Internat. J. Math. & Math. Sci.
VOL. ii NO. 4 (1988) 751-762
751
ON SOLUTION OF THE INTEGRAL EQUATIONS FOR THE
POTENTIAL PROBLEMS OF TWO CIRCULAR-STRIPS
C. SAMPATH and D.L. JAIN
Department of Mathematics
University of Delhi
Delhi-f10007, India
(Received May 28, 1986 and in revised form June 20, 1988)
ABSTRACT.
Solutions are given to some singular integral equations which arise in two-
dimensional Dirichlet and Newmann boundary value problems of two equal infinite cocircular strips
axial
in
various
branches of
potential theory.
For illustration,
these solutions are applied to solve some boundary value problems in electrostatics,
hydrodynamics, and expressions for the physical quantities of interest are derived.
KEY WORDS AND PHRASES.
Inflnite co-axial circular strips,
Chebychev polynomials,
surface charge density, kinetic energy, elliptic integrals of the first and the second
kinds.
1980 AMS SUBJECT CLASSIFICATION CODE.
I.
31A.
INTRODUCTION.
Recently many authors [I-5] have presented solutions of various two dimensional
boundary value problems of two infinite strips, by integral equation techniques [6Shall [9] has given the solution of the Fredholm singular integral equation of
8].
the first kind with logarithmic kernal{q +
where a and q are known constants.
logmla
si(0 01)I},- <
i, <
,
This type of singular integral equation governs
the solutions of various two-dimensional Dirichlet boundary value problems involving
an
infinite
circular
acoustic scattering.
strip
in
electrostatics,
hydrodynamics,
and
low-frequency
The solution to this singular integral equation is derived in a
closed form from those of some well known integral equations of Carlemann type [I0,
We present here solutions of Fredholm singular integral equations of the first
kind of the type
C. SAMPATH AND D.L. JAIN
752
the
where
function f(6)) is
known
inequality 0
B
I01
equations of
the
type
a 7, and
of
a
class C
q
I,
for
the
values
known constants.
are
of 0 satisfying
The
singular
the
integral
(I.I) govern solutions of various two-dimensional Dirichlet
boundary value problems of two equal infinite co-axial circular strips in potential
In the corresponding Newmann boundary value problems, the governing integral
theory.
equations are of the type
+
l(01)cosec
2
(6)
< 16)I <
f(0),
01)d01
(1.2)
’
where the unkown density function satisfies the edge condtlons
( )
( =)
o.
(.B)
Integrating by parts and using edge conditions (1.3), (1.2) becomes
-
/ +/
6)l)dO
f(6)),
IB
< 16)( <
(1.4)
c
(.5)
l’(O).
g(O)
where
>=or (o
g(o
We present here a simple technique of solving integral equations of the types
1. I)
(1.2)
and
For
illustration,
this
technique
is
applied
solve
to
an
electrostatic Dirichlet boundary value problem and a hydrodynamic Newmann boundary
value problem of two equal infinite co-axial circular strips.
We have also solved the two-dimensional problems of scattering of a low-frequency
incident plane acoustic wa. by two equal infinite co-axial soft and rigid circular
strips by the integral equation technique.
The plan of this paper is as follows.
This work will appear separately.
In section 2, we first present a simple
technique of solving integral equations of the type (I.I) without reducing it to some
well known integral equations of the Carleman type [10, II, 12].
This is achieved by
reducing the solution of (I.I) to that of two Fredholm singular integral equations of
the
first
kind
with
(ii) (cosy- cosx)
-I
(i)
kernels
0
<
x,y
<
7.
(Const.) +
logl2(cosx- cosy)l,
0
<
x,y
<
7, and
The unknown and known functions are both even
degree functions in each of these two integral equations.
The first of these two
equations readily yields the Fourier expansion of the unknown even degree funtion over
the interval 0
<
y
<
,
when the well known expansion of the the kernel
[1,7] and the
Fourier expansion of the known even degree function over the interval 0
made in it.
<
x
<
n are
Similarly, we obtain the series expansion of the unknown even degree
function of the second integral equation in terms of Chebychev polynomials T (cosy) of
n
the first kind when we use the series expansion of its known even degree function in
terms
of Chebychev polynomials U (cosx)of
n
the second kind.
The solution of
this
second integral equation contains an unknown constant which is evaluated by making its
solution
to
satisfy an appropriate inner edge condition.
