,

517.3, 519.6
. .
,
. , . . . .
: , , , .
Abstract. The article examines a problem of electromagnetic field diffraction on a dielectric body located in a rectangular resonator. The problem is reduced to volume singular integral equation on the body. The author has considered a numerical collocation method for solving the equation and presented the formulas of matrix coefficients for the collocation method.
Key words: boundary value problem, electromagnetic scattering, integral equations, numerical method.

, . . . , . . . . . , . . (Ansis, Quikwave ..) . . . (A. . , . . , . . [1-3]).
1.
. P = { : 0 < xi < a, 0 < x2 < b, 0 < X3 < c} -
. Q (Q - ), (3 X 3 )- (). ()
Q, 1 () Q, Q [3].
dQ Q -. , Q , dQ = 0 . Q (> 0), (> 0).
E, L ioc () (, , E, H L (Q)), exp(-iW), - . - jE L2 ioc () .
3
R grad, div, rot .
ї () :
rotH = -E + jE ,
rot E = , x . (1)
E, H :
Et = 0;
= . (4)
(1)-(4) E [3]
(()
E() = () + jge (r) -1 E(y)dy +
Q ^
v
Q
f ()
,
+graddiv j(Ge (r) -^--1 E(y)dy, Q. (5)
V
j
, E() \ Q
( (\
E() = E () + jge (r) -1 E(y)dy +
Q P"
+graddiv| ()

-1
(), \ .
:
()
() = 0 () - 70 |

-1
0
( )<, ,
I - .
= diag (, , [3]
2'(! + 0) =0=1 ' ' ^ ' ^ ( ' ^) ^

∞ = '
] . I ] I
0081 ---1 | 8 | ~^~2 I 0081 - X

=
. I 1 ( 3)8( ( )) 3 < >
2 1' [ (
2 '(1 + 0 )
I | (3 )' 8 ( ) > ;
1 I . I
8I -------1 0081 ----------2 8I --------_1 X
1=0 8( ) V ) V ) V
8(3 )" 8( ( )) 3 < ,
I 8
0081 ----2 ^, , ,
1 [8 ( ) 8 ( ( 3 )), 3 > ;
=
. .
8I 1 8 ^ 2 8I
1 =! 8 ( ) V ) V ) V
(
3 ) ( ( )) 3 <,
X I
2 ) !(3 ) ( ( 3)) 3 >.


]2 ]2 , 2
]+ ()- "
, 1 > 0 .
[4, 5] :
1
2
3
= , ( x2, 3): ї1 ^ ^ ї1 + 1, ї2 ^ 2 + ! ї3 ^ ^ ї3 + 1 ,
!
/, 2, 3 ,
0 ^ ^ /0 ( \ ) = 2 008 () 8 (2 )
=1 =1
8 (2 (ї2 + 0,5))- 8 | -2 2) 008 (( (ї1 + 0,5)) 8 ) ~~) +
+ 2~2 ^1∞ (23) 8{2) 8(2 (2 + 0,5)))I-22] ; =1 2 )
-2 = ∞ ^ ^ ^ ( 3) 8(1)008(2 )
1 =4 -
=1 =1
8 ((1 ( + 0,5))-8 1 008 (2 ( + 0,5))^
2
H /0 (2) 8 (1) 8 (ї1+ 0,5)) 8 12 1];
=1 0
]=4
/. ( 3 ) ---------8
=1 =1
(1) 8 (2 )
8(1 {1 + 0,5)) -8| - ]8(2 (ї2 + 0,5))^ ).

/ (3 ) =
-2 8(3 ) 8
(
. -- - * ))* [

8( )
, ] < ї^ ;
2 8 ( ( - 3 ))) ) (3 + ї3*3 ) ] ^I (3 - ї3*3 )1
8( )
2 8 (3 ) 8 ( )
8
-
^ + * +
8
(3 - ї3*3 - *3 )

< <( + 1)*;
28 ( ( - 3 ))
8( )
8
*3 ]] 3
ї3*3 + 8I
>( + 1)*3,
/ 110 (3 ) = 4 /00 (3 ) , /20 (3 ) = 4 /0 (3 ) ,

