Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных
Встановлено асимптотичні оцінки розв'язків сингулярно збурених крайових задач з початковими стрибками. We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps....
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| Опубліковано в: : | Український математичний журнал |
|---|---|
| Дата: | 2013 |
| Автори: | , , |
| Формат: | Стаття |
| Мова: | Російська |
| Опубліковано: |
Інститут математики НАН України
2013
|
| Теми: | |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/165321 |
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| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных / К.А. Касымов, Д.Н. Нургабыл, А.Б. Уаисов // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 629–641. — Бібліогр.: 11 назв. — рос. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859538262654189568 |
|---|---|
| author | Касымов, К.А. Нургабыл, Д.Н. Уаисов, А.Б. |
| author_facet | Касымов, К.А. Нургабыл, Д.Н. Уаисов, А.Б. |
| citation_txt | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных / К.А. Касымов, Д.Н. Нургабыл, А.Б. Уаисов // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 629–641. — Бібліогр.: 11 назв. — рос. |
| collection | DSpace DC |
| container_title | Український математичний журнал |
| description | Встановлено асимптотичні оцінки розв'язків сингулярно збурених крайових задач з початковими стрибками.
We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps.
|
| first_indexed | 2025-11-25T23:47:37Z |
| format | Article |
| fulltext |
© !. ". !"#$%&', (. ). )*+,"-$.. ". -. *"/ #&' , 2013
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5 629
*(! 917.926
!. ". !#$%&'( (/0 -1 2314231565 5 24730565 !38. 039. :0-13, ";231<, !3837=130),
). *. *+,-#.%/ (>41<=:=. ?@=. :0-1, A3;B<6@C?30, !3837=130),
". 0. 1#2$'( (/0 -1 2314231565 5 24730565 !38. 039. :0-13, ";231<, !3837=130)
"345678749:3!4: 8;:* !4 <:= :*4>
!<":?8@ A")"94 3 *"9 "BC*D5 3!"9!85
)B> B4*:@*DE )4FF:<: *;4"BC*DE 1<"?*:*4@
3 5"BD5 6"<"5:7<85 6 <4 6<84A?8)*DE
We obtain asymptotic estimates for solutions of singularly perturbed boundary-value problems with initial jumps.
'=130@D;40@ 3=52E1@15F0i @9i065 C@8DÕG86iD =50?:;GC0@ 8H:C4057 6C3I@D57 83B3F 8 E@F316@D525 =1C5H6325.
(;G B@D@;J0@ K5C@6@?@ 6;3==3 =50?:;GC0@ D@82:L400<7 83B3F H<;5 C38C3H@130< MNN4615D-
0<4 3=52E1@15F4=654 241@B<, E@8D@;GOL54 =1C@51J C3D0@24C0<4 EC5H;5P405G = ;OH@I 1@F-
0@=1JO [1 Ð8].
'24=14 = 142 B;G K5C@6@?@ 6;3==3 =50?:;GC0@ D@82:L400<7 6C34D<7 83B3F D<H@C 03B;4-
P3L4?@ 241@B3 B;G E@=1C@405G C4K405I 5;5 57 3=52E1@15F4=657 EC5H;5P405I H48 EC4BD3C5-
14;J0@?@ 5==;4B@D305G @638<D341=G D4=J23 831C:B0514;J0<2. "03;58 E@638<D341, F1@ 6 13652
83B3F32 2@P0@ @104=15 5 6C34D<4 83B3F5, B;G 6@1@C<7 73C3614C0@ 03;5F54 03F3;J0@?@ =63F63.
)35H@;44 @HL54 C48:;J131< D M1@2 03EC3D;4055 E@;:F40< D [9 Ð11].
&B036@ D :638300<7 C3H@137 C3==231C5D341=G =;:F3I, 6@?B3 23;<I E3C3241C =@B4CP51=G
1@;J6@ EC5 =13CK4I EC@58D@B0@I. Q=14=1D400@ D@8056341 D@EC@= @ D<B4;4055 0@D<7 6;3==@D
6C34D<7 83B3F, 524OL57 03F3;J0<I =63F@6. /2400@ M1@ 5 GD;G41=G 94;JO 03=1@GL4I C3H@1<.
1. 6'$G#H'(I# J#K#L2. +3==2@1C52 ;504I0@4 B5NN4C40953;J0@4 :C3D04054 D<=K4?@ E@-
CGB63 = 23;<2 E3C3241C@2 EC5 EC@58D@B0<7
L! y! ! " ! ! r An+r (t )
dn+r y
dtn+r
r=1
m
# + Ak(t)
dky
dtk
k=0
n
# ! = !h(t), 0 ! t ! 1, (1)
= 6C34D<25 :=;@D5G25
di y
dti t=0
! = !! i , i = 0,!l " 1 ,
di y
dti t=1
! = ! i , i = 0,! p ! 1, (2)
?B4 ! > 0 Ñ 23;<I E3C3241C, ! i , ! i Ñ E@=1@G00<4, An+m (t) = 1, m+ n = l + p.
' 03=1@GL4I C3H@14 :=1303D;5D3O1=G 3=52E1@15F4=654 @94065 5 73C3614C C@=13 EC@58D@B-
0<7 C4K405G 83B3F5 (1), (2) EC5 ! " 0 , D<B4;G41=G 6;3== 6C34D<7 83B3F, 524OL57 03F3;J0<I
=63F@6.
