16
2006. 2
. .
1. .
- y
x t B y H x t A y H
L 2[ ] (, ) [ ] ( , )
1
0
0
1(, , ) [ ] (, , ) ] ,
[ ) ,
(t x K t s x H y K t s x y ds
F (1)
- :
).
( ) , , 0 ( ), ( ) , , 0
( x 0 x y' x 1 x
y t (2)
> 0 – , (t,x) G – , y y(t,x, ) – - ,A(t,x),B(t,x),F(t,x),Ki(t,s,x) i(x) )
1 , 0
(i – , G {(t,x):0
)}
( )
( , 1
0 t 1 t x 2 t , :
]]
[ [ ] [ , )
, ( ]
[y et x grady H2 y H H y
H ,
- )) , ( , 1 ( ) ,
(t x Q t x
e )
,
( x
y t
grady y , Q(t,x)
G.
, :
I. A(t,x),B(t,x),F(t,x),Ki(t,s,x) i(x) )
1 , 0 ( , ) ,
(t x G i .
II. A(t,x),Q(t,x) s(x) - :
G x t x
t Q x
t
A(, ) 0, (, ) 0, (, ) , 0
, – .
III. 1(t) 2(t) -
) , (t dt Q
d , -
(1),
. 1 ) 0 ( , 0 ) 0
( 2
1
IV. 1
:
1 0
1(, , ) [ ( , , )
) , ( ) 1 , , (
s
x t K x s t x K s x A s t K
) , (
) , (
1 ]
) , (
) , ) ( , , ( )
xd A
x B
d x e
A x x B
t
K x . (3)
- :
0
0] (, )
[ ) ,
(t x H y B t x y
A (4)
1
0
0 0
0
1(, , ) [ ] (, , ) ]
[ ) ,
(t x K t s x H y K t s x y ds
F ,
(1) = 0
:
) ( ) ,
( 0
0 t x x
y . (5)
: 1) -
) ,
0(t x
y (4), (5); 2)
(1), (2); 3) ,
) , , (t x
y (1), (2)
) ,
0(t x
y G
. ,
(1) -
. , -
[1].
2. .
1. I–IV.
(4), (5) -
G .
. , -
:
x s y H x s t K x t F x t
z(, ) ( , ) ( (, , ) [ ( , )]
1
0
0 1
ds x s y x s t
K0(, , ) 0( , )) . (6) (4), (5)
17
A d tB
e x t x
t
y 0 ( , )
) , (
0(, ) ( ( , ))
+ e d
A
z BA d
t t
) , (
) , (
0 ( , )
) ,
( , (7)
) ,
(t (t,x) ,
[2].
(7) (6),
1
0
), , ( ) , , ( ) , ( ) ,
(t f t K t s z s
z (8)
) , , (t s
K (3),
) , (t f
1
0 0
0( ) [ (, , )
) , ( ) ,
(t F t K t s
f
1 ]
) , (
) , ) ( , ,
( A s
s s B
t
K e ds
A d sB 0 ( , )
) , (
. (9)
I–IV, -
(8)
G z(t, ). -
, (7), , -
(4), (5)
G. 1 .
-
(4), (5). -
) , , (t s
R K(t,s, ).
) , (t z (8)
1
0
) , ( ) , , ( ) , ( ) ,
(t f t R t s f s ds
z . (10)
(10), (9),
) , ( ( ) ( ) , ( ) ,
(t t 0 t
z
) ) , ( ) , , (
1
0
ds s s t
R , (11)
) ,
(t x (t,x) -
:
1
0
1
0 ]
) , (
) , ) ( , , ( ) , , ( [ ) ,
( A
t B K t
K t
0
) ) , (
) ,
exp( ( d d
A
B ,
1
0
) , ( ) , , ( ) , ( ) ,
(t F t R t F d . (12)
, (7), (11),
) ,
0(t x
y (4), (5)
:
t t
d A t B
d A B
A e e
x t
y ( , )
) , (
0 ) , (
) , ( 0
0 ( , )
) 1 ( ) , (
d ds s s R
1
0
) ) , ( ) , , ( ) , ( (
d A e
t d
A tB
0
) , (
) , (
) , (
) ,
( . (13)
3. . ,
) , , (t x t
, (t,x) G -
:
0 ) , ( ] [ ) , ( ]
2[
x t B H
x t A
H , (14)
, 0 ) , ,
(t x t t(t,x,t) 1,0 t t 1. (15)
1. I–III.
) , ,
(t x t G
:
e K t K
t x K t
t x
t (, , )
, )
, ,
( ,
t t
e K x K
t x t, , )
( , (16)
K > 0 > 0 t,x .
