becomes
(x,y;X) - £:1(x,y;X)!
(5,9)
JBSTITtJIN T
BOUNDS.
jxj
MLL'1111 (N)
We will now substitute the expres-
sions just found for the corresponding quantities in (5.2) which is
b
LID
<
-u(x)I
Iu(x)
-Dl
4 ID11A-
I
jf(t)1
dt.
D(X)
and
IDIIDmI
1
Let
max 1f(x)I
for
a
x
b.
Since
are constants, we may bring them from under the integral sign.
stituting the bounds we have just found, (5.5),
DQ.)
Now sub-
(5.7), (5.8), and (5.9),
the inequality becomes
<
¡u(x)
IXIM.CL!
(1'T) ..n..
I
(N) +n.(N) Ixji:[.n.1i(N)
I
*u(x)1
(b-a)
IDIIDI
L
j
or
(5.10)
u(x)
-u(x)I
<
r.Û(N)A(N) +fl(N)t11(i)
1P
I
L
The quantity
it
IDI
IDI IDmI
may be left in this expression because
will be computed as part of the solution but a lower bound must be
found. for
IDI.
In order to get this lower bound,
consider (5.8) which
16
is
¡D-.Dj
Since
IDI -
ID_Dj,
¡DI
we may write
IDI -
II
(5.11)
or
IDI
_fl..(N)
Providing that the qu.antity
may be substituted for
IDI
=
IDI
IDI
-_O(N)
is positive,
in (5.10) without disturbing the inequali-
There is no reason to believe that for a particular value of
ty.
this will be true.
with
hand,
However, £)_(N)
and its limit is zero as
ni,
ID1
ni
D(?)
becomes infinite,
¡DI.
On the other
Since we have
there will exist a positive integer
O,
ni
is positive, monotone decreasing
will converge to a fixed constant
assumed that
it
such
p
that
IDI > .fl(i)
for
ni
> p
This means that in order for this substitution to be valid,
m
mast be taken large enough to insure that
is very close to zero,
practice,
¡DEI
this would reTxire that
m
>.O..(N).
If
be very large.
the computation will indicate the point at which the
exoression is valid.
Assuming that the substitution is valid, (5.10) becomes
(5.12)
¡u(x)
<
.n.$
(N) .Ç1(N)
+
-u(x)j =
IDI{IDm1 _fn(H)}
This is the final forni for the error bound.
D(X)
In
TABULATIONS AND A NUMEIRICAL EXAIPLE
fl
The function
intervals of
0.05
.
It
has been tabulated for
(x)
tabulate.
at
i
was tabulated by Tricorni for use in his error
bound which was mentioned in the introduction
The functions
x
O
£1.
m
(x)
and
1(x)
Ç).
(8, p. 28).
are not convenient to
rn
That is, they must be conwuted for each value of
Por
rn.
this reason, the coefficients in the series expressions for ft(x)
and
.û.
1(x)
are tabulated below.
to determine the value of
m
ll(x)
and all preceding terms.
The series for
fl
Similarly, to obtain
flmt(x)
at intervals of
fl'2(x),
0.1
fl.(x),
from
and all preceding terms.
The second table is a tabulation of fl.(x),
('
may be used.
by omitting the term containing
omit the term containing
fl).(x),
(x)
and 1i'6(x)
fl'(x),
for
O
fl(x),
x
18
COEFFICIENTS OF TH
SERIES FOR
fl(x)
fl(x)
fl(x)
1.000000
1.000000
x
1.000000
X2
x3
.ç(x)
fL'(x)
x]2
0.0062311-
0.036333
2.000000
x]-3
0,002795
0.016928
1.000000
2.598077
1L
0.001209
0.007591
0,866026
2.666667
X]-5
0.000506
0.0032811.
.0,666667
2.329238
x]-6
0.000205
0,001375
O.)4658118
1.800000
x]-7
0.000081
0.000558
0.300000
1,26011.06
x]-8
0.000031
0.000220
0.180058
0.812698
x]-9
0.000011
0,0000811-
0.101587
o.1817o
x
0,0000011.
0,000031
0,05)1.211.1
0.275573
x21
0,000001
0,00001].
x1°
0,027557
0,111.7196
x22
X11
0.013382
0,0711805
x5
0.0000011-
'9
TABLE OP IUI1ICAL VALUES
X = O
z = 0.1
x
0,2
X
0.3
xi
O.0
x
0.5
fl.(x)
1.000000
1.110938
1.28166
1.2O181
1.638873.