R@1C4H:42 D<E@;0405G =;4B:OL57 :=;@D5I:
10. S:06955 Ai (t) !Cn+m+1([0,1]) , i = 0,!n + m, h(t) ! C([0,1]).
20. S:0695G An (t) :B@D;41D@CG41 04C3D40=1D: An(t) ! 0, 0 ! t ! 1.
630 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
30. (@E@;0514;J0@4 73C3614C5=15F4=6@4 :C3D04054
µ
m + An+m!1(t) µm!1 +É + An+1(t) µ + An(t)! = !0
52441 m C38;5F0<7 6@C04I µ1, É ,!µm = @1C59314;J0<25 D4L4=1D400<25 F3=1G25, EC5F42
m < l . R:=1J l ! m = n1.
40. #EC3D4B;5D@ J0 = det ! ij " 0, ?B4 M;42401< ! ij = uj 0
i " 1( )(0) , j = 1,!n, i = 1,!l , ! l+i, j =
= uj 0
(i ! 1)(1), j = 1,!n , i = 1,! p , =@=13D;40< 03 @=0@D4 N:0B324013;J0@I =5=142< C4K405I u10 (t),
u20(t), … ,!un0(t) :C3D0405G
L0y = 0. (3)
R:=1J W(t) Ñ DC@0=6530 N:0B324013;J0@I =5=142< C4K405I :C3D0405G (3), 1@?B3
W (t) ! 0, t ![0,1].
50. /2441 24=1@
! 0 ! = ! " i #1
Ji
(n1)(0)
J0
+ $i #1
Jn1+ i
(n1) (0)
J0i=1
p
% #
Jn1+ i
(n1) (0)
J0
Wt
(i #1)(1, s)
W(s)An(s)
h(s)ds
0
1
&
i=1
n2
%
i=1
n1
% # " n1 ! ' !0 , (4)
?B4 Jk
(n1)(0) Ñ @EC4B4;514;J n -?@ E@CGB63, E@;:F400<I 58 J0 83240@I k =1C@65 03 =1C@6:
u10
(n1)(0),!É !,!un0
(n1)(0) , @EC4B4;514;J Wt
(q)(t, s) E@;:F341=G 58 W (s) 83240@I n-I =1C@65 03
=1C@6: u10
(q)(t),!u20
(q)(t),!É !,!un0
(q)(t).
2. F+HK#&MHG#/NH#O $2$GM&# ,MPMH2Q. )3CGB: = :C3D040542 (1) C3==2@1C52 =@@1D41=1-
D:OL44 @B0@C@B0@4 D@82:L400@4 :C3D04054
L!y! = !0. (5)
BM&&# 1. +(,&- ./012)3)/ (,21.45 10 Ð30. 6178% 825 9()8%$3)&%2-)1: ,4,&3$/ #3;3 -
)4: !yi (t, !), i !=!1,!n + m, ,4)7(25#)1 .1<$(=3))171 18)1#18)171 (#%.)3)45 (5) ,0#%.3824./
,238(>=43 %,4$0&1&4?3,"43 0#4 ! " 0 0#38,&%.23)45:
!yi
(q)(t, !)! = !ui0
(q)(t) + O(!), i = 1,!n, q!=!0,!n + m "1,
!yn+r
(q) (t, s)! = !
1
! q
exp 1
!
µ r
0
t
" (x)dx
#
$
%
&
'
( un+r (t)µ r
q (t) +O(! )( ), (6)
r = 1,!m, q = 0,!n + m ! 1,
783 u10(t),!u20(t),!É !,!un0(t) Ñ 9 ()8%$3)&%2-)%5 ,4,&3$% #3;3)4: (#%.)3)45 (3), un+r (t) ≠
≠ 0, t ! [0,1].
Доказательство ;422< 04E@=C4B=1D400@ =;4B:41 58 58D4=10@I 14@C42< T;4850?4C3Ð
-5C6?@N3Ð):3I@03 (=2., 03EC524C, [7, =. 29Ð34]).
' 63F4=1D4 N:0B324013;J0@I =5=142< C4K405I :C3D0405G (5) D@8J242
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 631
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
yi (t, ! )! = ! !yi (t, ! ), i = 1,!É !,!n, yn+s (t, ! )!! = !! n1 !yn+s (t, ! ), s = 1,!É !,!m , (7)
?B4 n1 = l ! m , p = n ! n1 = n2. #@=13D52 @EC4B4;514;J 'C@0=6@?@ W(t, !) =5=142< C4K4-
05I (7). # :F41@2 (6) 5 (7) D<04=42 N:06955 ! n1 exp 1
!
µ i (x)dx
0
t
"
#
$
%
&
'
( , i = 1,!…!,!m , 83 8036
@EC4B4;514;G W(t, ! ) . A@?B3 E@;:F52
W(t, !)! = !!" exp 1
!
µ(x)dx
s
t
#
$
%
&
'
(
) *(t, !),
?B4 µ = µ1 +…+ µm, ! = n1m, ! (t, " ) = det ! ij (t, " ) Ñ @EC4B4;514;J n + m( )-?@ E@CGB63, M;4-
2401< 6@1@C@?@ 524O1 D5B
!ij (t, ")! = !
u j0
(i#1)(t) +O("), j = 1,É , n, i = 1,É , n + m,
1
"i#1
u j0 (t)µ j#n
i#1 +O(")( ), j = n +1,É , n + m, i = 1,É , n + m.