Then we illustrate this
INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS
753
(I.I), when the known function f(0)
technique to solve the integral equation
in it is
Lastly, we explain how equation
of a particular form, for our subsequent analysis.
(1.2) can be transformed to the form of integral equation (i.i)
and
is
therefore
solvable by the above technique.
In Section 2 we apply the integral equation technique given in Section
the
two-dimensional
electrostatic
Dirichlet
of
problem
value
boundary
to solve
two
equal
infinite co-axial perfectly conducting circular strips in a free space, when the total
Section 3 is devoted to the study
charge per unit height of the two strips is unity.
of the two-dimensional hydrodynamic Newmann boundary value problem of uniform flow of
an inviscid homogeneous liquid streaming past two equal infinite co-axial fixed rigid
circular
The
strips.
expression
for
the kinetic
energy
per unit
of
height
the
secondary fluid flow is derived.
2.
SOLUTIONS OF INTEGRAL EQUATIONS.
We present here a simple technique to derive solutions of equations (i.I) and
(1.2).
To solve equation (I.I) we first substitute in it [II]
INTEGRAL EQUATION (1.1):
fl () + f2 (0);
f(0)
g2(O),
(2.1)
and 2 represent the even degree and the odd degree parts of the
where the subscripts
These
respectively.
functions
corresponding
gl (0) +
g(O)
readily
substitutions
decouple
the
integral equation (I.I) into the following two integral equations
f g2(
for
the
loglma 2 (cosO
sin( 0
gl (){2q +
E
)log
determination
substitutions
cos
01
cos 0
( +
sl
-
=
of
gl(0)
{(cosE
{(nOsE
i)
and
cos0
l)l}d01
E
d01 f2 (0)
We
g2(0).
=f (O)
<
first
0
<
E
0
<
a
<
(2.2)
(2.3)
solve
equation
cos=)cosy + (cosE + cos=)},
(2.2).
The
(2.4)
cos=)cosx + (nose + cos=)},
(2.5)
reduce the equation (2.2) to a simple form
G
0
where
Gl(Y)
R(0)
l(y){
A + log
[(cosE
[cosE
21cosx- cosy I}
dy
cos)g I(8 l)sinh]/sin81
cos0)(cos0
cosa)]
1/2
F
l(x),
0
<
x
<
,
[gl(Ol)R(Ol)l/sin01’
(2.6)
(2.7)
(2.8)
C. SAMPATH AND D.L. JAIN
754
2q +
A
fl(0),
Fl(x
log[a2(cosB
cosa)/2],
(2.9)
and in (2.7) we have used the relations
sin
cos
cos0
(cosB
cosa
I/2
I)
cos
=
cos0
readily derivable from the substitution (2.4).
Lastly, the well known formula [7]
0
n
n;1
(2. lO)
cosa
cosny
cosnx
--2 r.
log 2 cosy- cos x
I/2
cosa
cosB
<
x,y
and the Fourier expansion of the known even degree function
=a
Fl(X)where
a
+ E
z o
F
n
l(cosnx
readily yield the solution
G l(y)
b
O
+ Z
dx, n
<
x
(2.11)
Fl(x)
(2.12)
7,
(2.13)
0
of the equation (2.6).
Gl(Y)
This is given by
(2.14)
b cosny,
n
n=l
bo ao/(2A), bn -(nan)/
where
<
0
a cosnx,
n
n
<
and the known coefficients a
n
n
substituting the above value of
I,
n
(2.15)
O, are defined by the relations (2.13). Finally,
Gl(Y) in the relation (2.7), we obtain the required
solution of the equation (2.2)
gl(01
[b
0
+ Z
n=l
b cosny] [sin0
n
I]/R(o I)
B
Now we take up the solution of the equation (2.3).
of equation (2.3) with respect to 0 and obtain
a
g2(0
<
O
<
a
(2.16)
We first differentiate both sides
)sln0
B cos0- -cos0
d01 f2 (0)’
B
<
0
<
(2.17)
a,
where the integral in the left hand member is to be inerpreted as a Cauchy principal
value.
When we make the substitutions (2.4) and (2.5) in the above equations, we get
C2(Y)
cosy
where
and
F2(x)
cosx
G2(Y)
g2(01)
F2(x)
f2’(0),
dy
slny
F2(x),
0
<
x
<
[2g2(Ol)R(01)]/[cosB
is a known even degree function of x.
