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Решение одной нелокальной задачи для гиперболического уравнения в замкнутой форме.

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??? ????????? ? ???????? (3). ????????, ??? u y0 ( x) ???????????? ????? ? ??????????????,
????? ????? z0 , ? ??? u y0 ( z0 ) = 0. ??? ????? ????????? ????????, ??? Wj ( X ) ???????? ???????? ?????? (6), (7).
????????????????? ??????
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
??????? ?.?., ????????? ?.?. ? ????????? ?????????? ?????????? ???????? ????????????? ??????? ????? //
????. ?? ????. 1969. ?.185. ?.739?740.
??? ?. ???????? ? ?????????????? ???????????? ???????????? ??????. ?.: ???, 1970, 184 ?.
Penkin O. About a geometrical approach to multistructures and some qualitative properties of solutions // Partial Differential Equations on Multistructures, ed. by F. Ali Mehmeti, J.von Below and S.Nicaise / Lect. Notes Pure Appl.
Math. 2001. V. 219. P. 183?192.
Penkin O.M. Second-order elliptic equations on a stratified set. Differential equations on networks // J. Math. Sci. (N.
Y.) 119 (2004), ?. 6. P. 836?867.
?????? ?.?., ??????? ?.?. ? ?????? ???????????? ?????? ??????? ?? ?????????????????? ?????????? //
?????. ???????, 2000. ???. 68. ? 6. C. 874.
?????? ?.?., ???????? ?.?. ? ???????????????? ???????????? ??? ????????????? ????????? ?? ???????
????????????? // ????. ???. ?. 360. ? 4. 1998. ?. 456?458.
Nicaise S., Penkin O. Poincare-Perron's method for the Dirichlet problem on stratified sets // J. Math. Anal. Appl. 296
(2004). ? 2. P. 504?520.
John F. Partial Differential Equation. Springer Verlag, 1986. 250 p.
??????? ?, ????????? ?.?. ????????????? ???????????????? ????????? ? ???????? ???????????? ???????
???????. ?.: ?????, 1989. 464 ?.
?????? ?., ??????? ?. ???????????????? ???????. ?.: ???, 1980, 304 c.
????????? 2.10.2006 ?.
??? 517.956
?. ?. ???????
??????? ????? ??????????? ?????? ??? ???????????????? ?????????
? ????????? ?????
????????? ??????????? ??????? ?????? ??? ?????????????? ???????????????? ????????? ? ???????
D , ??????? ???????????? ????? ??????????? ???? ???????? ? ??????? ? ?????? ??????????????.
?????????????? ????????????? ? ?????????????? ??????? ??????????? ?????? ??????? ? ???????
???????????? ???????????? ????????????? ?????????.
?????????? ?????????
LU =| y |2 m U xx - sign( y )( yU yy + aU y ) = 0,
(1)
1
1 - 2m
??? m > ,
< a < 1 , ? ??????? D = D1 И D2 , ??? D1 ? ??2
2
?????? ??????????? ??????? ????????? ??????????? ?????????? x
? y , ???????????? ????????????????
2 m +1
2 m +1
2
2
(- y ) 2 = 0, BC1 : x +
(- y) 2 = 1
2m + 1
2m + 1
????????? (1) ??? y < 0 ? ???????? J є AB = {x : 0 Ј x Ј 1} , D2 ?
???????? ??????????? ??????? ????????? ??????????? ??????????
x ? y , ???????????? ????????????????
AC1 : x -
2 m +1
2 m +1
2
2
y 2 = 0, BC2 : x +
y 2 =1
2m + 1
2m + 1
????????? (1) ??? y > 0 ? ???????? J .
?????? ????????? ???????????: Q0 ( x ) ? Q1 ( x) ? ????? ??????????? ????????????? ????????? (1), ????????? ?? ????? x О J
? ???????????????? AC1 ? BC2 ??????????????; ( I 0a+, b ,h f )( x)
AC2 : x -
? ? ?. 1. ???????
D
15
? ( I1a-, b ,h f )( x ) ? ????????? ??????????? ???????? ???????-????????????????? ? ??????????????????? ???????? ?????? F ( a, b, c; x) , ????????? ? [4] ? ??????? ??? ??????????????
a , b ,h ? 0 < x < 1 ???