$
%
&
'
&
&
A4E4CJ 5==;4B:42 3=52E1@15F4=6@4 EC4B=13D;4054 @EC4B4;514;G !(t, "). +3=6;3B<D342
!(t, ") E@ D=42 D@82@P0<2 250@C32 E4CD<7 n =1@;H9@D. '=4 M15 250@C< 524O1 0:;4D@I
E@CGB@6 E@ ! . R@M1@2: D =:224 EC@58D4B405I D=47 M157 250@C@D 03 57 3;?4HC35F4=654
B@E@;0405G B@2505C:OL52 F;40@2 ! (t, " ) H:B41 1@ EC@58D4B4054, D 6@1@C@2 3;?4HC35F4=6@4
B@E@;04054 52441 035240JK5I E@CGB@6 E@ ! . ' =5;: (6), (7) E@CGB@6 ?;3D0@?@ F;403 M;4240-
1@D @B0@?@ 5 1@?@ P4 =1@;H93, 830523OL4?@ = n + 1( )-I E@ n + m( )-O =1C@65, 03F503G = E4CD@I
=1C@65 :H<D341 E@ 24C4 C@=13 0@24C3 =1C@65. #;4B@D314;J0@, 3;?4HC35F4=6@4 B@E@;04054
?;3D0@?@ 250@C3 n-?@ E@CGB63 w! (t), C3=E@;@P400@?@ D ;4D@2 D4C7042 :?;:, 52441 035240J-
K5I E@CGB@6 E@ ! . R@M1@2: B@2505C:OL5I F;40 @EC4B4;514;G ! (t, " ) E@;:F341=G : 20@P4-
0542 250@C3 w! (t) 03 4?@ 3;?4HC35F4=6@4 B@E@;04054, 6@1@C@4 2< @H@803F52 F4C48 D(t, ! ).
A@?B3, 5=E@;J8:G EC4B=13D;405G (6) 5 (7), B;G !(t, ") E@;:F342
!(t, ")! = !w" (t) D(t, ") + O
1
"#$1
%
&'
(
)* , (8)
?B4 ! =
2n + m" 1( ) m
2
. &EC4B4;514;J w! (t) 5 3;?4HC35F4=6@4 B@E@;04054 D(t, !) D E4CD@2
EC5H;5P4055 EC5 B@=131@F0@ 23;<7 ! EC4B=13D52< D D5B4
w! (t)! = !W(t) + O(!), D(t, ! )! = !(" 1)n(n+1) 1
! # $(t)%(t) 1+ O(! )( ), (9)
?B4 n(n +1) Ñ F410@4 F5=;@; !(t) = un+k (t)µ k
n (t)k=1
m" ; ! (t) Ñ @EC4B4;514;J '30B4C2@0B3
B;G 6@C04I µ1,!É !,!µm , 6@1@C<I @1;5F40 @1 0:;G 03 0 ! t ! 1, 136 636 M15 6@C05 E@E3C0@
C38;5F0< 5 @1;5F0< @1 0:;G. #@?;3=0@ EC@94B:C4 @EC4B4;405G N:06955 un+1(t),!…!,!un+m(t)
136P4 @1;5F0< @1 0:;G 03 @1C4864 0,1[ ]. A4E4CJ, :F51<D3G EC4B=13D;405G (9), 58 (8) 52442
632 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
!(t, ")! = !
1
"#
W (t) $(t)%(t) 1+O(")( )! & !0 .
A@?B3
W(t, !)! = !
!"
!#
W(t) $(t)%(t) exp 1
!
! µ(x)dx
s
t
&
'
(
)
*
+
, 1+ O(!)( )! - !0. (10)
3. F+HIR2O !'P2 2 -,#H2LH%M S+HIR22.
8T,MKM/MH2M 1. @()"A45 K t, s, !( ), 10#38323))%5 0#4 0 ! t, s ! 1, )%</.%3&,5 9()" -
A43: B1;4, 3,24 1)% 01 t (81.23&.1#53& 18)1#18)1$( (#%.)3)4> (5) 4 0#4 t = s Ñ )%?%2--
)/$ (,21.45$
K ( j ) s, s, !( )! = !0, j = 0,!n + m" 2, K (n+m" 1) s, s, !( )! = !1.
#EC3D4B;5D3 =;4B:OL3G 14@C423.
7M',M&# 1. +(,&- ./012)3)/ (,21.45 10 Ð30. 6178% 0#4 81,&%&1?)1 $%2/C ! 9()"A45
B1;4 K t, s, !( ) 0#4 0 ! t, s ! 1 ,(=3,&.(3&, 384),&.3))% 4 ./#%'%3&,5 91#$(21:
K (t, s, !)! = !W (t, s, !)/W (s, !), (11)
783 W (t, s, ! ) Ñ 10#38324&32- n + m( )-71 01#58"%, 012(?%3$/: 4< .#1),"4%)% W (s, ! ) <%$3-
)1: n + m( )-: ,"4 9()8%$3)&%2-)1: ,4,&3$1: #3;3)4: y1(t, ! ), y2(t, ! ), É , yn+m(t, ! )
(#%.)3)45 (5).
BM&&# 2. D,24 ./012)3)/ (,21.45 10 Ð30 , &1 0#4 81,&%&1?)1 $%2/C ! > 0 9()"A45
B1;4 K t, s, !( ) 0#4 0 ! s ! t ! 1 0#38,&%.4$% . .483
Kt
(q)(t, s, !)! = !!m Wt
(q)(t, s)
W(s)An(s)
+
!n"1
!q
#(q)(t, s, !)
#(s)
+ O ! +
!n
!q
#(q)(t, s, !)
#(s)
$
%&
'
()
'
(
)
$
%
& , (12)
783 Wt
(q)(t, s) Ñ 10#38324&32- 4< (4), % !(q)(t, s, ") Ñ 10#38324&32- m-71 01#58"%, 012(-
?3))/: 4< ! (s) <%$3)1: m-: ,"4 )% ,"(
exp 1
!
µ1(x)dx
s
t
"
#
$%
&
'(
un+1(t)µ1
q(t)
un+1(s)µ1n(s)
,!É !,!!exp 1
!
µm(x)dx
s
t
"
#
$%
&
'(
un+m(t)µm
q (t)
un+m(s)µm
n (s)
.
