Proc. Indian Acad. So. (Math. SO.),Vol. 106,No. 3, August 1996,pp. 261-270.
9 Printed in India
Solution of singular integral equations with logarithmic and
Cauchy kernels
A C H A K R A B A R T I and T S A H O O
Department of Mathematics, Indian Institute of Science,Bangalore560012, India
MS received23 January 1996;revised 20 April 1996
Abstract. A direct method of solution is presentedfor singularintegralequations oftbe first
kind, involvingthe combinationof a logarithmicand a Cauchy type singularity. Two typical
cases are considered,in one of which the range of integration is a singlefinite intervaland, in
the other, the range of integration is a union of disjoint finite intervals. More such general
equations associated with a finite number (greater than two) of finite, disjoint, intervals can
also be handled by the techniqueemployedhere,
Keywotdg Waterwaves;scattering; Cauchy kernel; logarithmickernel.
1. Introduction
The singular integral equation
f~f(t)IKln Ix~-~t t + 1 + 1 ] dt = o(x), x~ L,
xwt x-t
(K > 0, a known constant),
(1.1)
with L representing either (i) a ~ingle finite interval (a, b), or (ii) a union of two disjoint
finite intervals of'the type (a, b) u(c, d), where 0 < a < b < c < d, has been considered for
solution by employing a direct method which reduces such integral equations to an
equation having only a Cauchy tylde singularity.
Singular integral equations of the form (1.1), where the unknown function f(x)
possesses square-root singularities at the end points of L (see Mandal [3]) occur in the
study of scattering and radiation of surface water waves, in the linearized theory, by
vertical barriers possessing either (i) a single gap, in the case when L = (a, b) or (ii) two
gaps, in the case when L = (a, b)u(c,d). Recently, Banerjea and Mandal [6] have
presented a method of solving eq. (1.1) when L = (a,b), by reducing it to a singular
integral equation over two disjoint intervals, with a Cauchy type singularity, and
employing a function theoretic approach t o t h e resulting equation. Banerjea and
Mandal [6] have also mentioned that eq. (1.1) cannot be reduced to an equation with
a Cauchy-type singularity, of a type which has been possible to equations of this form,
(see Ursell [2]) where L represents a single interval of the form (0, a) or (b, oo), in which
case, the end behaviours of the unknown function f(x) have a different structure as
compared to the ones required for (1.1).
In the present paper, we have shown that a method, similar to Ursell's [2] is
applicable to the integral equation (1.1), which reduces the equation to the one which
involves only a Cauchy type singularity and which can be solved by using well-known
results available in the books of Muskhelishvili [1] and Gakhov [41 as well as in
a recently published paper of Chakrabarti and George [5].
261
262
A Chakrabarti and T Sahoo
The all-important identity that has helped in arriving at the closed form solution of
the integral equation (1.1) by our approach is the one as given by the relation
In x - e
=2x
t 2 ~ x 2,
0<x<e.
(1.2)
In the next section, we take up the solution of (1.1) in the two cases (i) and (ii)
described above, and present the solutions in closed forms.
2. Methods of solutions
Case (i). L =
(a, b)
In this case, we look for the solution of (1.1) along with the end point requirements
f(x)~.
~O(]x-a[ -1/2)
[O([x
x~a,
x--*b.
as
as
b] -1/2)
(2.1)
We set, as in the work of Ursell [2],
F(x) = fbf(t)dt
for
a< x< b
(2.2)
so that
F(b)=O
F'(x)=-f(x)
and
for
a<x<b.
(2.3)
Substituting f o r f ( x ) from (2.3) in (1.1), we obtain the singular integral equation
f l t~_
;t(t)dt
x2 = n,( x),
a < x < b,
for
(2.4)
where
2(x) =
F'(x) + KF(x)
d[ f.
