MMET'98 Ilroceedings
I9
k
Galer in methods in solving integral equations with
applications t o scattering problems
A. Chakrabasti
Department of Mathematics, Indian Institute of Science
Bangalore - 560012, India
ethods of obtaining approximate solutions of integral equations and their applications to
e examined . Two typical problems, one
propagation of two dimensional surface
boundary value problems (see Sneddon [6]),of Mathematical Physics
ucing them to those of solving integral equations of various types and
me specially simple situations that exact closed form solutions of the
be determined completely, and, in the cases of integral equations with
"4 01: otherwise, only approximate solutions of certain types can
Uy. Of all such approximate methods for solving integral equations,
see Jones [4], Evans and Morris [2],[3],Banerjea and Mandal [I],
others) appear lo be extremely powerfd, in the sense that certain
ccuracy can be recovered with appropriate choice of certain sets of
be described in section 2 of the present paper.
ajor mathematical ideas behind the Galerkin methods in section 2.
n 3, ic__wo_ diaer ent mathematical problems of scattering, occurring
and in the theory of water waves respectively, and have reduced
problenis to those of solving & a
integal equations of first kind, with &r
4, we have presented the approximate soiutionrs of the intea-ai
ated in section 3, by employing just or,e term Galerkin approximations and
ave derived approximate results for certain special quantities of practical
the problems considered in section 3.
nes considered in the present work, the principal
of solving some linear operator equations (linear
red here) of the type
( U ) ( z )= qz), x E A,
(2.1)
where L is (B linear operator from a certain inner product space S to itself and A c IR (can
be Etn,in general), where f and 1 are real valued functions. It may also be required (as in
the problems considered here) to determine the inner product :
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Whenever the mathematical problems at hand are expressible in the forms of the two relations
(2.1) and (2.2) we can resolve them, approximately, by utilizing the following senses and
:A---.
LUGCXD.
Definition: A real valued function F ( z ) E S is said to solve the equation (2.11, approximately,
if and only if
[LF,A] =: [A, LF] R i [A, E ] ,
(2.31
where the symbol
sense that
M
means ”approximately equal to” and we shall write:
Wf,4
f
M
F , in the
ILK 4
for all X(z) E S.
Then, using the approximate solution F of the equation (2.l), we can derive an
imate value of the inner product [ f ,11, as given by the relation (2.2),in the form:
Cf,4 = [ K I ] .
In the Galerkin methods which can be successfully utilized for many problems (especially
for the p~-obIemsconsidered here), we express the approximate solution F ( z ) , in the form:
(2.6)
where {$?(x>)~+ denotes a set of n linearly independent functions (not necessarily orthogonal) in S and c3’s are n constants to be determined. as desired below.
Using the relation (2.6) in the relation (2.3), after choosing X(x) = $k(z), €or a fixed
k (I 5 k 5 n,),we obtain the following set of approximate linear relations
(2.7)
Treating the above approximate relations (2.7) as a set of n linear equations, we can determine the constants ej’s ( j = 1 , 2 , . . . , n) and then the determination of the approximate
solution F ( z ) , can be completed by using the relation (2.6).
Also, the approximate evaluation of the inner product [ f , l ] can be completed and we
obtain
12
As an example, by taking n = 1 only, we obtain
It is obvious from the above discussion that varieties of Galerkin methods can be developed
by varying $ j ’ s and n.
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concentrate only on " single-term" Galerkin approximations.
we sha.1 now make the followkg observations:
We have
4, (4
+
[.f,21 = [461 = 11, f l [l,f - PI,
( i ) [F: LF]M i4;:
( i i i ) [E, f - F ] w [J,f, f j - 2[LF:F ] [F,
L F ] , (by using (2))
(i?]) [.f - F, L ( f - F ) ] Fz [ L f ,f ] - 2 [ W , F ] [F,L F ] .
+
+
By using t ne results (iii)and ( i v ) we find that
a,
11, .f' - P
I = [f - F, L(.f -
(2.10)
and then cne of the following two cases hold good.