(13)
Доказательство. /8 (11) @EC4B4;52 Wt
(q)(t, s, !). # :F41@2 (6) 5 (7) D<04=42 83 8036
@EC4B4;514;G Wt
(q)(t, s, !) N:06955 ! n1 exp
1
!
µi (x) dx0
s
"
#
$%
&
'(
, i = 1,É ,m. A@?B3 E@;:F52
Wt
(q)(t, s, ! )! = !! " exp
1
!
! µ(x)dx
s
t
#
$
%
&
'
(
) * (q)(t, s, ! ), (14)
?B4 !(q)(t, s, ") Ñ @EC4B4;514;J n + m( )-?@ E@CGB63, E@;:F400<I 58 ! (s, " ) 83240@I n + m( )-I
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 633
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
=1C@65 03 =1C@6: y1
(q)(t, !),!É !,!yn
(q)(t, !),!vn+1(t, s, !),!É !,!vn+m(t, s, !) , EC5F42 N:06955
vn+ j (t, s, !) EC5 B@=131@F0@ 23;<7 ! EC4B=13D52< D D5B4
vn+ j (t, s)! = !
1
!q
exp
1
!
µ j
s
t
" (x)dx
#
$
%
&
'
( un+ j (t) µ j
q (t) +O(!)( ) , j = 1,!m, q = 0,!n + m )1.
(15)
+3=6;3B<D3G !(q)(t, s, ") E@ M;4240132 n + m( )-I =1C@65, E@;:F342
!(q)(t, s, ")! = !y1
(q)(t, ") Mn+m,1(s, ") +É + yn1
(q)(t, ") Mn+m,n1
(s, ") +
+ vn+1(t, s, !) Mn+m, n+1(s, !) +…+ vn+m(t, s, !) Mn+m, n+m(s, !) . (16)
/==;4B:G 14E4CJ 3;?4HC35F4=654 B@E@;0405G Mn+m, j (s, ! ) 136 P4, 636 5 EC5 D<D@B4 N@C-
2:; (8 ) Ð(10), 037@B52
Mn+m, j (s, !)! = !
("1)n+m+ j !m#(s)$(s)
!% µk(s)k=1
m&
Wnj (s) + O(!)( ) , j = 1,!…!,!n, (17)
?B4 Wnj (s) Ñ @EC4B4;514;J n ! 1( )-?@ E@CGB63, E@;:F400<I 58 @EC4B4;514;G W(s) D<-
F4C65D30542 n -I =1C@65 5 j -?@ =1@;H93; ! (t) Ñ @EC4B4;514;J '30B4C2@0B3 B;G 6@C04I
µ1,!É !,!µm .
"03;@?5F0<2 @HC38@2 B;G Mn+m,n+ j (s, ! ) E@;:F342 EC4B=13D;4054
Mn+m, n+ j (s, ! )! = !(" 1)2n+m+ j ! n+m" 1#(s)W (s)
! $
%mj (s)
un+ j (s) µ j
n (s)
+O(! )
&
'
(
)
*
+, j = 1,m , (18)
?B4 ! mj (s) Ñ @EC4B4;514;J m ! 1( )-?@ E@CGB63, E@;:F400<I 58 @EC4B4;514;G '30B4C2@0B3
! (s) D<F4C65D30542 m-I =1C@65 5 j -?@ =1@;H93.
# :F41@2 (6), (7), (15), (17), (18), 3 136P4 N@C2:;< An (s) = (!1)m µ k (s)k=1
m"
C38;@P4-
054 (16) EC5241 D5B
!(q)(t, s, ")! = ! "
m#(s)
"$
Wt
(q)(t, s)%(s)
An (s)
+
"n&1
"q
%(q)(t, s, ")W (s) + O " +
"n
"q
W (s)%(q)(t, s, ")
'
()
*
+,
*
+
,
'
(
) ,
(19)
?B4 ! (q)(t, s, " ) Ñ @EC4B4;514;J m-?@ E@CGB63, 6@1@C<I E@;:F341=G 58 @EC4B4;514;G !(s)
83240@I m-I =1C@65 03 =1C@6: (13).
A4E4CJ, 5=E@;J8:G EC4B=13D;4054 (19) 5 N@C2:;< (8), (14), 58 (13) EC5 B@=131@F0@ 23;<7 !
E@;:F342 5=6@2:O @9406: (12).
.4223 B@638303.
8T,MKM/MH2M 2. @()"A44 !k(t, ") , k = 1, n + m , )%</.%>&,5 7#%)4?)/$4 9()"A45$4
"#%3.1: <%8%?4 (1), (2), 3,24 1)4 (81.23&.1#5>& 18)1#18)1$( (#%.)3)4> (5) 4 "#%3./$
(,21.45$
634 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
!k
( j "1)(0, #)! = !1 0#4 k = j , j = 1, l ,
! k
( j " 1)(0, #)! = !1 0#4 k ! j , j = 1, l , k = 1, n + m ,
(20)
! k
( j " 1)(1, #)! = !1 0#4 k = l + j , j = 1, p,
!k
( j "1)(1, #)! = !0 0#4 k ! l + j , j = 1, p, k = 1, n + m .
+3==2@1C52 @EC4B4;514;J J(! ) = det " ij , i, j = 1, n + m , ?B4 M;42401<
! ij ! = !yj
i " 1( )(0, #), j = 1, n + m, i = 1, l ,
!ij ! = !y j
(i"l"1)(1, #), j = 1, n + m, i = l +1, l + p ,
D =5;: (6) 5 (7) EC4B=13D52< D D5B4
! ij ! = !uj 0
i"1( )(0) + O(#), j = 1, n, i = 1, l ,
!l+i, j ! = !u j0
(i"1)(1) +O(#), j = 1, n, i = 1, p ,
(21)
! i,n+ j ! = !