/ (3 )
4' ()' ,
! / \ 111 11
( ' ) V 2
\ _ 33 _ I , 3 < 33;
4 ' ( ( _ 3 )) , 3 _ 33
^ ( ')
8

3 + 33
4 ' (3 ) 3 _ 33 _ 3 ( ' )
X
2 _ 3 _ 33 _ 3

3/3 < 3 <(/3 + !)3;
4 ' ( '( _ 3 )) ) 3 1 , ., 3
( ' ) 'V 2 | V V3^3 + 2
3 >(/3 + 1)3;
X = ^1 = 2 = 1 ^ =2 = 1 = 2
XI =, 2 = ~~, 11 =, 12 = ~-, 1 =, 2 =_-

1 = 711, 2 = 72^ 1 = 2 = 22 ,

\2 / \2 1 ( 1 , 2
I +1 --1 _ ^2 .
I V I
12 3
/, / , / -

23
21 2) 2) 22 22 22 23 23
2 21 31 1^ 2 32 13 23

2

2/ =^8 ∞ (3 )' ?
2- = -2 '-' 08(1)' sin (2)
1 =1=1 '
2

X[2 {2 + 0,5) I 2 I'(11 (/'1 + 0,5))

2
_ 0 ( )
- = ~ 7 3 ) ' sin(1 )' 0(2 )
2 1
1 =1 =1 I;
2

Xsin(2 (2 + 0,5))' I 2 I'(( (/'1 + 0,5))sin

2

= ^ /^)' sin(^1)' (2 )
2

Xsin(m#2 (2 + 0,5))-sin| j cos(n( ( + 0,5))" ) n1 ^
2 1 1 ,)sin,
2 -j \ ^ y 2
2of -sy y f () ) . ( X )x
5^=~ y ^-yS 'cos (nXi)-sln (X2)X
1 A n=1 m=1 *nm
Xsin(n((i, + 0,5))sin| H, I cos(mH2(/'2 + 0,5))si") H2 ^
isin
2 *j v -v- ' >) y 2
2G72 -s y y fm (3 )m ( X) , X )X
~~2 = 7 yy----------- 2~ sin (X1 )COs (mX2 )X
2 b n=1m=1 n -ynm
Xsin(n( (/'1 + 0,5))sin^-2H1 j cos(mH2(/'2 + 0,5))sin^ m2 ,
G -sy y d_fim() , ( X, , ( X )X
^=^b -^------------------sin (nX1 )sin (mX2 )X
D A n=1 m=1 * nm
Xsin (/'1 + 0,5) sin iH1 | cos(mH2 (/'2 + 0,5))si"' m2 I
isin
2 *j v -v- ' >) y 2
G s y; y;d_flm() , X) ( X )X
=y y--------------cos (nX1 )sin (x2 )x
1 3 , , m v2
1 3 n=1 m=1 i.
nm
X sin (((/'1 + 0,5)) sin y n21 jj sin {mH2 (/2 + 0,5))sin ^ m2 j;
2G/3 s y y d _ flm (x3) ( X ) , X )x
' = yy---------- .2 sin (nX1 )cos (mX2 )X
23 bn , , n v
2 3 n =1 m=1 n 1.
nm
Xsin(n1 (/'1 + 0,5))sin^n1 jjsin(m2 (/'2 + 0,5))sin^m2 j;
2G/3 s y; y; d2_ flm (3) ( X ) X )X
2^ = ~2 yy-----------------2 sin ( nX1 ) sin (mX2 )X
3 n n=1 m=1 n m V
nm
Xsin( + 0,5)) sin^n~~jsin(jmH2 (2 + 0,5))sin^mH2j
d _ f∞m ( ) - 3 fnm ( ); d _/ ( ) - 3 / (); d2_/ () - 3
f ( ) ;

^ _ / ( 3 )
^ _/ (3 )
_2'
( (3 ))
| 33 2 I ' 2
( ' )
3 < /33;
( (3 33 )) + ( (3 + 33 ))
26| ( ' ) 2 ( ')
(3 )( ( _ 3 _ 33 ))
( ' )

3/3 < 3 <(/3 + 1)3;
( ( '( 3 )))

\
( ')
3 >(3 + 1)' 3;
4 (3 \ 2 | ^ 33 2 |
^( ')
3 < /33;
( ((3 + 3 )) ( (3 + 3 )) ( ') ( ' )
2^ ( (_/33 _3 )) ^
(3 )
^( ')

33 < 3 <(3 + 1)3;
( '( 3 ))'
4 ' 6| ^ 2 | ^33 + 2 | ( ' )
3 > (/3 +1);
4 ^ ( 3 [ ^ J ^
( ' )
7 3
-33-
3 < /33;
^ ( (33 ^ 3 ) ) ^ ( (33 ^ 3 ) )
( ' ) ^ ( ' )

(
3 ) (
(-/33 ~3 ))
( ' )

< 3 <(3 + !)3;
4^ [ ^^ I
^ 33 + ] ( '(-3 ))
( ' )

3 >(( + 1)3-
(5) . , .
2.
= / (, / X) : X ^ X X , . Xn = / . Xn (Xn - X), : X ^ Xn - , .
2 / <2/ , / 67 = 0
/ ^ 7 2 = . / () /
/ . :
= I1, ,
^ [0, ^ .
Xn : Xn = {/,...,} . , :
Vx X lim inf ||x-x|| = 0.