1.921110
flx)
0.000000
0,000938
0.008166
0.030181
0.078871
0.1711011.
flax)
0.000000
0.000005
0.000171
0.001398
0.006379
0.0211811-
ftx)
O.000000
0,000000
0.000003
0.000011.7
0.000379
0.001939
1.228901
1.529652
1.930203
2.1171325
3.213360
0.000000
0.028901
0.129652
0.330203
0.671325
1.213360
0.000000
0,000253
0,0011.395
0.O24376
0.0811.966
0.230507
0.000000
0,000002
0.000092
0.001133
0.006905
0.028680
x = 0.6
x = 0.7
x = 0.8
x = 0.9
x = 1.0
2.291210
2.7811.7115
3,11.511.618
11.,380827
5.68611.1111.
0,331210
0,5911.711.5
1.01)4618
1.670827
2.68611.14
0.0577)-1-8
0.137631
0.298111.6
0.6020914.
1.153752
0.007527
0.0211.011.2
0.0668511.
0.1675811.
0.38790)4
4.21169118
5.7103511.
7.817878 10.907159 15.519250
2,011.6911.8
3.310351!.
5.217878
8.107159 ]2 519250
0.535611.0
1.122630
2.189775
}-1..058717
7.25)4507
0.093803
0.2608511.
0.611.5896
1.11.67622
3.125269
.çLt(x)
..04x)
.fl.,!(x)
QQQ
Por an example of the aDplication of this error bound, we
will consider the boundary value problem
d.2u
+
Xu
+
=
g(x)
u(l) = O
u(0)
0,
&X2
The corresponding integral eQuation iS
pl
u(x)
where
f(x) + XJ
=
K(x,t)
=
K(x,t) u(t)
dt
(x(1-t)
for
O
x
t
t(l-x)
for
O
t
x
K(x,t) g(t)
dt
and
ri
f(x)
=
Jo
Por this kernel, we find that to satisfy
in
R, we rtust take
and
max If(x)j
=
M
P
=
for
0.25
O
It
x
i
M
K(x,y)I
will be assumed that
N
Therefore
jX
X
=
i
m(b-a)0.25
and the error expression becomes
(6.i)
lu(x)
-um (x)
l(O25)(025)
0.25
+ fl(O,25)mm+i(O,251
lDm(l)I{ÌDm(l)i
This expression will be evaluated for
m =
2,
-ft(0.25)}
)4,
m
6
and for this pur-
pose we will need the following quantities
(6.2)
fl
(0.25) = i.329177
n_l (0,25) = 1.715269
fl2(O.25) = 0.016677
fl(O.25)
fl(o,25) = O.O0O5a
fl.(o,25) = 0,002125
= 0.000013
fl1,(O.25) = 0.000059
= 0.052890
D2(1)
=
0,8)41661
D)4(1)
=
0,8)41)471
D6(i)
=
0,8)41)471
Gonsidering these values, it is seen that the error
expression is valid for
m
2
That is,
>
fl(o.25)
Substituting the values as given by (6.2) into (6.1),
obtain
Iu()
lu(x) - u)4(x)
fu(x)
0.0356 F
ua(x)
u6(x)
0,00133 F
¡
L
0.000035 F
we:
22
B
IBLIOGRAPHY
1. Evans,
G.. C
A review of the book Linear Integrai Equations,
We V. Lovitt.
The Apierjcn metheratjca1 monthly
34:124-2-l5O.
1927.
1r
2.
Prank, Phillip
and. Richard. von iises.
Die Differential i.n
Integraigleichungen der Mechanik und Physik.
2d. ed.
Vol. 1.
Braunschweig, Friedrich Vieweg und. Schn
1930.
3,
Frecihoiri,
..
916p.
Sar une classe d'cjuations fonctionnelles.
27:365-390.
1903.
Ivar.
Acta mathematjca
)4
::.
Hu1debrand P. B,
Methods of aDplied mathematics.
Prentice-hall,
1952.
U32p.
Lalesco,
Trajan.
Introduction
la thgorie des quations
Ìaris, A, He'man et fils,
1912.
152p.
intgra1es.
6.
Lovitt, William Vernon.
Linear intera1 equations.
i4cGraw-Hill,
192)4.
253D.
7.
Tricorni, Pranceso.
New York.
Sulla risoluzione nuaerica delle equazioni
integrali di Fredhoirn.
Atti della reale accademia
nazionale di lincei, Rondiconti 33, 1°.
emestre
)#83-)486.
8.
New York,
192)4.
Francesco.
Ancora sulla risoluzione numerica delle
equazioni integrali di Predhoim.
Atti della reale
accademia nazionale dei lincei, Rendiconti 33, 20.
Tricorni,
Seraostre :26-30.
192)4.