1{
f ( x ) = ~ xx e -xx
for
;t(u)er'du
X
and
a < x < b, so that
1
a<x<b
for
h(x) =
~x 0 ( x ) - KF(a)ln x - a
~O(Ix-al -I/2)
as
x~a,
;r
[O(lx-bl-1/2)
as
x~b.
for
(2.5)
a<x< b
(2.6)
with
(2.7)
The solution of the integral equation (2.4) is well-known and is given by (see Gakhov [4])
4xfbo
2Cx
;,(x) = R(x---) n2
tR(t)h(t)dt
R~Z-~-i)
for
a < x < b,
(2.8)
where
R(t) =
x/ii "~- a2)(b 2 - t 2)
for
a < t < b,
(2.9)
and C is an arbitrary constant.
We next mention the following results to be obtained by standard methods (see
Gakhov [4-] problem 18, pp. 81)
fba CR(t)g(t)dt
2 R - ~ ' ~ Z2)- =x R(x)
1 t2-~
v A ( x ) /dfb
o g(t)dtA(t)
for
a < x < b,
(RI)
Singular integral equations
263
where
~-b g(t) .
~
C==Jo-~)ot,
u------t2 _ 7 = -~
-
tR(t)dt
t 2 _U
- a~
k(x)=~/b2_x2
u2
for
a<x<b,
(a < u <
b),
(R3)
u2)(b2--u 2) ua +
-~/(a 2
2 ~ g
for
--
__
for
_
(0<u<a),
_
(R4)
R(t)ln
t+a t 2 - u 2 = ~ a + n
t2
u2
for
a<u<b.
(R5)
Using the above mentioned results (R1) to (R5), from the relation (2.8) we obtain
A(x)
~t)~---~
for
a<x<b,
(2.10)
where
Q=2
C-
+
-
and
-x 2
, a<x<b.
(2.11)
From the relations (2.2), (2.5) and the expression (2.10), we obtain the relation
connecting the constants Q and F(a) as given by
"
R(u)
a(u) j ~ a(t) t = - u = J du.
(2.12)
Also, from relations (2.5) and (2.10) we finally obtain the complete solution of(1.1) in the
case when L = (a, b), as given by
d{
fx
f(x) = ~xx e -Kx /,
+
tte/~u
([Q-(2K/Tr)F(a)G(a,b,u)]
\
R(u)
I a O(t)dt'~
~(u) f : a-,-,-Z
/ ~ - >- -),o u ~,) for a < x < b ,
{2.13)
where
~__- -Q.
The solution (2.13) agrees, in principle, with the one obtained by Banerjea and Mandal
[-6] recently.
A Chakrabartiand T Sahoo
264
The solution of (1.1) can also be obtained, in this case, by setting
f(t)dt
if(x)=
a<x<b.
for
(2.14)
a
Proceeding in a similar manner as discussed above, we find the following alternative
form of the solutionf(x) as given by
d{
fl
([O.-(2K/Tz)ff(b)G(a,b,u)]
f(x)=dx e-K~ uer"
R(u)
fl 2fft)~u2]au;
1 g(t)dt ]
for
a<x<b,
(2.15)
where the constants (~ and/~(b) are related as
~ I 9(t)dt'~.
eK~i(b)=;iuer'([Q-(2K/~)i(b)G(a'b'U)]R(u)
+ ~A(u)2 J b
A]tjt~-_---~'2) ou.
(2.16)
Next, we will discuss the method of solution of (1.1) when L is the union of two
disjoint finite intervals.
Case (ii).
L = (a, b) ~ (c. d)
In this case, we desire to solve (1.1) where
O(Ix-a1-1/2)
O(Ix-bl -u2)
f(x)~ o(Ix_cl-1/2 )
O(Ix-dl -'z)
We
as
x --*a,
as
x --~ b,
as
x~c,
as
x ~d.
(2.17)
set
f(x)=
~fl(x)
~f2(x)
for
for
x~(a,b),
xe(c,d)
(2.t8)
~Ol(x)
for
for
x~(a,b),
xe(c,d).