(2.18)
cosa],
(2.19)
(2.20)
Next, when we substitute
INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS
x
in
(2.18),
cos
-I
(X), y
cos
(Y),
(2.21)
we obtain
G (cos
2
-I
Y) dY
F2(cos-Ix)
I/2(Y-X)
-I (I _y2)
The
-I
755
above
integral
be
can
equation
expansion of the known function F
<
-I
readily
X
<
I.
(2.22)
solved when we
substitute
it
in
the
in terms of the Chebychev polynomials U (X) of the
2
n
second kind
F2(cos IX)
Z
c U (X),
n n
n:l
<
-I
X
<
I,
(2.23)
and use the well known formula
T (Y)dY
(2.24)
y2)I/2(y
-I (I
n--O,
0
n
X)
n
n-I (X),
I,
U
where T (Y) is the nth degree Chebychev polynomial
n
solution of the equation (2.22) is given by
G2(Y
+
ATo(cOsy
I
Cn-ITn(csy)’
n=l
0
of
<
the
y
<
first
,
f
Cn
Lastly,
sinx
sln(n + l)x
0
the
(2.25)
where A is an arbitrary constant and the constant coefficients c
expansion (2.23) of the known function F
Thus
kind.
n
are defined by the
These values are
2.
F2(x)dx,
n
(2.26)
0.
(2.19) and (2.25) lead to the required solution of the equation
relations
(2.3)
(cos-
g2(01)
r.
cos)[A +
c
n
<
n-I cosny]/R(0
where relation (2.4) gives the value of cosy in terms of cos 0
0
<
(2 27)
The value of the
I.
constant A in (2.27) is readily obtained by using the edge condition satisfied by the
density function
g2(01)
g2(O1
The value of
G2
y
O([cosB -cosO 1]
g2(01)
O, as y
as
0
This edge condition is
+ 0
(2.28)
0+, and hence
l
n
g2(01)
1/2),
i=.
given by (2.27) satisfies this edge condition if in (2.25),
A----
Therefore,
at the inner edge
c
(2 29)
n-I
2- cos
cosa)
l
n=l
Cn_ I(I
cosny)]/R(0 I),
<
01 <
a.
(2.30)
756
C. SAMPATH AND D.L. JAIN
Since,
cs01’)
Yl (cosBCSB _-cosa
sin2
infinite
g2(01)
for
relations
(2.1),
where all B’s
various
(2.16)
and
the above equation,
of
right member
satisfies
the
edge
required
(2.30)
the
yield
condition
solution g(0
required
+
loglma
I)
above
Finally,
of
<
equation
[0[ <
B,
siq(0 01)[}d01 [Bl0 + BllCOS0] + [B200 + B21sin0],(2.31)
Its solution is also very useful
are known constants.
problems
boundary value
the
therefore,
(2.28).
We illustrate the above results by solving the integral equatlon,
g(01){ q
+
the
in
series
expression
(I.I).
is a common factor of all the terms occuring in the
or
presented in our subsequent analysis.
in
solving
Comparing
equation (2.31) with (I.I), we have, in this case,
fl(0)
Blo + BllCOS0, f2 (0)
Fi(x)
[Bio + Bil(COS8
G1(Y
I_.{
B200 + B21sin0,
+ cosa)] +
(cosB + cosa)]
(sln01)[
gl(l
BI0 + "(
E
+
(cosB
B
cosy) +
=-I
cos
cos0 I)
I/2
+
relation g(0
I)
gl(01)
+
01 <
(2.34)
cosa)](1
cos2y)}(2.35)
gl,g 2
(2.36)
a,
cosa)]
cosa)I/2
B21cos01}/(cosO
8
< e <
(2.37)
a
from equations (2.36) and (2.37) in the
we get the required solution of equation (2.31).
g2(02),
the above illustration, when
cosa) cosy},
[B21 (cos
<
B
{[B20 + B21 (cos
and putting the values of the functions
(2.33)
1,2,
i
+ cosa)}
l)Bll (cos
-BIICOS01}/R(01)
g2(01
cosa)cosx,
(cos8
[B20 + 1/2B21 (cosS + cosa)](1
Cz(y
(2.32)
In
0, we readily obtain the solution of the equation,
-<0<a,
g(81)( q
+
loglZa
siq(8 81)
de
(Blo
+
BllCOSO)
+
(B20 8
co
s01]
+
B21 sinS),
(2.38)
81 <
(2.39)
in the form
,-5.
g(Ol)
"
sin
,’2tcos
"’Bl{
(I___
’1 B I0+ B 11 AI +
l)cos
2
l{B20 + B21 [sin2 b + csOl ]}II(csol csa)ll2’
-a
<
a’
where
A
2[q + log(a
sin
a)],
(2.40)
757
INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS
which agrees with results derived by Shail [9] by applying tedious inversion formulas.
