517.9
. .
1
. . .
: , , , .
Abstract. The author suggests a collocation method to solve problems on surface eigenwaves and leaky eigenwaves of weakly guiding dielectric waveguide in the half-space. This method is investigated theoretically and practically.
Key words: dielectric waveguide, eigenvalue problem, integral equations, collocation method.
[1-3]. [4, 5]. ( ) ( ) . .
[3]:

1.
u = ().
(1)

( () )u( ) = J (; X, y)u(y)dy, . ;
(2)
1 09-01-97009.
(; , ) = 2- ( ( | - |) - 0 ( | - y * |))())(),
- - R+ = {-= < < ^, 2 > 0}, R ; - -
2 2
* 2 ()_
(., ., [6]), y = (y1,-y2), p () = 2---------2^ > 0; n -
n+ -
; > 0 -
, n+= max(), = 2 -2- > 0 -

; > 0 - -
2 2 2
; > 0 - ; X = w -0 (+ _ ) > 0 , 0 (-) - () . > 0 X ().
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C ()
|| ||C () = sup| ()|. (3)

, C(). , , ., [8]. jh
, ^ , h = 0 , i ^ j. max diam^ jh) ^ h ,
Kj^Nh
Nh
Nh - . = ^ j h -
j=1
, Sh = jh}^=1 - , ^jh - jh, j = 1,..., Nh . , dist(,Sh) ^ 0,
h ^0, , dist(,Sh) = min | -^jh |.
^ j ,hSh
E = C () Eh = C (S h), Sh , || Uh |E = max | Uh (^ j h) |, Uh Eh . -
h 1jNh
Ph : E ^ Eh E Sh : PhU Eh - (p^)(^ j h) = (^ j h), j = 1, ., Nh .
( )^( ) [4] , || ^ | ^ , ^ 0. : -------> . ---------------> , | (,, ) (,)1^,
1],
^ 0.
, || , ) = 1, (, ) ( (,)
: ^ ) : || || ^|| || , ^ 0 .
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=1
() = 1 , ] () = 0 .
(1) , () ^- . :
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]=1 ] ,
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- , :

( )(^,) =
]=1 ],
( )( ) (, ) [4] (, ), -------------> ,
----> . ----> . , ( )/(0) -
(,) (,) , -----> , , , || || < !;, (0, ), , -
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{ (^',, ^. (6)
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, 0, .
() = {X > 0: 3 0 , = XTu} , $() - .
1. 0 ^ sp(T) X $(), X ^0 0. , ) X 0 0, 0 (). (), -
2
0 () , | X X0 .
. (, ) - . (,) - ( 2.2 [4]). , 4.2 [4] , ( )(0) (, ) (, ) 0 = XT
5.2 [4]. (7) -
5.3 [4]. .
2.
- [3]:
\\ 2, ∞.
(7)
2
, (),
2 0 ~
(1) [4] ()
II ||2,0() = | () | + | () | + ()


() = | |.

(8)
(, ) = I X()T(, );
((,))() = | (,;,)(), ;

(, ; , ) = 4 (01) ((, ) | |) 01) ((, ) | * |)) ( ) ( ),
%((0,e) = \W2n0-0 -2 ; ^) - (., ., [6]), . >0 , ї 2) - ln %(), :
02) = (: -2 < arg() < 3/2, Im(())< 0}.
(8) E = C () (3). , - - [9].
, . 1, [9] j - j, - n, - -n
= ^ j , - j=1
= (^ j }=1, j h - j . :
II un lien = max I un, j I, un C ,
C 1jn
un j j - un . ,
E En , pn L(E, Cn) .
N - . N', N' .. N. Zn ^ Z, n N', n , n N'. [9] (un }neN' un Cn E, ||un - pnu^Cn ^ 0, n N'. : un ^ , n N'. , un ^ u, n N', Cn , max | un j - u (, j) |^ 0, n N'.
1<j<n J
(8) -
n
u(n)(x) = un jj(x) Lra(), (10)
j=1
Q Qn , :

uni = ^unj j KZ>i,)d, i = 1,, (11)
j=1
T A - , C :