( I 0a+, b ,h f )( x) =
x -a - b
ж
( x - t )a -1 F з a + b , -h ,a ;1 G(a ) т0
и
x
tц
ч f (t ) dt
xш
(a > 0) ,
(2)
t-xц
(1 - x) -a - b
ж
(t - x)a -1 F з a + b , -h ,a ;
ч f (t ) dt (a > 0) .
т
G(a ) x
1- x ш
и
??? ?????????????? ??????????? ??????????? ??????????? ????????? ????? [5].
????? 1. ????? 0 < -a < l < 1 , ? b < min[0,h + 1] . ???? j ( x) О H l [0,1] , ??
1
( I1a-, b ,h f )( x) =
(3)
( I 0a+, b ,hj )( x ) , ( I1a-, b ,h j )( x) О H min[ l +a ,- b ] [0,1].
????? 2. ?????
0 < a < l < 1,
l -a <1 ?
r ( x) = x m , ??? 0 Ј m < l - a + 1 . ????
j ( x) О H 0l ( r ;[0,1]) , ??
( D0a+j )( x) О H 0l -a ( r ,[0,1]).
??? ????????? (1) ?????? ??????? ??????.
?????? 1. ????? ??????? U ( x, y) ?? ??????????:
1) LU є 0 ? ??????? D = D1 И D2 ;
2) U ( x, y) О C ( D) З C1 ( D \ J ) З C 2 ( D \ J ) ;
dU
dU
,n 2 ( x) = lim
, xО J ;
y
®
0
+
dy
dy
4) t 1 ( x) = lim U ( x, y ) , t 2 ( x) = lim U ( x, y) , x О J ;
3) n 1 ( x) = lim
y ® 0-
y ® 0-
y ®0+
5) ??????????? ??????? ??????????:
t 1 ( x ) = t 2 ( x) , n 1 ( x ) = n 2 ( x) ;
6) ??????????? ???????:
A1 I 0a+,b ,- a - b t 2 b -1U [Q0 ] + A2 I 0a++ b ,b +1-2 b , - a - bU [t ,0- ] + A3 I 0a++1-b ,b, - a -b U y [t ,0 -] = j1 ( x) ,
*
*
*
*
*
B1 I1a- ,b , - a - b (1 - t ) 2 b -1U [Q1 ] + B2 I1a- + b ,b +1-2 b , -a
*
-b
*
*
(4)
(5)
*
U [t ,0+ ] + B3 I1a- +1- b ,b , - a - b U y [t ,0+ ] = j2 ( x) ,
(6)
??? A1,2,3 , B1,2,3 ? ????????? ???????????? ?????????; j1 ( x) ? j 2 ( x ) ? ????????? ???????,
?????, ???
j1 ( x) О H l [0,1] , j2 ( x ) О H l [0,1] ,
0 < a* + b < l1 < 1 , 0 < a + b < l2 < 1 .
1
(7)
2
??????? 1. ????? - b < a < b , 0 < b < 2 b , - b < a * < b , 0 < b* < 2b , A2 № -
G(2 b )
A1 ,
G( b )
G(2 b )
B1 , ??????? j1 ( x ) ? j2 ( x) ????????????? ???????? (7), ????? ?????? 1 ????????G( b )
?? ?????????.
??????????????. ??? y < 0 ????????? (1) ????? ???
B2 № -
???
(- y ) 2 m U xx + yU yy + aU y = 0 .
(8)
1
- m < a < 1 ?????????? ??????? U ( x, y ) ????????? (8), ??????????????? ????????
2
U ( x,0) = t 1 ( x ) , 0 Ј x Ј 1 ,
(9)
lim ( - y) U y ( x, y ) = n 1 ( x ) , 0 < x < 1 ,
a
y ®0 -
??? t 1 ( x) О C 1 ( J ) З C 2 ( J ) , n 1 ( x) О C 1 ( J ) , ?????? ???????? [1, 2]
U ( x, y ) =
16
2
1
й
щ b -1
G(2 b )
m0 m0
b -1
t
x
+
(1
2
t
)(
y
)
к
ъ t (1 - t ) dt 1
2
т
G ( b ) 0 лк
2 ыъ
(10)
2
1
й
щ -b
m0 G(1 - 2 b )
m0 m0
1-a
-b
(
y
)
n
x
+
(1
2
t
)(
y
)
к
ъt (1 - t ) dt.
1
2
т
2G (1 - b )
2
0
лк
ыъ
??? y > 0 ????????? (1) ????? ???