" n1
" i#1 un+ j,0(0)µ j
i#1(0) +O(" )( ) , j = 1,m, i = 1, l ,
! l + i ,n+ j = !
! n1
! i " 1 un+ j ,0 (1)µ j
i " 1(1) + O(! )( ) exp 1
!
µ j (x)dx
0
1
#
$
%
&
'
(
) = o(! N ), j = 1,m, i = 1, p.
YB4=J N Ñ ;OH@4 031:C3;J0@4 F5=;@.
A4E4CJ = :F41@2 (21) H:B42 C3=6;3B<D31J J(! ) E@ D=42 D@82@P0<2 250@C32 E4CD<7 n
=1@;H9@D. '=4 M15 250@C< 524O1 0:;4D@I E@CGB@6 E@ ! . A@?B3 D =:224 EC@58D4B405I D=47
M157 250@C@D 03 57 3;?4HC35F4=654 B@E@;0405G B@2505C:OL52 F;40@2 J(!) H:B41 1@
EC@58D4B4054, D 6@1@C@2 3;?4HC35F4=6@4 B@E@;04054 52441 035240JK5I E@CGB@6 E@ ! .
";?4HC35F4=6@4 B@E@;04054 250@C3 J0(! ) E@CGB63 n , C3=E@;@P400@?@ D E4CD<7 n =1@;H-
937 5 =1C@637 = 0@24C325 1 ! k ! n1, l +1 ! k ! l + n2 , 52441 035240JK5I E@CGB@6 E@ ! , 136
636 D =5;: (6), (7) E@CGB@6 M;42401@D @EC4B4;514;G J(! ) , 830523OL57 =1C@65 = E4CD@I E@
l -O 5 =1@;H9< = n + 1( )-?@ E@ n + m( )-I, :H<D341 E@ 24C4 C@=13 0@24C3 =1C@65.
#;4B@D314;J0@, B@2505C:OL5I F;40 @EC4B4;514;G J(!) E@;:F341=G : 20@P40542 250@C3
J0 (! ) 03 4?@ 3;?4HC35F4=6@4 B@E@;04054 D(! ), 5 E@M1@2: @EC4B4;514;J J(! ) D E4CD@2 EC5-
H;5P4055 C3D40 EC@58D4B405O @EC4B4;514;4I J0 (! ) 5 D(!):
J(!)! = !J0(!) D(!) + O
1
!"1#1
$
%&
'
() .
YB4=J ! 1 =
m(m " 1)
2
, @EC4B4;514;5 D(! ) 5 J0(! ) EC5 B@=131@F0@ 23;<7 ! 524O1 D5B
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 635
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
J0(! )! = !J0 + O(! ), D(! )! = !(" 1)mn2
1
! #1
$(0) %(0) + O(! )( ), (22)
?B4 J0 ! 0, ! (0) =
s=1
m" un+s(0)µs
n1 (0) 5 !(0) = ! µ1(0),!…!,!µm(0)( ) @1;5F0< @1 0:;G.
A@?B3 B;G @EC4B4;514;G J(!) = :F41@2 (22) EC5 B@=131@F0@ 23;<7 ! 03 @1C4864 0 ! t ! 1
52441 24=1@ EC4B=13D;4054
J(! )! = !(" 1)mn2 J0
! #1
$(0)%(0) 1+O(! )( )! & !0 . (23)
#EC3D4B;5D3 =;4B:OL3G 14@C423.
7M',M&# 2. +(,&- ./012)3)/ (,21.45 10Ð40. 6178% 0#4 81,&%&1?)1 $%2/C ! > 0 7#%-
)4?)/3 9()"A44 !i (t, "), i = 1, n + m , )% 1<"3 [0,1] ,(=3,&.(>&, 384),&.3))/ 4 ./#% -
'%>&,5 91#$(21:
! i (t, " )! = !Ji (t, " )/J(" ), i = 1, n + m , (24)
783 Ji (t, !) Ñ 10#38324&32-, 012(?3))/: 4< J(!) <%$3)1: i -: ,"4 9()8%$3)&%2-)1:
,4, &3$1: #3;3)4: yk(t, ! ), k = 1, n + m, (#%.)3)45 (5).
&124152, F1@ N:0695G !@K5 K t, s, !( ) 5 ?C305F0<4 N:06955 ! i (t, " ), i = 1, n + m , 04
83D5=G1 @1 D<H@C3 N:0B324013;J0@I =5=142< C4K405I :C3D0405G (5).
BM&&# 3. D,24 ./012)3)/ (,21.45 10Ð40, &1 7#%)4?)/3 9()"A44 !i (t, "), i = 1, n + m, )%
1<"3 0 ! t ! 1 4$3>& %,4$0&1&4?3,"43 0#4 ! " 0 0#38,&%.23)45
! i
(q)(t, " )! = !
Ji
(q)(t)
J0
# " n1#q $ 1
(q)(t)
$ (0)
Ji
(n1)(0)
J0
+ O " + " n1#q+1 $ 1
(q)(t)
$ (0)
%
&
'
(
)
* , i = 1, n1 , (25)
! n1+m+i
(q) (t, " )! = !