(2.19)
and
g(x)= [g2(x)
Then, defining
Fl(x)=fbxfl(t)dt
for
a<x<b,
(2.20)
so that
F'1(x) = - f l (x)
for
a < x < b,
F~ (b) = 0
(2.21}
and
F2(x)=flf2(t)dt
for
c<x<d,
(2.22)
so that
F'2(x)=-fz(x)
F 2 (at) = O.
for
c<x<d
and
(2.23)
Singular integral equations
265
We reduce (1.1) into the following new equation
fZ).,(t)dt_~_~+ f ; 22(t)dtt
2 --x 2 = .q(X)2x for xe(a,b)u(c,d),
(2.24)
2~(x)=F'l(x)+KFl(x ) for a < x < b and
fi2(x) = F'z(X) + KF2(x) for c < x < d,
(2.25)
where
and
xe(a, b) • (c, d),
(2.26)
along with the end conditions that
~O(Ix-al -v2) as x-~a,
)'t(x)~ [O(lx-b}-l/2) as x-~b
(2.27)
~O(Ix-c1-1/2) as x-,c,
[O(]x-- d]- 1:2) as x--,d.
(2.28)
and
22(x) ~
The solution of the integral equation (2.24) along with the end conditions (2.27) and (2.28)
can be derived by using an analysis similar to the one available in the paper of Chakrabarti
and George [5], with appropriate modifications (see Appendix) and we find that
21(x)
- 2(Al X2 + B1)x
R, (x)/~2(x)
2x F ~ b ~l(t)Rl(t)"2(t)dt
n2 LJo R ~ )
- f l R'(t)R2(t)O2(t)dt 1
R 1(x)/~z (x)( t2 -- x2) J for a < x < b
(2.29)
and
)-2(X)
2(A,xZ + B1)x 2x [ ~ b ~,(t)Rl(t)R2(t)dt
RI(x)R2(x) + ~
f l R:(t)R2(t)O2(~dt l
Kl(X)R2(x)(t2 x2)_] for c<x<d,
LL
-
(2.30)
where A 1, B 1 are arbitrary constants,
xe(a,b) and
~2(x)=~(x) for xe(c,d),
R l ( x ) = x/(x2-a2)(b2--x 2)
for a<x <b,
(2.32)
[ ~ / ( x z - a 2 ) ( x 2 - b 2) for
Rl(x)=(x/(a2-x2)(b2 x 2) for
(2.33)
01(x)=~(x)
for
x>b,
0<x<a,
R2(x)= x/(x2-c2)(d2-x z)
for c <x <d,
~.,~(d2 - x2)(c2 - x 2) for x < c
/~2(X)= (x/(x2--c2)(x2-d 2) for x>d.
(2.31)
(2.34)
(2.35)
A Chakrabarti and T Sahoo
266
We then use the following results to simplify the expressions for ;t 1(x) and
fl Gl(X2,r162
-x2
fbR,(t)~2(t)ln t-- p~ t~--x
,~2(X):
(R6)
,
where p = a or c, and
Gt(x,{'=2If'tRl(t'-R2(t){t2-@x2+ ~2-~t2}dt 1,
xe(a,b)
for
R,(t)~2(t)gl(t)dt
2- x
f boRl(x)R2(x)(t
2)
or
xe(c,d)
Ca
Rl(X)RE(X) +/~2(x)
J . Al(t)(t 2 - x 2)
for a < x < b,
(R7)
where
C3 = fb
J~ ~ gitl J( t ) d t
and
a~(x) =
x~ - a2
-~
for
a<x<b,
A2(x) Ia/~l(t) O2(t)dt"
fl Rl(t)R2(t)g2(t)dt _
C~
a<x<b,
R, (x)R2(x)(t2 - x 2) R,(x)R=(x) Rl(x) J , A2(t)(t 2 - x2) '
(R8)
where
C4= Ia ~ g 2 ( t ) d t
,Jc
2[ t)
and
X] d 2 -- X 2 '
fac R2(t)Rl(t)ln ~t - q d t~
a<x<b,
= flG2(x,~)d~
~2_x 2 ,
(R9)
where q = a or q = c and
for
xe(a, b) or
for
xe(c,d).
fbo Rl(t)R2(t)g,(t)dt
R ~
Ca
---Z~) - R~(~)g~I~)
A3(x) I b/~2(t) gl(t)dt
+ R2(x ) J ~ A - ~ ( ~ - - - x -5)
for
where C a is the same as defined in (R7) and
c <x <d,
(R10)
Singular integral equations
a3(x) =
267
/~-Z-~2
~iR,(t)R2(t)g,(t)dt
for
C,
c < x < d,
A4(x ) fd ff~t(t)
R 2 (x)/~ 1(x)(t 2 - x*) --- R2(x)R 1 (X) -
2c A-
g2(t)dt
(t
for c < x < d,
(Rll)
a<x<b
(2.36)
where C 4 is the same as defined in (R8) and
A,(x)=
~L
~/d ~ _ x 2
for
c<x<d.