INTEGRAL EQUATION (1.2).
equivalent form (1.4).
ddo
f
+
We present here the method of solving equation (1.2) or its
Equation (1.4) can be rewritten in the form
f g(O)loglXn (0
O1 )idOl
Iol
<
f(O), l
<
,
(2.41)
which on integration of the both raembers of this equat[on yields
-B
a
(2.42)
where p’(0)
f(0) and C in an unknown constant.
also of the form (I.I), with f(0)= C-
I)
solution g(0
1/4P(0),
Since the above inegral equation is
q =-log2, and a
can be derived as explained in Section 2.1.
unknown constant C occuring in the even degree part
g(0 I)
-
gl(01)+ g2(01),
relation
S
can be
obtained
therefore, its
Lastly, the value of the
of this solution
gl(01)
by putting this value of
-B
a
f-a + f g(O1)dO
gl (O1)dO
gl(01)
i,
in
the
a
f
+
-a
S I’(O1)dO
B
(2.43)
O,
where we have used relation (1.5) and the edge condition (1.3)).
3.
ELECTROSTATIC POTENTIAL PROBLEM.
We consider the electrostatic problem of two equal infinite co-axial perfectly
conducting strips charged in a free space so that the total charge per unit height on
the two strips is unity.
In cylindrical polar co-ordinates (r,O,z), the two strips
are defined by r
-, <
a,
<
0
-B, B
<
0
<
a, B
O,
a
<
7,
z
<
.
The electrostatic potential (r,O) of this boundary value problem is given by
6(r,
where g(0
I)
O)
-a
the
-
a
+
f
g(O1)log[r2 + a 2
2arcos(O
01)}1/2 dO1,
(3.1)
is the unknown surface charge density per unit are defined by
g(01)
Since
f
of
value
the
r=a+
(3.2)
r=a-
assumes
potential
strips, using this boundary condition in
-B
S
a
+
S g(l)lOgl2a sin
(0
(3.1),
o )lao
a
value,
constant
say
o’
on
the
two
we obtain the integral equation
o
<
Iol
<
,
(3.3)
C. SAMPATH AND D.L. JAIN
758
The solution of this equation is readily obtained from that of equation
BI0 =-o/a, BII
setting q--O,
0, in (2.36) and (2.37).
020 B21
(2.31), by
Therefore, the
solution of equation (3.1) is given by
g,(o,)=- ;[,o/(=
g(o,)
A
where
log[a2(cos13
2
,2)]l,nO, l/R(0,),
Io,
<
<
,,
(3.4)
(3.5)
cosa)/2].
in the above expression for the charge
o
of the strips, it is given that the total charge per unit height on each
Finally, to evaluate the unknown constant
density g(8
I)
of the strip is unity and therefore g(8
f
a
+
g(0 I)
d01
I)
must satisfy the condition
(3.6)
2
We substitute the value of g(B I) from (3.4) in the above equation, and get
A
o
(3.7)
2’
and, therefore,
When 13
0, we
<
g(01)
4.
the
obtain
a, -a
strip r
0
<
,/2 cos
1
result
[9]
for
the
circular
<
z
,_
cosa)l/2
a(cosO
limiting
corresponding
a,
<
-a
<
01 <
(3 9)
(x.
HYDRODYNAMICAL PROBLEM.
We consider the problem of uniform flow of an inviscid homogeneous liquid in the
-
icos + jsinT of velocity U, streaming past two fixed rigid strips r
direction n
a,
<
0
-,
4 (r,0) of
,s(r,
O)
where
B
I)
0
<
a, 0
13
<
a
The secondary velocity potential function
7.
this two-dimensional problem is given by
a
I(01
and I(0
<
-B
f
a
+
f
Cs(a
3
{-rllog[r
I(0
+
2
+ r
01 -Cs(a-, 01)
2
[9]
2rrlcos(O
13
<
1011 < ,
01)]
1/2 }dO
r =a (4.1)
(4.2)
satisfies the edge condition
I(l)
I(e=)
0.