Jn1+i
(q) (t)
J0
#
" n1
" q
$ 1
(q)(t)
$ (0)
Jn1+m+i
(n1) (0)
J0
+ O " +
" n1+1
" q
$ 1
(q)(t)
$ (0)
%
&
'
(
)
* , i = 1, n2 , (26)
! n1+i
(q) (t, " )! = !" i Jn1
(q)(t)
J0
# i
# (0)
$
" n1$1
" q
# i
(q)(t)
# (0)
+ O " +
" n1
" q
# i
(q)(t)
# (0)
%
&
'
(
)
*
%
&
'
'
(
)
*
*
, i = 1, m, (27)
q = 0,!n + m ! 1,
783 Jk
(q)(t) Ñ 10#38324&32- n -71 01#58"%, 012(?3))/: 4< J0 <%$3)1: k -: ,"4 )%
," ( u10
(q)(t),!u20
(q)(t),!…!,!un0
(q)(t); ! i Ñ 10#38324&32- m-71 01#58"%, 012(?%3$/: 4< 10#383-
24&325 E%)83#$1)8% ! (0) <%$3)1: i -: ,"4 )% ," ( µ1! 1(0),!…!,!µm! 1(0) ; ! i
(q)(t) Ñ
10#38324&32- m-71 01#58"%, "1&1#/: 012(?%3&,5 4< 10#38324&325 ! (0) <%$3)1: i -:
,"4 )% ,"(
exp
1
!
! µ1(x)dx
0
t
"
#
$
%
&
'
(
un+1(t) µ1
q(t)
un+1(0) µ1
n(0)
,!É !,!exp
1
!
! µm(x)dx
0
t
"
#
$
%
&
'
(
un+m(t) µm
q (t)
un+m(0) µm
n (0)
. (28)
636 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
Доказательство. +38;@P52 @EC4B4;514;J Ji
(q)(t, !) , i = 1, n1, q = 0, n + m! 1, E@ M;4-
240132 i -I =1C@65, =@=1@GL4I 58 N:0B324013;J0@I =5=142< C4K405I yk
(q)(t, !) , k = 1, n + m,
q = 0, n + m !1, : C3D0405G (5):
Ji
(q)(t, !)! = ! ("1)i + j
j =1
n
# yj
(q)(t, !)Mij (!) + ("1)i +n+ j
j =1
m
# yn+ j
(q) (t, !)Min+ j (!), i = 1, n1 . (29)
+38;@P52 @EC4B4;514;J Mij (! ) E@ D=42 D@82@P0<2 250@C32 E4CD<7 n ! 1 =1@;H9@D.
RC5 B@=131@F0@ 23;<7 ! B@2505C:OL5I F;40 @EC4B4;514;G Mij (!) E@;:F341=G :20@P40542
250@C3 Jij (! ) E@CGB63 n ! 1, C3=E@;@P400@?@ D E4CD<7 n ! 1 =1@;H937 5 =1C@637 = 0@24C325
1! k ! n1 "1, l ! k ! l + n2 " 1, 03 4?@ 3;?4HC35F4=6@4 B@E@;04054 D(! ). A@?B3 B;G @EC4B4;5-
14;G Mij (!) D E4CD@2 EC5H;5P4055 =EC3D4B;5D@ EC4B=13D;4054
Mij (!)! = !("1)mn2 Jij (!)D(!) + O
1
!#1"1
$
%&
'
()
$
%&
'
()
,
EC5F42 250@C Jij (!) D =5;: (6), (7) EC5 B@=131@F0@ 23;<7 ! 52441 D5B
Jij (!)! = !Jij +O(!). (30)
YB4=J Jij Ñ B@E@;0514;J0<I 250@C M;424013 uj
(i ! 1)(0) D @EC4B4;514;4 J0 . A@?B3, 5=E@;J-
8@D3D (22), (30), @EC4B4;514;J Mij (!) EC5 B@=131@F0@ 23;<7 ! 2@P0@ EC4B=13D51J D D5B4
Mij (! )! = !
(" 1)mn2
! #1
$(0)%(0) J0
Jij
J0
+O(! )
&
'(
)
*+
. (31)
"03;@?5F0@ 1@2:, 636 H<;3 E@;:F403 @94063 (31) B;G Mij (! ), 041C:B0@ E@63831J, F1@
Mi,n+ j (!)! = !
("1)mn2 +n1"i"n2 #(0)$1j (0)
!%1un+ j (0) µ j
n1 (0)
Ji
(n1) +O(!)( ) , i = 1, n1, j = 1,m , (32)
?B4 Ji
(n1) Ñ @EC4B4;514;J n -?@ E@CGB63, 6@1@C<I E@;:F341=G 58 J0 83240@I i -I =1C@65 03
=1C@6: u1
(n1)(0),!É !,!un(n1)(0) ; !1j (0) Ñ @EC4B4;514;J m ! 1( )-?@ E@CGB63, E@;:F342<I 58
@EC4B4;514;G '30B4C2@0B3 ! (0) D<F4C65D30542 E4CD@I =1C@65 5 j -?@ =1@;H93.
/8 C38;@P405G (29) = :F41@2 (31) 5 (32) 037@B52
Ji
(q)(t, !)! = !
("1)mn2 J0#(0)$(0)
!%
×
×
Ji
(q)(t)
J0
!
"n1
"q
#1
(q)(t)
#(0)
Ji
(n1)
J0
+ O " +
"n1+1
"q
#1
(q)(t)
#(0)
$
%&
'
()
'
(
)
$
%
& , i = 1, n1 . (33)
/=E@;J8:G (24), (23) 5 (33), E@;:F342 B;G N:06955 ! i
(q)(t, " ) =@@10@K4054 (25).
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 637
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
/==;4B:G N:06955 !n1+m+i
(q) (t, "), ! n1+ i
(q) (t, " ) 303;@?5F0@ 1@2:, 636 H<;3 E@;:F403 @940-
63 (25), B;G N:0695I ! n1+m+ i
(q) (t, " ), ! i
(q)(t, " ) ;4?6@ E@;:F342 @94065 (26) 5 (27).
