Proc. Indian Acad. Sci.(Math. Sci.),Vol. 106,No. 2, May 1996,pp. 177-182.
9 Printed in India
A note on a class of singular integro-differential equations
A CHAKRABARTI and T SAHOO
Department of Mathematics,Indian Institute of Science,Bangalore560012, India
MS received21 August 1995
Abstract. A simplifiedanalysisis employedto handlea class of singular integro-differential
equations for their solutions.
Keywords. Integro-differentialequation.
1. Introduction
The problem of solving the singular integro-differential equation, as given by
u(x). .l . v(s)ds
. .
+ 9(x),
7Z
S-- X
(0 < x < ~),
(1.1)
where u(x) and v(x) represent two linear differential expressions of the forms
u(x) = ~ akfk(x)
(1.2)
k=0
and
v(x)= ~ bkfk(x),
(1.3)
k=O
in which ak'S and bk'S are complex constants,fk(x) represents the kth derivative of the
unknown function f(x) with prescribed initial valuesfk(0), along with the conditions
thatfk(oo) = 0, for k = 0, 1, 2 ..... n, and 9(x) represents a known differentiable function
arises in a natural way while solving a class of mixed boundary value problems of
mathematical physics (see [-3, 5, 6]).
Varley and Walker [7] have discussed a general method of solving (i.1) by
converting it to a singular integral equation of second kind with a Cauchy Kernel, and
have discussed a method of solution of the resulting singular integral equation which
avoids the complication (see [4]) of calculating various singular integrals appearing in
the final form of the solution.
In the present investigation, we have employed a straightforward analysis which
simplifies the work of Varley and Walker I-7] slightly and demonstrates clearly the
underlying difficulties.
A detailed procedure is explained to examine (1.1) for its solution under the
assumptions that
v(x).~O(x~'), as x ~ O
(1.4)
v(x),~O(x~),
(1.5)
and
as x ~ oo
with - 1 < real(fl) < 0 and real(~) > - 1.
177
178
A Chakrabarti and T Sahoo
2. The detailed method of solution
With the aid of the Laplace transform V(p) of the function v(x) as defined by
V(p)= f ~ v ( x ) e x p ( - px)dx,
(2.1)
(p > O)
equation (1.1) can be shown to be equivalent to the following singular integral equation
(as in the work of Varley and Walker (1989))
1 f ~ V(q)dq
U(p) + n o ( q ~
- G(p),
(2.2)
where, V(p), U(p) and G(p) are the Laplace transforms of the functions v(x), u(x) and
g(x) respectively.
The expressions for V(p) and U(p) are given by
U(p) = A(p)F(p) - Al(p),
V(p) = B(p)F(P) - Bl(p),
(2.3)
with
A(p) = a.p" + a,_lp n-1 + ... + a 1 + a o
A l ( p ) = ( a . p "-1 + a , _ l p "-a + ... + a l ) f o
+ (a.p"- z + a._ lP"- 3 + ... + a2)f 1 + . . . + a . f , _ x,
B(p)=b.p" + b . _ l p "-1 + . . . + b 1 +bo
and
Bt(p) = (b,p'- 1 + b,_ lp "-2 + .-- + b~)fo
-k-(b,p n- 2 q-bn_lp n-3 + "" + bz)fl + . . . b . f . _ ~ ,
where f , = fk(0) for k = 0, 1, 2 ..... n.
Using the relation (2.3), equation (2.2) can be cast into the form
A(p) v(p)+ "(p)
~t
V(q)dq
( q _ p)
C(p),
p> 0
(2.4)
where
C(p) = a 1(p)B(p) - Ba (p)A(p) + G(p)B(p).
By the Abelian theorem on Laplace transforms (see [1]) and using the relations (1.4)
and (1.5), we obtain
V(p),~O(pl--~),
as p-~ oe
(2.5)
V(p),~O(p~),
as p--,0.
(2.6)
and
The above behaviours of V(p) assure that the integral in (2.4) exists and is finite.
We then discuss, in detail, the method of determination of the function V(p).
We consider the general case of (1.1) for which the pair (a o, b0) has the property that
either a o or b o or both of them are non-zero; the same is assumed to hold good for the
pair (a,, b,) also.
Singular integro-differential equations
179
Defining
q, tz) = ~
1 ff
V(t)dt
t- z
(z=p+ip,,i2=_l)
(2.7)
and using the Plemelj's formulae (see [-2]) from (2.7), we obtain that
(2.8)
~ + (p) - ~O_(p)= V(p)
and
lffV(t)dt
+ (p) + ~b_ (p) = i~
(2.9)
(t -- p)'
where ~ + (p), ~O_ (p) are the limiting values of ~ (z) as z = p + ip' approaches a point p of
the positive real axis, from above and from below respectively.