(4.3)
759
INTEGRAL EQUATIONS FOR THE POTENTIAL PROBLEMS
Using the boundary condition
r =a
where
91(r,0)
r =a
-Ur cos(0
T) is
-B
the
2
llcosec g(0
0
To solve this integral equation,
we
f
f
+
I(0
velocity
potential
of
function
the
incident
(4.1), the integral equation [9]
uniform flow Un, we obtain from
lld01
8,aUcos
y), B
(0
I01
<
<
(4.5)
a,
integrate the left member by
first
parts and
obtain by using the edge conditions (4.3)
f
where g(0)
can be
+
f
g(8 l)cot
I’(0).
(8
8
-4,aUcos (8
I)d81
V), B
<
181
<
,
(4.6)
The above integral equation is of the type (1.4), and therefore
solved by the method given in the Section 2.2.
We first rewrite integral
equation (4.6) in the form (2.41) which on integration yields
f
+
f
g( 81)l og s in
(8
where C is an unknown constant.
type
(2.31),
log2, a
with q
I,
8
l)ldS1
C
), B
2.aUsin (8
<
181
<
a, (4.7)
Fortunately, the aboe integral equation is of the
BIO
C,
BII
2aUsinY,
B20
0,
B21
-2aUcosY.
Therefore, substituting these values of the constants in equations (2.36) and (2.37),
we obtain the required solution of the above integral equation in the form
g(01)
g,(01)
+
g2(01)
(4.8)
cosa)18}.
(4.10)
where
and
6
log{(cosI3
The value of the unknown constant C occuring in the value of
(4.9) is obtained by putting this value of
gl(81)
gl(81)
given by equation
in the relation (2.43).
This yields
the required value of the unknown constant C is given by
C
-aU sin
"(
(cosS + cosa).
(4.11)
We substitute this value of C in (4.9) to obtain the value of the function
g1(81)
aU
sinvls+/-n 811[(cos
+ cosa)-2 cos
81]IR(01),
<181<
gl(81)
as
(4.12)
C. SAMPATH AND D.L. JAIN
760
Using formulas (2.19) and
A
=
A
expression for
g2(01)
is given by
[C- 2aU cos Y(B cos y + A cos 2y)], 8
g2(01)= R(I
where
(2.25), the
(cos8
(cos8 + cos), cos
cos=), B
81
<I01
<
(4.13)
A cos y + B,
(4.14)
and the unknown constant C is evaluated by using the condition (1.3) which yields
A
+ J2
2aU cos Y [B J
C
/Jo’
(4.15)
(cos ny) de
where
0, I, 2,
n
R(0
(4.16)
We substitute this value of C in (4.13) to obtain the following value of
J1
2aUR(ol)COSY
g2(01
2aUcos Y
Sgn
(0)
Sgn 8 I[B(]--- cos y) + A
o
[_2
01
A
+ -j--(BJ
+A J2
o
J2
(--o
cos2y)]
+ B cosO
cos
2
01]
8
<
[01[
The results (4.8), (4.12) and (4.17) yield the required solution g(0
equation (4.6).
o
where
for
we
gl(01
Slnce g(0)
f t(o
have
I’(0), 8
o +f
used
the
[011
<
relation
(
Iol
(4.8).
f8 gl(01
I)
<
(4.17)
a,
of the integral
a, therefore
g(o) o I’
satisfies the relation
g2(Ol).
d01
(
Since
( e)
the
(4 12)
expression
O thereforef g2(01) d01
the expression (4.18)
for I(0) satisfies all the required edge conditions (.3) if
0.
We may remark here that there is no need of obtaining the solution I(0) of the
(4.5) for finding the expressions for the physical quantities of interest.
equation
These expressions can be readily derived from the value of the function g(0
given by
relations (4.8) to (4.12). For instance, the kinetic energy per unit height (K.E.) of
the secondary fluid flow is given by
--
K.E. =_1 aP
Ua_P2
Ua__2_p
2
+
I)
-8
a
+
8
I(01)[
( )cos( o
(rl, 01
8r
dO
rlfa
y)d o
-8
./"
g(O )sin(O
y)dO
Uap {d cosy
d2sinY}
where we have used the boundary condition (4.4), the relation g(0)
(4.19)
l’(O), the edge
INTEGRAL EUATIONS FOR THE POTENTIAL PROBLEHS
(4.3), and
conditions
and d
constants d
2
p is the density of the ho,aogeneous liquid.
d2
=
+ J2]/J o’
2auUA cosT[BJ
(4.20ab)
gl (O1)esOldO
gl(01)
We substitute the values of the functions
(4.17) in (4.20) to obtain
d
The values of the
are given by the relations
g2 (O1)sinO IdOl,
dl
761
d
cosa)2sinT.
