− to obtain separately SIE for particular cases of the considered mixed boundary
value problem − the first and the second boundary value problem;
− to study dependences of Fourier coefficients on the slot width for various
opening angles of cone using the Gaussian quadrature formulas in case of a cone with
a slot.
Method of singular integral equations in the solution of the first and the second
boundary value problem. For the solution of the problem one can use the pair of
integral Kontorovich-Lebedev transforms with respect to the radial coordinate [24,25].
Using these transforms the potential
1
u
is found in the form of the Kontorovich-
Lebedev integral:
()
1
1
2
0
()
2
,
i
mm
m
K
qr
u
sh
bU
d
r
+
+
=−
=
(2)
where
()
0
0
,
1
,
,
ˆ
(,,),
0,
ˆ
(,,)(),
;
mnm
n
mi
mnm
n
X
U
Y
+
+
=−
+
+
=−
=
(3)
0
0
()
1/2
,
,
1/2
(cos)
ˆ
(,,)
()
;
(cos)
mnN
inNm
i
mnm
mnm
mnN
i
P
X
x
e
d
P
d
+
+
−+
++
+
−+
=
=
0
0
()
1/2
,
,
1/2
(cos)
ˆ
(,,)
()
;
(cos)
mnN
inNm
i
mnm
mnm
mnN
i
P
Y
y
e
d
P
d
+
+
−+
++
+
−+
=
−
=
−
()
i
K
qr
is the Macdonald function,
1/2
(cos)
mnN
i
P
+
−+
is the Legendre function of
the first kind,
m
b
are known and
0
,
mnm
x
+
,
0
,
mnm
y
+
are unknown coefficients;
0
mN
m
=+
,
0
m
is the closest integer to
mN
,
12
12
−
.
To find coefficients
0
,
mnm
x
+
,
0
,
mnm
y
+
it is necessary to use boundary
conditions and additional conditions which are introduced in every specific case on the
surface of the conical gratings. Let us consider important particular cases of the
boundary condition (1).
The first boundary condition on the strips. In this case in the boundary
condition (1) we assume
1
=,
12
0
g
=
=
==
and obtain:
0
u
=, (4)
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