Case
(2.):
If L is 3 positive semi-definite linear operator, i.e. if [h,Lh] 2 O , for all h E S ,
then
[I, q 5
p, s:,
(2.11)
an
Case
(L):
E L is a negative
-_
semi-definiie linear c~pemtor,i.e. if
[h,Lh] 5 0 . for all h E
S,
I
-
with
(i)
< IC?
such that
(1) = 0, on LU = O and
(e-ixz
(iii)
4)
+
x = a,
Re"%) sin
(ii)
a&
-= 0 ,
dY
(7j
, as x
,
as x
--+00,
ony=Oandy=b
-+
X
-00,
>0
x > o (a known constant)
(a known constant)
I
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MMET’98 Proceedings
(Note that R and 7’ are unknown complex constants to be determined).
along with the edge condition that ~4 possesses a square-root singularity at the edge y = d.
Note: The forms of # as given by (zzz), suggest that
-A
2
- - T+2p = o
_,A2=
a2
along with the equation (2.1).
Reduction to two integral equations.
Setting
(-1
7r.z
4(x,y, z> = qi(g7z> sin \ a
(3.2)
with
according as z > 0
x
< 01,
with
(3.4)
conditions of the problem1 are met with? except the two conditions ( A )and ( B )of (iv),which
lead to the following DUAL SERIES RELATIONS, for the determination of the constants
a, :
and
-KO
+
00
K,,u,
cos
(y)
= 0, for d
< y < b.
(3.6)
n=Q
These dual relations can be easily reduced to two intes-a1equations, in the following manner:
i
Firstly, setting the left side of the relation (3.6) as equal to -aoK&g(y), and noting that
2
g(y) = 0, for d < y < b, we can easily determine the Fourier coefficients, in terms of g(y)
and then the relation (3.5) easily gives rise to the integral equation:
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory
MMET'S
with
CO
nny
n-1
We also f
3 that a quantity N tan be defined as :
tend approach, setting the lek side of the relation (3.51, as equal to
__.-
zf% (1 2
--
,nd noting that S I ( ~ )= 0, for 0 < y < d , we can again determine the Fourier
easily in terms of gI(y) and then the relation (3.6) gives rise to the following
iation:
with
We also f
5- that
(3.12)
such that
Kq5+--0,
a4
ay
---CO
o n y = O a n d -mcox~t:m,
< z < 00, ( K > 0, a known constant)
(ii)
Kharkov, I:
aine, VUth International Conference on Mathematical Methods in Electromagnetic Theory
M M E T 9 8 Proceedings
84
e- K Y + ~ P Z(e-imz + Rei-), as LI: --+CO,
~ ~ - K y + i p z- i m x
e
, asx-+co,
where p = K sin(a),m = K cos(a),(0 < Q < 5)
( R and T are unknown complex constants tobe determined).
d,
(222)
(4 4, vd
0, 8s Y
+
CQ
along with the edge conditions that 0 4 possesses a square root singularity at the edge y = a,
ensuring uniqueness of the solution of the problem.
Reduetian to inteErd equations
(3.14)
with
(3.15)
where k l = (k2 pz)t and R = (1 - T ) and b ( k ) are unknowns, we find that all the
conditions of the problem are satisfied, except the conditions ( A )and (C) of ( i i ) ,givislg rise
to the following "DUAL INTEGRAL EQUATIONS" for the determination of the function
+
b(k):
__
/" b(k)(k cos(ky) - K sir;(ky))
---- d k
A
+ M2
+ im(R - 1)e-"g
= 0, (0
< y < a),
(3.16)
and
(3.17)
in which
R is also an unknown constant.
separate integral equations, by
The above dual integral equations can be seduced to
employing a trick, similar to the one used for the problem 1, along with the use of the
Havelock's expansion theorem (see Ursell [7]). In fact, this has already been done by Evans
and Morris [2].