.4223 B@638303.
4. "H#/2G2LM$I'M T,MK$G#(/MH2M 2 'RMHI# ,MPMH2O. +3==2@1C52 =50?:;GC0@ D@82:-
L400:O 6C34D:O 83B3F: (1), (2).
7M',M&# 3. +(,&- ./012) 5>&,5 (,21.45 10Ð40. 6178% 0#4 81,&%&1?)1 $%2/C ! > 0
#3;3)43 ,4)7(25#)1 .1<$(=3)) 1: "#%3.1: <%8%?4 (1), (2) )% 1<"3 [0,1] ,(=3,&.(3&,
384),&.3))1 4 ./#%'%3&,5 91#$(21:
y(t, !) = " i#1$i (t, !) + %i#1$l +i (t, !)
i =1
p
&
i =1
l
& Ð
Ð ! l +i (t, " )
i =1
p
# 1
" m ! Kt
( j $1)(1, s, " )h(s)
0
1
% ds+
1
" m ! K (t, s, " )h(s)
0
t
% ds. (34)
Доказательство. +4K4054 y(t, ! ) 6C34D@I 83B3F5 (1), (2) H:B42 5=631J D D5B4
y(t, ! )! = ! ci" i (t, ! ) +
1
! m
K (t, s, ! )h(s)
0
t
#
i=1
n+m
$ ds , (35)
?B4 ci Ñ 0458D4=10<4 E@=1@G00<4. )4E@=C4B=1D400@I EC@D4C6@I 2@P0@ :H4B51J=G, F1@ N:06-
95G y(t, ! ), 83B3003G E@ N@C2:;4 (34), GD;G41=G C4K40542 :C3D0405G (1). (;G @EC4B4;405G ci
E@B=13D52 (35) D 6C34D<4 :=;@D5G (2). A@?B3 = :F41@2 6C34D<7 :=;@D5I (20) @B0@803F0@ E@-
;:F52
ci = ! i"1 , i = 1, l , cl + j ! = !! j "1 "
1
#m ! Kt
( j "1)(1, s, #)h(s)
0
1
$ ds, j = 1, p.
R@B=13D;GG 57 D (35), E@;:F342 (34). QB50=1D400@=1J C4K405G B@638<D341=G @1 EC@15D0@?@.
A4@C423 B@638303.
7M',M&# 4. +(,&- ./012)3)/ (,21.45 10Ð40. 6178% 0#4 81,&%&1?)1 $%2/C ! > 0 825
#3;3)45 y(t, ! ) "#%3.1: <%8%?4 (1), (2) 4 371 0#14<.18)/C )% 1<"3 0 ! t ! 1 ,0#%.3824./
1A3)"4
y(q)(t, ! ) ! " !C # i $1 + %i $1 + max0" t " 1
h(t) +
i=1
n2
&
i=1
n1
&
'
(
)
)
! n$q max
0" t " 1
h(t) +
+ ! n1" q exp "
#t
!
$
%&
'
()
* i " 1 + +i " 1 + max
0, t , 1
h(t) + * n1
i=1
n2
-
i=1
n1
-
$
%
&
'
(
)
.
/
0
0
, q = 0, n + m" 1. (36)
Доказательство. /8 N@C2:; (12), (25) Ð(27) D =5;: :=;@D5I 10Ð30 52442 =;4B:OL54
@94065:
638 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
Kt
(q)(t, s, !) ! " !C!m 1+ !n#1#q exp #
$(t # s)
!
%
&'
(
)*
%
&'
(
)*
, q = 0,!n + m #1, (37)
!i
(q)(t, ") ! # !C 1+ "n1$q exp $
%t
"
&
'(
)
*+
&
'(
)
*+
, i = 1,!n1, q = 0,!n + m$1,
!n1+m+ i
(q) (t, ") ! # !C 1+ "n1$q exp $
%t
"
&
'(
)
*+
&
'(
)
*+
, i = 1,!n2, q = 0,!n + m$1, (38)
! n1+i
(q) (t, " ) ! # !C" i 1+ " n1$1$q exp $
%t
"
&
'(
)
*+
&
'(
)
*+
, i = 1,!m, q = 0,!n + m$ 1.
&9405D3G C4K4054 (34) = :F41@2 (37), (38), E@;:F342 @9406: (36).
A4@C423 B@638303.
A4E4CJ @EC4B4;52 D<C@PB400:O 83B3F:. -48 63657-;5H@ B@E@;0514;J0<7 C3==:PB405I 2<
04 2@P42 =N@C2:;5C@D31J 6C34D<4 :=;@D5G B;G 04D@82:L400@?@ (D<C@PB400@?@) :C3D0405G
L0y! = !h(t), (39)
E@;:F342@?@ 58 (1) EC5 ! = 0 . A3654 B@E@;0514;J0<4 C3==:PB405G 2< 2@P42 E@;:F51J 58
14@C42< 4. /8 @94065 (36) EC5 q = 0 =;4B:41, F1@ EC4B4;J03G N:0695G B;G y(t, ! ) EC5
! " 0 04 H:B41 =@B4CP31J ! n1 , ! n1+1,!É !,!! l " 1, 136 636 6@MNN595401< EC5 ! i , i = 0, n1 ! 1,
5 ! i , i = 0, n2 !1, 524O1 E@CGB@6 O(1), 3 EC5 !n1
,!!n1+1,!É !,!! l "1 Ñ E@CGB@6 O(!) EC5
! " 0 . #;4B@D314;J0@, 6C34D<4 :=;@D5G B;G C4K405G y(t) 04D@82:L400@?@ :C3D0405G (39)
@EC4B4;GO1=G = E@2@LJO 6C34D<7 :=;@D5I (2), =@B4CP3L57 !i , i = 0, n1 !1, 5 !i , i =
= 0, n2 ! 1:
diy
dt i t=0
! = !! i , i = 0,!1,!É !,!n1 " 1 ,
di y
dti t=0
! = !! i , i = 0,!1,!É !,!n2 " 1, (40)
?B4 n1 = l ! m , n2 = n ! n1 = p. )5P4 E@63P42, F1@ :C3D04054 (39) 5 6C34D<4 :=;@D5G (40)
B4I=1D514;J0@ @EC4B4;GO1 D<C@PB400:O 83B3F:.