Using the results (R6) to (R11), we obtain
-2[A1 x2 +Ql + Hl(x)]x
R,(x)~,(x)
,~l(X) =
2Xp
~-5 l(x)
for
and
22(x) =
2[A1 x2 + Q1 + Hl(x)]x
+~P2(x)
~l(x)R2(x)
c<x<d,
for
(2.37)
where
Qa = Bx +
C 3 -
C 4
n----T--,
Hl(x)=-~{Ft(a)fi [G2(x'r162
+ F2(c)fl [G2(x'~)-Gl(x'~)]d~
(2.38)
Ax(x)fb R2(t)g,(t)dt A2(x)fd~,(t)g2(t)dt for a<x<b
P*(x)=~2(x) J , ~ ) + R l ( x
) J~ ~
)
(2.39)
and
fb
R2(x)Jo A,(t)(t
2-x 2) Rt(x)~ ~ Z - ~ )
for
c <x <d.
(2.40)
Now using the relations (2.25), we find that
eX"Ft(a)=fler~2,(u)du
and
(2.41)
and
f~(x)=
-K~
, for
f2(x)=d[e-X"fler';~2(u)du
]
for
a<x<b
and
c<x<d,
where 2t (x) and 22(x) are given by the expressions (2.36) and (2.37) respectively.
(2.42)
268
A Chakrabarti and T Sahoo
The relations (2.42) completely solve (1.1), in this case, if the two relations in (2.41) are
utilized to determine the constants F~(a) and F2(c) in terms of A~ and Q~, which remain
arbitrary. Ahemative forms of the functions fx and f2 can also be derived, as has been done in
the ease (i), where the solution has been expressed through the relations (2.15) and (2.16).
3. Cenclusion
A unified approach has been developed to solve the singular integraJ equations of water wave
problems, which involve a union of disjoint finite intervals, at the end points of which the
unknown function is required to satisfy square-root type integrable singularities. The cases of
single as well as double intervals have been analyzed here in detail. For more number of
intervals (greater than two), the method is similar, even though a little more involved.
A~mxUx
Solution of (2.24)
Equation (2.24) can be cast into the form
T12"(~/)+ 7~2*(r/)=g~(r/) for rle(a~,b~)
(A1)
for )le(cl,dl)
(A2)
and
7"12"(r/)+ T22*(r/)=~(r/)
after using the transformations
x2=,.
21(N/-0, ~.,(0
2
- /'2('~-0, a2=al,
e--c
0,(x/~), rle(al, b,) '
a2=d,.
b2=b~ . c2=q
g~2(tl) g2(~r~), ne(Cl,dl )
=
.
(A3)
2 V/~
and employing the operators T 1, T 2, T1 and T2 as defined by the relations
T t2~'tr/)=
T2)'*0/)=
T~2~'(~/)=
*
f
f
b'2~(0d~
., e--r/
for
~le(a,,bl),
a, ).2*(~)dr
. ~-r/
for
~le(c,,d~),
;t~'(~)dr
., e - r /
for
r/~(al,b~),
-_[a'2*(0d~
for
t/~(Cl,dl).
We then define the inverse operators T [ ~ and T~ ~ as given by
Dl
T[ ' 2"()/)= a3(r/----)
T~(A3(~/)2*(~/)) for
rr2 A30/)
02
T2(A4(~/)'R'*(~/))
T 2 ' 2~(t/) = A,(~/----)
n2A,t(r/) for
~le(al,b~) and
~le(c,,dl),
(A4)
Singular integral equations
269
where D 1 and D 2 are two arbitrary constants and
Aa(q)=~/(q-a~)(b~-r/)
for ~le(a~,b~) and
A4(n)=x/(q-q)(dt-q)
for qe(c~,dt).