-
(11)
y 2 mU xx - yU yy - aU y = 0 .
???
(12)
1
- m < a < 1 ?????????? ??????? U ( x, y ) ????????? (12), ??????????????? ????????
2
U ( x,0) = t 2 ( x) , 0 Ј x Ј 1 ,
(13)
lim yaU y ( x, y ) = n 2 ( x) , 0 < x < 1 ,
(14)
y ®0 +
??? t 2 ( x) О C 1 ( J ) З C 2 ( J ) , n 2 ( x) О C 1 ( J ) , ?????? ???????? [1, 2]
2
1
й
щ b -1
G(2 b )
m0 m0
b -1
t
x
+
t
y
(1
2
)
к
ъ t (1 - t ) dt +
1
2
т
G ( b ) 0 лк
2 ыъ
U ( x, y ) =
2
1
й
щ -b
m0 G(1 - 2 b ) 1-a
-b
m0 m0
+
y
n
x
(1
2
t
)
y
ъt (1 - t ) dt.
1к
2
т
2G (1 - b )
2
кл
ъы
0
2 m - 1 + 2a
4
????? b =
, m0 =
.
2(2m + 1)
2m + 1
??? ???
m0
m0
ж
ц
ж
ц
x ж x ц2 ч
1+ x ж1- x ц 2 ч
з
з
Q0 ( x) = з ; - з ч ч , Q1 ( x) = з
;з
ч ч,
з 2 и m0 ш ч
з 2 и m0 ш ч
и
ш
и
ш
??, ????????? (11), (15), ??? U (Q0 ) ? U (Q1 ) ?????:
+
(15)
(16)
1- 2 b 1
m G(1 - 2 b ) ж x ц
G(2 b ) 1
U [Q0 ( x)] = 2
t 1 ( x - xt )t b -1 (1 - t ) b -1 dt - 0 2
з ч
т
2 G (1 - b ) и m0 ш
G (b ) 0
тn ( x - xt)t
1
-b
(1 - t ) - b dt , (17)
0
G(2 b )
U [Q1 ( x)] = 2
t 2 (1 - (1 - x)t )t b -1 (1 - t ) b -1 dt +
G ( b ) т0
1
1- 2 b
1
m G(1 - 2 b ) ж 1 - x ц
-b
-b
+ 0 2
з
ч
т0 n 2 (1 - (1 - x)t )t (1 - t ) dt.
2 G (1 - b ) и m0 ш
?????????? (17), (18) ? ????????? (2), (3), ???????
U [Q0 ( x)] = M1 ( I 0b+,0, b -1t 1 )( x) - M 2 x1-2 b ( I 01-+b ,0,- bn 1 )( x) ,
1- 2 b
U [Q1 ( x)] = M 1 (1 - x)
b ,1- 2 b ,- b
( I1-
t 2 )( x) + M 2 ( I
(18)
(19)
n )( x) ,
1- b ,2 b -1, b -1
12
(20)
-2 b
G(2 b )
1 G(1 - 2b ) ж 2m + 1 ц
, M2 =
з
ч .
G( b )
2 G(1 - b ) и 4 ш
?????????? U [Q0 ( x)] ? U [Q1 ( x)] ? ??????? ??????? (5), (6), ?????
??? M 1 =
A1M 1 ( I 0a+,b , - a - b t 2 b -1 I 0b+,0, b -1t1 )( x) - A1M 2 ( I 0a+,b ,- a -b I 01-+b ,0,- bn 1 )( x) +
+ A2 ( I 0a++ b ,b +1- 2 b , - a - bt 1 )( x) + A3 ( I 0a++1-b ,b , - a - bn 1 )( x) = j1 ( x),
a* ,b* ,- a* - b b ,1- 2 b ,- b
11-
B1M 1 ( I
I
*
*
a* , b* , - a* - b
1-
t 2 )( x) + B1M 2 ( I
*
*
*
(1 - t )
I
*
+ B2 I1a- + b ,b +1- 2 b ,- a -bt 2 + B3 I1a- +1- b ,b ,- a -bn 2 = j 2 ( x).