-aU(cosg
2
from equations (4.12) and
g2(01)
and
(4 21ab)
Finally, relations (4.19) and (4.21) give rise to the required expression for Kinetic
energy and this is given by
pa2U 2 A
K.E.
+ A
{[
o
where the definite integrals J
A +
+ ]
J2
o
(BJI +A J2
A
-]
cos2},
(4.22)
n=O,l,2 are defined by relation (4.16) and A and B
n
are defined in the relation (4.14).
We derive now some interesting lmltng results from the formula (4.2) for K.E.
g
When
=,
O, J
n
0,I,2 and
Jt J2
j
j
o
-
and therefore the formula (4.22)
o
yields the followlng corresponding limiting expression for the K.E.,o
infinite circular rigid strip r
ra2U2.sin2
o
a, -a
[1 + cos
This seems to be a new result.
<
0
<
a
a2(a 2 al)
I and a8
<
02 in case of
< y < al,
<
0, a 2
pU
0
(4.23)
0,
O,
a
=,
such that
the two equal parallel co-planar infinite rigid
z
<
and it is given by
2 2
2
2
<
in case of the
=:
we obtain the corresponding limiting expression for
the Kinetic Equation
strips x
<
z
cos2Y],
Similarly, when in formula (4.22) we let e
aa
a,
al [(l+c 2 )-2E’/F’]cos 2 "f,
(4.24)
where
c
and
F(,
a2/a <
c) and
E(,
I, F’
F(,
(I _c 2
)I/2 ),
g’
E(, (I-c2) I/2 ),
(4.25)
c) are elliptic integrals of the first and the second kind [14].
We have also solved the two-dlmenslonal problems of scattering of a low-frequency
incident plane acoustic wave by the integral equation techniques presented here.
This
C. SAMPATH AND D.L. JAIN
762
work will appear separately.
ACKNOWLEDGMENT.
This
research work was
supported by a grant from the University
Grants Commission, New Delhi (India).
REFERENCES
A note on electrostatic problem involving two
I.
GOEL, G.C. and JAIN, D.L.
2.
GOEL,
3.
strips, Indian J. Pure Appl. Math. 7(1976), 809-816.
JAIN, D.L. and KANWAL, R.P. Acoustic diffraction of a plane wave by two co-
strips, J. Pure Appl. Math.
G.C.
and
JAIN,
D.L.
7(1976), 751-756.
Electrostatic
problems
of
two
co-planar parallel
planar parallel perfectly soft or rigid strlps,Can. J. Phys. 50(1972), 929-
939.
Diffraction of elastic waves by two co-planar and
4.
JAIN, D.L. and KANWAL, R.P.
5.
parallel rigid strips, Int. J. Engg. Sci. 10(1972), 925-937.
Approximate solutions of certain integral
KANWAL R.P. and SACHDEVA, B.K.
equations for the diffraction of two strips, ZAMP______, 24___(1973), 111-119.
6.
GAUTESEN, A.K. and OLMSTEAD, W.E.
On the solution of the integral equation for
the potential of two strips, SlAM J.
Math. _Anal_., 2(1971),
293-306.
MILLAR, R.F. A note on diffraction by an infinite sllt, Can. J. Phys.,
38(1960) ,38-47.
Finite Hilbert transform techniue for triple
8. SRIVASTAVA, K.N. and LOWENGRUB, M.
integral equations with trignometric kernels. Proc. R. Soc. Ednib. 39(1970),
7.
309-321.
9.
SHALL, R.
A class of singular integral equation with some applications, Int. J.
Math. Educ. Technol.,
10. CAP.LEMAN, T.
15___(1984),
Integrationsgrezen, Math.
11. COOKE, J.C.
359-374.
Uber die Abelsche Integralgleichung mit konstanten
Z., 15___(191 1),
11-120.
The solution of some integral equations and their connection with
Glasgow Math. J., 11__.(1970) 9-20.
12. WILLIAMS, W.E. A note on integral equations, Glasgow Math. J., 13(1972), 119-121.
Uniform streaming past a fixed semi-circular
13. LAL, BANSI and JAIN, D.L.
infinite strip, J. Math. Phy. Sci. 17(1978), 365-376.
Elements of the Theory of Elliptic and Associated
14. DUTTA, M., and DEBNATtl, L.
Functions with Applications, World Press, Calcutta (1965).
dual integral equations and series,