Pn : X ^ : ()(x) = (x1 ), x Q. , ()(x) x dQi, , X = L2 . = Pnf :
( )( xJ ) = f ( xJ ), j = 1,..., .
-

: = 2 ^ .
=1
, :

2 (Avk )(x ) = f (xj ), j = 1, ., .
=1
[4-7]. -
( !x - ^
(5). ,
( x)
Q ,
.

-
0

1
J =
0
E.
0
:
7 = |()() _ | (, )() _
_ & | (, )() = ,/0(), 2 . (6)
:
3 2 ^ () _ ^0 | & (, )^ (^ _
/=1 2
div* f Ge (x, J()dy = E∞l (x), l = 1,2,3. (7)
OXi J
l Q
^ 12 3
Jn = (Jn, Jn , Jn) :
n n n
J'n = Iatfl(X). Jn = Iblfi2(x), j3 = ^ckf(x),
k=1 k=1 k=1
fk - -ї.
fk . , Q - : Q = {x: < *1 <an, b < *2 <bn, c < X3 < cj} . Q :
klm = {x : x1,k < X1 < x1,k+1, x2,l < xl < x2,l+1, x3,m < x3 < X3,m+1},
a2 - ai b2 - & c2 - c1
x1,k = a1 +------k, x2,l = b1 +------l, x3,m = c1 +-----------m,
n n n
k, l, m = 0,..., n -1.
fklm , i = 1,2,3 :
/ = J1, x klm,
Jklm = | n
[∞, x nklm.
-
3
L2 = L2 L2 L2 .
Q (. 1). .
Q
. 1. Q , n^m
\, G^,
-3
I , . , , :
1 3
22 23 2
31 32 33 3 J
^ :
= 4 ( ); (8)

/ = (xi2,/3 ), = ( + ,5)/*!, -2 = ( + 0,5)h2, = ( + 0,5)/ ,
,I = 1,2,3; /-2,/3, 71,72,7 = 0,...,-1.
, . , , . , [8-18]. .

1. , . . / . . . - . : , 1998.
2. , . . / . . . - : - , 2009.
3. , . . / . . , . . // . - 2004. - . 44, 12. - . 2252-2267.
4. , . . / . . , . . , . . // . . - . - 2009. - 3. - . 71-87.
5. , . . / . . , . . // . . - . -
2009. - 4. - . 54-69.
6. , . . - / . . , . . // . . - . - 2008. - 2. - . 2-14.
7. , . . / . . , . . // . - 2010. - . 50, 9. - . 1587-1597.
8. , . . / . . , . // . . - 2004. - 5. - . 5-19. - ( ).
9. , . . / . . , . . , . . // . - 2005. -. 6. - . 99-108.
10. , . . web- - / . . , . . , . . // . . - . - 2007. - 4. -. 60-67.
11. , . . / . . , . . // . - 2008. - . 53, 4. - . 441-446.
12. , . . , / . . -, . . , . . // . . - . - 2009. - 1. - . 87-99.
13. , . . - , / . . , . . // . . - . - 2009. - 3. - . 59-70.
14. , . . / . . // . . - . -
2009. - 4. - . 48-53.
15. , . . - / . . , . . , . . // . . - . -
2010. - 1. - . 2-13.
16. , . . / . . , . . , . . // . . . - 2010. - 2. - . 32-43.
17. , . . , / . . , . . , . . //
. . - . - 2010. - 2. - . 44-53.
1S. , . . I . . II . . - . - 2010. - 3. - . SS-94.

- , , ,
Medvedik Mikhail Yuryevich Candidate of physical and mathematical sciences, associate professor, sub-department of mathematics and supercomputer modeling,
Penza State University
E-mail: _medv@mail.ru
517.3, 519.6 , . .
, I
. . II . . - . - 2011. - 2 (1S). - . 2S-40.