From (2.4) and the relations (2.8) and (2.9) we then obtain that
a(p) + iB(p) , , ,
C(p)
~ ~ +
~Pl - ~9_ (p) - Atp) - iB(p)'
p >~O,
(2.10)
which represents a Riemann Hilbert problem for the determination of ~O(z), which can
be solved as described below.
We first construct a sectionally analytic function n(z) in the z-plane cut along the
positive real axis (0, ~ ) satisfying the relation
n+(p) ~4(p)+iB(p)
n_ (p~) - A ( p ) - iB(p)'
pc(O, ~ ) ,
(2.11)
where n+ (p), n_ (p) are the limiting values of n(z) on the two sides of the positive real
axis.
Assuming that A ( p ) + i B ( p ) possesses zeros at the points /~l,tL2..... kt, and
A(p) - iB(p) possesses zeros at different points 21, 22 ..... 2. none of which lie on the
positive real axis and setting n(z) = znm(z), with 6 = (1/2~i)ln((a. + ib.)/(a. - ib.)), we
can recast the problem (2.11) into the form
m+(p) ( p - # l ) ( p - l t 2 ) . . . ( p - # . )
m_ (p~ - (p - 21)(p - 22)... (p - 2.)"
(2.12)
The solution of the problem posed by relation (2.12) can be easily written down as
(see [2])
ln(t - ~l)(t - #2)--. (t -/~.)
In re(z) = ~
l f]
(t-21)(tt__2zE)...(t-2.)dt+lnEl(z)
(2.13)
where E1 (z) is an entire function, and, using Plemelj's formulae we obtain that
in
i
m+(P) = j=l 2
\p-#J
+
~/
~
)"
After simplification this gives
Et(P) = f i o~J(o-2:) V:(o).
(2,151
180
A Chakrabarti and T Sahoo
where
Vj(p)=(p-2f]~
kp - I~j/
2hi
I ln(t/l#fl)dt "],
(t - 2~)(t -/~i)]
(2.16)
with ~j = -(1/2ni)ln(2Jl~i), 01i = arg(21), 02i = arg(pi),
Cj = exp (-(2J2ni- #j) I(0) + yi(ln [2 j l - i(2n - 01i- 02i)))
and
I(0) =
(t -
#A
We thus determine the limiting value of (l/n+ (p)), as given by
p~+~+~'+'"+~"(p-~l)(p-J,2)...(p-~n) Vl(p) V2(P)... Vn(p)
El(p)
(2.17)
1
n +(p)
Next, using the relation (2.11), we can recast (2.10) as
n+ (p)~J +(p) - n_ (p)~J_ (p) =
C(p) n_ (p)
A(p) - iB(p)"
(2.18)
giving
1 f]
C(t)n_(t)dt
n(z)~k(z) = ~-~
[A(t) - iB(t)](t -- z) + E2 (z)'
(2.19)
where E2(z) is an entire function.
By Plemelj's formulae from (2.19) we get
1
C(p)
n+(p)
2 A (p) -- iB(p) n_ (p)
+ (p) =
1
~ o~ C(t)n_ (t)dt
e2(p)
+n+(p)'
+ 2nin+ (p) Jo (A(t) -iB(t))(t-p)
p>0
(2.20)
and
r
1
C(p)
2 A ( p ) - iB(p)
(p) =
l
f]
C(t)n_(t)dt
+2nin_(p~)
(A(tSZ~_p)
E2(P)
k--n_(p), p > 0 .
(2.21)
The function V(p) is finally determined, by using the relations (2.20) and (2.21) in the
relation (2.8), and we find that
V(p)
2A(p)_iB(P)kn_(p) + l +-~i n~p)
•
f~
C(t)n_(t)dt
( 1
(A(t) - iB(t))(t - p) + E2(P) n~p)
(2.22)
Sinoular integro-differential equations
181
The integral in the relation (2.22) can be easily evaluated by considering the contour
integral
C(On(~)d~
fr
B(O(
- z)'
where F is a closed contour comprising of a circle of large radius along with a loop
around the positive real axis in the complex plane, with the assumption that
x, ~z ..... r are the n distinct zeros of B(~), which do not lie on the positive real axis and
B'(~i) # 0, forj = 1, 2 ..... n (the case of multiple zeros can also be dealt with). Then, by
application of the Cauchy residue theorem we obtain that (using the relation (2.11) also)
2 ~+
,-~ 9
C(p) , ,F . n-(p)]
,~ n(~,)C(~,)
=ff~n+tp) L1
+ 2 2~
,=1B'(r162
(2.23)
C(t)n_(t)dt
30
Using the result (2.22) in (2.23) we derive that
n(~:)C(~fl _2iEe(P) 3
B(p)[j=~
V(p) =
-
ej)
(a, - ib,)(p - 21)(p - 22)... (p - 2.)n+ (p)'
(2.24)
which on substituting for n+ (p) from the relation (2.17) gives
V(p) = n(p)p ~+~ Vl(p) Vz(p)... V,(p),
(2.25)
where
H (p) = B( p) [ET= 1n( r j) C (r j) - 2i( p - r
(p -
(p)]
(2.26)
i)E1 (p)
which is a rational function, with 7 = 71 + 72 + "'" + 7..