?rR
2
Firstly, setting the left side of the relation (3.16), as equal to -f(y),
and noting that
f(y) = 0, for 0 < y < a, we can determine b ( k ) in terms of f(y), by using Havelock's
expansion theorem, and then the relation (3.17) gives rise to the integral equation:
(3.18)
where
( k c o s ( k y )- Ksin(ky))(kcos(kt) - Ksin(kt))
dk'
0
kl(k2 K 2 )
along with the defining relation:
L(Y't) =
J
M
+
(3.19)
(3.20)
Kharkov, Ukraine, VIIth International Conference on Mathematical Methods in Electromagnetic Theory
'roceedings
85
;he relation
(3.23)
d
d
of the first of the relations (2.9).
r,we assume a "sing1e-term" galerkin approximation to the solution of the integral
IO), as given by
(4.3)
1
aine. VIIth International Conference on Mathematical Methods in Electromagnetic Theory
MMET'98 Proceedings
86
Again, for the solution of the integral equat'ions (3.18) and (3.21), we assume the following
"single-term" Galerkin approximations ( see Evans and Morris [a]) :
and
where
6
1 and
Ci are to be calculated by using the first of the relations (2.91, giving
and
5 . Some approximate resdts
In this section vie shall explain about the derivation ~f some approximate results for the
quantities H and A associated with the two problems I and 2. considered in section 3 , which
represent important quantities of practical interest in the thtr?ary of dectr-omagnetism (see
Jones[4]) and in surface water wave theory (see Evans amid Monk iZ]), respectively.
y using the relations (3-9)and the approximate solution for g(y) as given by the relations
(4.1) and (4.2). we can easily determine H czppmximatelgi. We find that in the particular
b
situation, when d = - and p,,
2
n
7
T
(i.e when
b
M --
<< E ) , we obtain
(5.1)
Similarly, by tising the relation (3.12), along with the approximate solution for gl(y> as given
by the relations (4.3) and (4.4), we can determine an approximate value for the quantity H .
b
nK
We find that when d = - and pn = -, we have
2
b
From the theory that has been explained in section 2, we find that the two results in the
relations (5.1) and (5.2) provide some upper and lower bounds respectively, for the quantity
7r
H , and, we find that the average of these two bounds gives the value -0.73--,
which,
&Ob
.n;
ac-cording to Jones 141, is very near the actual value -0.71-.
&Ob
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Also, by using the approximate solutions as given by the relations (4.51, (4.6), (4.7) and
(4.82, ink(> the relations (3.20) and ( 3 . 2 3 ) , we can determine two values of A , approximately,
which prcwlde some upper and lower boiinds, AI and A2 respectively, for this quantity. Then,
with the aid of the defining relation (3.20), we find that
The rpxnerical values of 1121 have been worked out, by Evans and Morris 121, by using
the two tsounds for AI and A, of A as described above, and we give below a representative
tabie for ot == 30°qfar the purpoce of completion of this article. The table clearly shows the
closeness of the boiinds of /HI, i.e. 1H1/ and I&/, which must be attributed to the particular
choice of the " single-term" Galerkin approximations as suggested in the relations (4.5) and
(4.6).
Ssneqea
s. and :da:;daI B.N.," S C
e4
vaves by
8
srrbn1erged
3vans D.11.and Mcxri~C,A. IY "The effect of a k ~ e vertical
d
barrier
cobl~q~ely
incident surface waves in deep water-' 2~I ~ s t .I V a t h ~ Applies.
.
pp. 198-204, (1972 a)
Evans D. V. and Morris C. A. N., "Complementary approximations to the
solution of a problem in water waves", J . Inst. Makhs. Applies, 18, pp. 1-9,
(1972 b)
Jones D. S., Theory of Eleetrc;emagnetisrrm,Pergamon Press, ( 1964)
Mandal B. N. aid. Das P., " Oblique diffraction of surface waves by a submerged
vertical plate", J. Engng. Math., 30, pp. 459-470, (1996)
Sneddon I.
IS. Mixed Boundaqy Value Problems, North Holland, (1966).
I
Ursell F., "The effect of a k e d vertical barrier on surface waves in deep water",
P ~ Q cCamb.
.
Phil. SOC.,
43,pp. 314-3232, (1947)
Kharkov, likraine, VIIth International Conference on Alathematicd Methods in Electromagnetic Theory