. ,
:
0 ) , , ( ), , , ( )]
, , (
[ t x t z t x t t x t
H . (17)
) , ,
(t xt (17), -
t
t
ds t s z t x
t, , ) ( , , )
( . (18)
(17), (18) (14), (15)
ds t s z t B z t A z H
t
t
) , , ( ) , ( )
, ( ]
[ , (19)
1 ) , , (t xt
z . (20)
, (19), (20) -
:
18
2006. 2
t
t
ds t s z s t T V t t
z (, , ) (, , ) ( , , ) , (21)
) , , (t s
T :
d B t V
t x t V s
t T
t
s
) , ( ) , , ( ) , , 1 ( ) , ,
( 1 , (22)
) , , (t xt V
V -
:
1
| , 0 ) , ( ]
[V A t V V t t
H . (23)
(23) , ,
[2], II,
1 0
), exp(
) , ,
( t t t t
K t t
V . (24)
(22), (24),
1 0
, ) , ,
(t s K s t
T . (25)
(21).
(21), (24) (25),
t t z (, , )
ds t s z t K
K t
t
t
) , , ( )
exp( . (26)
, -
,
) exp(
)]
, , (
[ t t
K K t x t
H . (27)
(27), , H[ (t,x,t)]
)) , ( , 1 ( ) ,
(t x Q t x
e grad ( , )
x y t
y y ,
(16). -
(27)
d t H
t x t
t
t
)]
, , ( [ ) , ,
( ,
(16). 1 .
4. .
2. I–IV.
) , , (t x
y -
(1), (2) G :
) , , 0 ( )
, , 0 ( )
, ,
(t x K y x K y x
y t
) , ( max
) , (
x t F K
G x t
t
t t x K y x K y x
y ( , , ) (0, , ) (0, , ) ) , , 0 ( )
, , 0 ( )
, ,
(t x K y x K y x
yx t (28)
) exp(
) , , 0 ( )
, ( max
) , (
x t y K x t F
K t
G x
t ,
K > 0 > 0 t,x .
. , -
(1) -
, .
x s y H x s t K x t F x t
z(, , ) (, ) ( (, , ) [ ( , , )]
1
0 1
ds x s y x s t
K (, , ) ( , , ))
)] 0 , (29)
) , , (t x
y (1), (2) -
) , , (t x t :
t
t d t B t t x
t y
0
0 1 (, , ) ( , ) )
1 ( ) ( ) , , (
t d t z t t t
t
) , , ( ) , , 1 ( ) ( ) 0 , , (
0
1 .(30)
(30) (29), -
) , , (t z
:
ds s
z s t K t
f t
z(, , ) (, ) ( , , ) ( , , )
1
0
, (31) )
, , (t s
K f (t, )
:
)]
, , ( [ ) , , ( 1 ( ) , , (
1
1 t s H t t
K s
t K
s
, )) , , ( ) , ,
0(t s t t ds
K
t K t
F t
f (, ) (, ) 1( ) ( , ) ds s B s t K s t
K (, , ) (, , ) ( , )) (
) (
1
0 0
0 , (32)
I–III (16), (27)
:
, 1 0
, ) , ,
(t s K s t
K
K K
t
f (, ) 0( ) 1( )
G x t t F K
G x
t (, ), ,
max
) ,
( . (33)
19 )
, , (t s
R – K (t,s, ).
(31)
ds s f s t R t
f t
z(, , ) (, ) ( , , ) ( , )
1
0
. (34)
(34) (33),
K K
t
z(, , ) 0( ) 1( ) G x t t F K
G x
t (, ), ,
max
) ,
( . (35)
(30)
t x (16), (35), -
(28). 2 .
5. .
) , , (t x
y (1), (2)
) ,
0(t x
y (4), (5)
3. I–IV.
G :
) , , 0 ( )
, ( ) , ,
(t x y0 t x K K y x
y t
) , , 0 ( )
, ( ) , ,
(t x y0 t x K K y x
yt t t
+K e 1 yt(0,x, )
t
,
) , , 0 ( )
, ( ) , ,
(t x y0 t x K K y x
yx x t
+K e 1 yt(0,x, )
t
, (36) K > 0 > 0 t,x .
. , -
(1) y(t,x, ) y0(t,x)
) , , (t x u
1
0
0
1(, , ) [ ] (, , ) ]
[K t s x H u K t s x u ds u
L
0 2[y (t,x)]
H . (37)
(37) 2 -
1, (36).
3 .
. 3 ,
G x t 1, ) 0
( :
) , ( ) , , (
lim 0
0y t x y t x ,
) , ( ) , , (
lim 0
0yt t x y t x ,
) , ( ) , , (
lim 0
0yx t x y t x .
1. .
. :
, 1981. 123 .
2. ., .
- -
// . , ,
. 2000. 2(21). . 100-106.
i i
- .
i i .
Summary
The estimations for the solutions of the Cauchy problems for the singular pertu rbed partial second order integral differential equations have suggesting.
Have proving that the solution of the perturbed problem going to the solution of the nonpertubed problem when small parameter going to zero.
517.938
, . 7.01.06 .