The analysis that has been explained above is applicable to (1.1) for the general case
when ak'S and bk'S a r e complex constants. But, for the particular case, when ak'S , bk'S are
real constants, we have 0 < 6 < 1. Setting 0 U = 0r, 02j = 2n - 0r and #j = 2-i, we obtain
?~ = (1 - (0Jn)), C i = 1 and using the relation (2.16) we find that
%(p)=exp[sinOjff~'r
-
zr
lntdt
]/
t 2 - 2 t c o s O i + 1 [(p_2j)(p_~)]l-~o~/2~)
(2.27)
This is equivalent to the result as obtained by Varley and Walker (1989).
Ultimately, by using the form (2.25), we find that
,~ H ( p )
V(p) ~ - ~ ,
as p--+ +
(2.28)
and
V(p)'~ H(p)p ~+6, as p-~0,
(2.29)
which help in expressing H(p) as
n--1
H(p)=
~
hip/,
(2.30)
j=-k
where k - 1 < 6 + 7 < k, k = 0, 1..... n - 1, when the relations (2.5) and (2.6) are also
utilized.
182
A Chakrabarti and T Sahoo
The determination of hj's can be completed by considering two separate cases as
described next. These cases take care of even different degrees of the two polynomials
A(p) and B(p).
Case 1. I f A(p) and B(p) are of the same degree n, the constants h o, h 1. . . . . hn can be
obtained by using the relation (2.4) along with the fact that r ~2 . . . . . ~ are the simple
zeros of B(~). The other constants h_ 1, h_ 2 . . . . . h_, will remain arbitrary.
Case 2. I f n > m where, n is the degree of A(~) and m is the degree of B(~), the
m constants h t , h 2 . . . . . h m can be determined by using the m zeros of B(~) and the
relation (2.4). The other n - m constants h m+ 1... hn- 1 can be determined by using the
asympototic behaviour of re(z) as z-o o% as has been discussed by Varley and Walker
[7]. The other constants will remain arbitrary.
The function v(x) can finally be determined by using the Laplace inversion formula
and we find that
v ( x ) = 2 ,=t
~' Ric~176176176
ff
l(s)e-XSds'
(2.31)
where Ri ei~' denotes the residue of V(p) at p = h i and
l(s) = (2rci)- 1 [V(se-i~) _ V(sei~)],
(2.32)
with the expression for V(p) as given by the relation (2.25). The unknown functionf(x)
can then be determined successfully, by using the the second relation in (2.3) along with
the convolution theorem for Laplace transforms.
Following are some observations on the method that has been presented above.
(i) We have avoided a lot of complication by calculating the Cauchy type integral in the
relation (2.22) directly. (ii) In our analysis, the determination of H(p) is simpler as
compared to that explained in Varley and Walker's [7] paper. (iii) The behaviour of the
function V(p) at the end points p = 0 and p = oo along with the analyticity property of
V(p) has helped us in arriving at the expression for H(p), as given by the relation (2.26).
Acknowledgement
One of the authors (TS) acknowledges the University Grants Commission, New Delhi
for the financial support received as a research student of Indian Institute of Science,
Bangalore.
References
[1] Doetsch G, Introduction to the theory and application of the Laplace transformation (Berlin, Hiedelberg,
New York: Springer-Verlag)(1974)
[2] Gakhov F D, Boundary value problems (New York: Dover Publication, INC) (1990)
[3] Holford R L, Short surfacewavesin the presenceof a finite dock. Proc. Cambridge Philos. Soc. 64 (1964)
1109-1129
[4] MuskhelishvilliN I, Singular Integral Equations (Groningen: Noordhoff) (1946)
[5] Spence D A, A flow past a thin wing with an oscillatingjet flap. Philos. Trans. R. Soc. 257 (1960)445-467
[6] Stewartson K, A note on lifting line theory. Q. J. Mech. Appl. Math. 13 (1960) 49-56
[7] Varley E and Walker J D A, A method for solving singular integro-differentialequation. IMA J. Appl.
Math. 43 (1989) 11-45