??? g > 0 ????? ??????? ?????????? [3]:
( I 0a+, b ,h I 0g+,d ,a +h f )( x) = ( I 0a++g , b +d ,h f )( x) ,
( I1a-, b ,h I1g-,d ,a +h f )( x) = ( I1a-+g , b +d ,h f )( x) ,
? ??? a > 0 ? ??????? ?????? [3]:
( I 0a+, b ,h f )( x) = x -a - b -h ( I 0a+, -a -h ,-a - b f )( x ) ,
a , b ,h
-a - b -h
a ,-a -h , -a - b
(21)
n )( x) +
2 b -1 1- b ,2 b -1, b -1
12
( I1- f )( x) = (1 - x)
( I1f )( x).
???????? ??????? (23)?(26) ??????????? ????????? (21), (22):
(22)
(23)
(24)
(25)
(26)
17
A1M 1 ( I 0a++ b ,b +1- 2 b ,- a -bt 1 )( x ) - A1 M 2 ( I 0a++1- b ,b ,- a - bn 1 )( x) +
+ A2 ( I 0a++ b ,b +1- 2 b , - a - bt 1 )( x) + A3 ( I 0a++1-b ,b , - a - bn 1 )( x) = j1 ( x),
a* + b , b* +1- 2 b , - a* - b
12
t )( x) + B1 M 2 ( I
B1M 1 ( I
*
*
+ B2 I1a- + b ,b +1- 2 b ,- a
*
-b
(27)
a* +1- b ,b* , - a* - b
1-
*
*
n 2 )( x) +
*
t 2 + B3 I1a- +1- b ,b ,- a -bn 2 = j 2 ( x).
(28)
- a - b , - b -1+ 2 b ,0
0+
?? ????????? (27), ?, ???????? ??????? (23), ??????????????? ?????????? I
??? ????????? ??? t 1 :
A M - A3 1-2 b ,2 b -1,0
1
t1 = 1 2
( I 0+
n 1 )( x) +
( I 0-+a - b , -b -1+ 2 b ,0j1 )( x ) .
(29)
A1 M 1 + A2
A1 M1 + A2
??????????? ??????? ?? (28) ???????? ????????? ??? t 2 :
*
*
B M + B3 1- 2 b ,2 b -1,0
1
t2 = - 1 2
( I1n 2 )( x) +
( I1--a - b , -b -1+2 b ,0j2 )( x) .
B1 M 1 + B2
B1 M1 + B2
????????? ??????? ?????????? (4), ??????? ?????????
A1 M 2 - A3 1-2 b ,2 b -1,0
B M + B3 1-2 b ,2 b -1,0
( I0+
n )( x) + 1 2
( I1n )( x) = f ( x) ,
A1 M 1 + A2
B1 M 1 + B2
(30)
(31)
???
1
( I 0-+a -b ,-b -1+ 2 b ,0j1 )( x) .
B1 M 1 + B2
A1 M 1 + A2
????? A2 № - M1 A1 , B2 № - M1 B1 . ?????? ????????? ???????????:
A M - A3
B M + B3
1
1
P1 = 1 2
, P2 = 1 2
, P3 =
, P4 =
.
A1M 1 + A2
B1M 1 + B2
A1M 1 + A2
B1 M1 + B2
f ( x) =
1
*
*
( I1--a - b , -b -1+ 2 b ,0j 2 )( x) -
(32)
(33)
???????? ? ????????? (31) ???????? I 02+b -1,1-2 b ,g :
P1n ( x) + P2 ( I 02+b -1,1- 2 b ,g I11--2 b ,2 b -1,0n )( x) = f1 ( x) ,
(34)
???
*
*
f1 ( x) = P4 ( I 02+b -1,1- 2 b ,g I1--a -b ,-b -1+ 2 b ,0j2 )( x) - P3 ( I 02+b -1,1- 2 b ,g I 0-+a -b ,-b -1+ 2 b ,0j1 )( x).
??????? ????????? (31) ????? ????????? ???????????????:
P1n ( x) + P2 ( D01-+2 b I11--2 bn )( x) = f1 ( x) .
???????? ??????????? [4]
(36)
a
sin(pa ) ж t - a ц j (t )
тa зи x - a чш t - x ,
p
????????? (36) ? ???? ??????? ????????????? ????????? ? ????? ????:
b
Daa+ ( I ba-j )( x) = cos(pa )j ( x) +
(35)
(37)
1- 2 b
sin(2pb ) ж t ц n (t )dt
(38)
т0 зи x чш t - x = f1 ( x) .
p
????? ???????, ??????????? ?????? ???????? ? ??????? ???????????? ????????????? ????????? (38). ????? ???????????
x1-2 bn ( x) = y ( x) ,
(39)
???????? ????????? (38) ? ????