(Tnun )(^-)=un,j j K(Si,)d, i = 1,, n; j= Q;
An () = I-\Tn (), An (): Cn Cn,
I - Cn
, {An }njv' A: E E, (8), :
un u, N Anun Au, N , (12)
||un|| < const, {Anun }nEN' P - ^ {un }n(=N'P - ^ (13)
, [9] {un }neN' , P -, N' N' N" N', {un }neN"' u E
o(A) A(e), (An) - An () An ()- ^
2^ (A),
{}neN , (An), , N {}neN - , ( An), , n N, (A) { }N - ; {vn }n<EN - ; ||un||cn = 1 , ( An); , An ( )un = 0, , un uo, n N (A) A(o)uo = , |u||e = 1
1-6 1 2 [9]
L pn : E
||pnu||cn ||u|E, N, Vu E
2. - () () . , [10, . 459].
3. () () . (): ^ (. 2.2 [4]) ().
4. {()}^ (). , (12). ,
|| - \\ ^11 - \ + |\ - \ . (14)
- ()
(10). , = (), : () ^ , ,
() -
\\AnPnu - PnAu\\Cn <||
IIE \\
(15)
(14) (15),
||Anun pnAu\cn |||lcn )Cn llUn PnU\cn +
+1 Cn llAll
||C
ECn\\ WL^E = 1,
u(n) - u
u(n) - u
L*
, n N
,
||A(P)||L ^e < c(p), ;
(16)
(17)
(18)
() - : () = 1 + sup J | K(, x, y) | dy.
xsQq
An () ,
||An ()|| Cn < |()
N, ^
(19)
, (12) (16)-(19).
(13). P - {Anun }nejv , N' N N" N, {Anun = un + Tnun}neN" P - w E . un Cn u(n) L() (10). || un C ^ const, || u(n) ||l ^ const n N" .
: () ^ [4, . 14]. ,
{ }"' - V . , {}"' - = ^ XV , (13) .
. (18) (19).
6. () , .. () ^. 3 [1]. .
, (5) (11) . . , , . 1 |^- | , -. 1 |^- | - : - = - (^-) (- \ - (^-)), - (^-) - , ^- . 1 |^- | > (^-) , - \ - (^-) |^- |
- .
> 0 (), (2), , , - j - ] , ,
(5) .
diag(| , 2,..., | ^ )/2, | -
. , (5) = () .
{ ()} " . , {( )} " {()}", .. () V ^ 0, "", V .
\\ - II ^ () - 1^ =1 , -

3.
[11].
(11) >0 () = 0, - , , - . , [12].
(5) (11) : 1) , ; 2) 1/2, , .
. 1 . = (), 1% = >0. X, = 0, . , X .
3
4
2

-2
4

. 1. () ()
. 2 3 .
N . = 1 N
X 6 . = 50,8596, N = 8096 , = 35,2225 , N = 8032 . . 1.
1
X 6 ; = 1

N 61 240 506 1059 2024 4236
0,3531 0,1693 0,1210 0,0863 0,0605 0,0432
X 6 39,3336 48,0972 49,5528 50,2392 50,5952 50,7702
1,8172 1,8956 1,7561 1,6377 1,4209 0,9432
0,2266 0,0543 0,0257 0,0122 0,0052 0,0018

N 64 320 664 1280 2656 4800
0,3896 0,1598 0,1125 0,0799 0,0562 0,0454
X 6 29,4901 33,6530 34,4707 34,8661 35,0785 35,1684
1,0720 1,7450 1,6866 1,5850 1,2929 0,7459
0,1627 0,0446 0,0213 0,0101 0,0041 0,0015
. 1 N : =| X X6 | /6 = /2 , - . , N . . 2.
2
X 4

N 240 506 1059 2024
0,1693 0,1210 0,0863 0,0605
X4 2,7616 - 0,9311/ 2,7897 - 1,0195/ 2,7978 - 1,0556/ 2,8020 - 1,0715/
1,8019 1,4209 1,1408 0,8241
0,0516 0,0208 0,0085 0,0030

N 320 664 1280 2656
0,1598 0,1125 0,0799 0,0562
X 4 2,6492 - 0,9372/ 2,6523 - 1,0056/ 2,6597 - 1,0224/ 2,6612 - 1,0389/
1,4703 1,0922 1,1777 0,5662
0,0375 0,0138 0,0075 0,0018
32. (
1 0 1
= 27.5:379
1-1 1
= 20.9^3 5 = 3-3.749:3
1

6
. 2. () () ; 0=1
2 (22), 2012 - .
. 3. () () . /, -
.
4 = 2,8042 -1,0803/, X4, , N = 4236 ^4 = 20,2. X4 = 2,6630 -1,0437/ N = 4800 ^4 = 15,2.
. . .

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Frolov Alexander Gennadyevich
, () Postgraduate student, Kazan
(Volga region) Federal University
E-mail: ekarchev@yandex.ru
517.9 , . .
/ . . // . . - . - 2012. - 2 (22). - . 3-15.