(A5)
We easily derive the following results (R1-R8), by using standard methods.
al+b. ]
Tl(A3(q))=~t
7"1(AB(t/) ~'2 2~(t/)) = n
-r/+~J
f
dt
for r/6(al,bt),
(ifl)
~
2*(t/)d~/- nT2(A 3 (r/)2~(t/)) for at < r / < b ~,
el
(if2)
where A3(r/) = ~/(r/- a, )(r/- bl) for q(~(aa,b,),
( 1
)
~Z T2(,~3(r/),~,,(q))
for qe(ct,dl),
(F,3)
( ~ 1 Tx(At(r/)g*(r/)) ) = - A3(~)
rr ~rt(Aa(rl)g,(rl)) for qe(ci,d,), (R4)
F'
lil
for ~s(q,dJ,
(ifS)
where A4(.) = ~(ct - ~)(dl -- ~),
?'2
1 ~,t(A3(r/)s
=~
Tt(As(~/) 7~,,(rt)91(rl))
- nTx(a3(rl)g, (q)),
7"2 A - ~ T2(A3(")A'(r/)g*(q))
=~
?'2(A*(r/) Xa(q)g*(r/))'
7"2 (A--~(~)) = n (1 + A~(~)) for qE(a,,bt).
(R6)
(if7)
(if8)
Applying the operator T~- t to eq. (A1) and using the result (R2), we obtain
~.*(q)= At
A3(r/)
r~?'2(A3(r/))+ Tt(A3(r/)O*(r/)),
n2A3(r/)
(A6)
where A a = D 1 + (1/n)I,,a, ~.~'(s)ds is an arbitrary constant.
Using the expression (A6) along with the results (if3) and (if4), we can rewrite (A2) in the
form
T2(,~3(q)2~(q) ) = At x _ 1/r Tt (Aa(r/)g|*(r/)) + A3(t/)g~(t/) for t/a(ct, d t).
(AT)
270
A Chakrabarti and T Sahoo
Applying the inverse operator T~- 1 to eq. (A7), we obtain, after using the results (R1), (R5),
that
;~*(,1) =
AIq+ B1
T2Ca,(,7)A3(,1)a~(n))- ~',(a3(,7)A,Cn)~(,7))
a,in) 3,~(,7)
n ~A~in) a,(,7)
for tle(ct,dx),
(A8)
where B, (= D2 -(al/2)(g +dl) + ~-2S~:A3(q)g*(q)dr/)is an arbitrary constant.
The relations (A6) and (A8), after using the results (R5) to (R8) ultimately give
- A t ~/-- e I
XI'(~) = a3(n)h,(rt)
Tt(A3(q)h,(~/)g*(r/))- 7"2(A,(rl)h3(r/)g~(t/))
~2A,(~)A3(~)
for a t < r / < b t .
(Ag)
Going back to the original variables x, t, ~-t and 22, we get the results (2.29) and (2.30) used
in the paper.
Acknowledgements
The authors are thankful to the referee for his comments and suggestions to improve the
presentation of the paper. TS acknowledges the University Grants Commission, New
Delhi, for the financial support as a research student of Indian Institute of Science. TS also
acknowledges the partial support of CSIR, New Delhi during the period when the paper
was revised.
References
I-11 Muskhelishvili N I, Sinoular inteoral equat/on (Noordhoof, Holland) (1963)
[-2] Ursr F, The effect of a fixed vertical barrier on surface waves in ~
water, Proc. Camb. Phil. Soc. 43 (1947)
374-382
[3] Mandal B N, A note on the diffraction of water waves by a vertical wall with a gap, Arch. Meclt 39 (1987)
269-273
I-4] Gakhov F D, Boundary va/ue prob/ems (Dover Publications: New York) (1990)
[51 Chakrabarti A and George A J, Solution of a singular integral equation involving two intervals arising in the
theory of water w a v ~ Appl. Math. Left. 7(5) (1994) 43-47
[6] Banerjca S and Mandal B N, On a singular integral equation with logarithmic and Cauchy kernel, Int'l J.
Math. Educ. Sci. Technol. 26(2)(1995) 267-313