1
b y (t )dt
ay ( x) + т
= f2 ( x) ,
(40)
p 0 t-x
???
a = P1 - P2 cos(2pb ) , b = P2 sin(2pb ) , f 2 ( x) = x1-2 b f1 ( x ) .
(41)
??????? ??????????????????? ???????????? ????????? ?? ???????? ???????
1
b y (t ) dt
ay ( x) + т
= f2 ( x) ( 0 < x < 1 )
(42)
p 0 t-x
P1n ( x) - P2 cos(pb )n ( x) + P2
1
????? ?????? ? ?????? H 0* = H * [0,1] З C[0,1) ????????????? ???????, ???????????? ? ?????
x = 0 . ????? ???????, ??? a 2 + b 2 № 0 . ?????????
a( x) - ib( x)
G( x) =
= eiq ( x ) , q ( x) = arg G ( x) .
a ( x) + ib( x)
18
(43)
??? ????????? (42) ??????? G( x ) , ?, ?????????????, ? arg G( x ) , ????? ???????????. ??????? ???????? arg G ( x ) ???, ????? 0 Ј q (0) < 2p , ?, ?????????????, ? 0 Ј q (1) < 2p .
?????? ????????? (42) ??? ???????? ?????? ??????? ?????
йq (1) щ
c =к
ъ =0.
л 2p ы
?????????? ?????? ????? ????????????? ?????????
*
*
*
*
f 2 ( x) = P4 x1- 2 b ( I 02+b -1,1- 2 b ,g I1--a -b ,-b -1+ 2 b ,0j2 )( x) - P3 x1- 2 b ( I 02+b -1,1-2 b ,g I 0-+a - b , -b -1+2 b ,0j1 )( x).
????????? ??????????? ?????????? ??????-???????? [3], ????????? (44) ? ????
f 2 ( x) = P4 x1- 2 b ( D01-+2 b I1--a - b , -b -1+2 b ,0j2 )( x) - P3 x1- 2 b ( D01-+2 b I 0-+a - b , -b -1+ 2 b ,0j1 )( x ).
(44)
(45)
?????????? ?????? ?????????. ????? j2 ( x ) О H [0,1] , - b Ј a Ј b , 0 < b < 2b , ????? ??*
*
%
?????? ????? 1 ( I1--a - b , -b -1+ 2 b ,0j2 )( x) О H l2 [0,1] , ??? l%2 = min(l2 - a * - b , b* + 1 - 2 b ) . ????? ????0 < 1 - 2 b < l% < 1 ,
l% + 2b - 1 < 1 ,
?????
??
?????
2
?????
????,
???
l2
2
1- 2 b - a* - b , - b* -1+ 2 b ,0
0+
11- 2 b
1- 2 b
(D
I
j 2 )( x) О H
l%2 + 2 b -1
*
*
2
[0,1] . ?????????? ??????????????? ?????? ?????????. ???
??? x
ОH
[0,1] , ?? ??????? f 2 ( x ) ??????????? ?????????????? ??????.
????? ???????, ??????????? ????????? (42) ????? ?????????????? ???????, ? ?????? ? ??????? ???????????? ?????? ?????????? ? ???????????.
????????????????? ??????
1.
2.
3.
4.
5.
??????? ?. ?. ????????? ?????????? ????. ?.: ????. ??., 1985. 304 ?.
??????? ?. ?. ????????? ?????? ????????? ? ??????? ???????????. ?.: ?????, 1981. 448 ?.
????? ?. ?., ?????? ?. ?., ??????? ?. ?. ????????? ? ??????????? ???????? ??????? ? ????????? ?? ??????????. ?????: ????? ? ???????, 1987. 688 c.
Saigo M. A remark on integral operators involving the Gauss hypergeometric functions // Math. Rep. Kyushu. Univ,
1978. Vol. 11. ? 2. P. 135?143.
Saigo M., Kilbas A.A. Generalized fractional integrals and derivatives in Holder spaces // Transfom Methods and Special Functions, Sofia 94 (Proceeding of International Workshop). Sci. Cult. Tech. Publ., Singapore. 1995. P. 282?293.
????????? 5.08.2006 ?.
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