R@ 303;@?55 = (11), (24) B;G 83B3F5 (39), (40) DD4B42 N:0695O !@K5 5 ?C305F0<4 N:06955
K (t, s)! = !
W (t, s)
W (s)
, ! k(t)! = !
Jk(t)
J0
, k = 1,!É !,!n, (41)
?B4 @EC4B4;514;5 J0 , W (t), W(t, s) DD4B40< D E:0614 1, Jk (t) Ñ @EC4B4;514;J 58 ;422< 3
EC5 q = 0 .
&F4D5B0@, F1@ K t, s( ) Ñ N:0695G !@K5: L0K (t, s) = 0 EC5 0 ! t, s ! 1, K ( j ) s,s( ) = 0,
j = 0, n ! 2, K (n! 1) s,s( ) = 1, 3 ! k (t), k = 1, n , Ñ ?C305F0<4 N:06955 6C34D@I 83B3-
F5 (39), (40):
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 639
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
L0! k (t)! = !0 , ! k
( j )(0)! = !1 EC5 j = k !1, ! k
( j )(0)! = !0 EC5 j ! k " 1, k = 1, n1,
!n1+i
( j ) (1)! = !1 EC5 j = i ! 1, ! n1+i
( j ) (1)! = !0 EC5 j ! i "1, i = 1, n2.
7M',M&# 5. +(,&- ./012)3)/ (,21.45 10Ð40. 6178% "#%3.%5 <%8%?% (39), (40) )% 1<"3
0 ! t ! 1 4$33& 384),&.3))13 #3;3)43
y(t)! = ! ! i"1#i (t ) + $i"1#n1+i (t )
i =1
n2
% " #n1+i (t )!
Kt
(i"1)(1, s)
An(s)
h(s) ds
0
1
&
i =1
n2
%
i =1
n1
% +
K(t, s)
An(s)
h(s) ds
0
t
& .
(42)
Доказательство 14@C42< 303;@?5F0@ B@638314;J=1D: 14@C42< 3.
Y324152, F1@ :=;@D54 50 D =5;: (41), (42) EC5241 D5B ! 0 = " n1 # y (n1)(0) $ 0 .
7M',M&# 6. +(,&- ./012)3)/ (,21.45 10Ð40. 6178% 0#4 81,&%&1?)1 $%2/C ! > 0 ,0#%-
.3824./ 1A3)"4
y(q)(t, ! ) " y (q)(t) ! # !C ! + ! n1" q exp "
$t
!
%
&'
(
)*
+ ! n+1" q%
&'
(
)*
, 0 # t # 1. (43)
Доказательство. R:=1J y(t, ! ) " y(t) = u(t, ! ). A@?B3 58 (1), (2) @10@=514;J0@ u(t, !)
E@;:F342 83B3F:
L!u! = ! f (t, !), f (t, !)! = !O(!), 0 " t " 1,
(44)
diu
dt i t=0
! = !0, i = 0,!n1 ! 1,
diu
dt i t=1
! = !0, i = 0,!p !1,
diu
dti t=0
! = !! i " y(i ), i = n1,!l " 1.
RC5240GG 14@C42: 4 6 6C34D@I 83B3F4 (44), E@;:F342 5=6@2<4 @94065 (43).
A4@C423 B@638303.
A3652 @HC38@2, 58 14@C42< 6 04E@=C4B=1D400@ =;4B:41, F1@ C4K4054 y(t, !) =50?:;GC0@
D@82:L400@I 83B3F5 (1), (2) EC5 =1C42;4055 23;@?@ E3C3241C3 ! 6 0:;O =1C4251=G 6 C4K4-
05O y(t) D<C@PB400@I 83B3F5 (39), (40):
lim
!"0
y(q)(t, !)! = !y(q)(t), 0 # t # 1, q = 0,!1,!É !,!n1 $1, (45)
lim
! " 0
y(q)(t, ! )! = !y(q)(t), 0 < t # 1, q = n1,!É !,!n1 + n2. (46)
Y324152, F1@ EC4B4;J0<4 E4C47@B< (46) 04 GD;GO1=G C3D0@24C0<25 D @6C4=10@=15 1@F65
t = 0 .
5. 8T,MKM/MH2M H#L#/NH'-' $I#LI# ,MPMH2O. ' =5;: 14@C42< 6 5 EC4B4;J0<7 C3-
D40=1D (45), (46) 52442
640 !. ". !"#$%&', (. ). )*+,"-$., ". -. *"/#&'
ISSN 1027-3190. !" #. $%&. '(#)., 2013, &. 65, * 5
lim
! " 0
y(q)(0, ! )! = !y (q)(0), q = 0,!n1 # 1, lim
! " 0
y(n1)(0, ! )! = !y (n1)(0) + $ ,
(47)
y(n1+ j )(0,!)!= !O !" j( ), j = 1,!n2 + m" 1,
?B4 ! Ñ 046@1@C3G D4;5F503, y(t, ! ) Ñ C4K4054 83B3F5 (1), (2), y(t) Ñ C4K4054 D<C@PB40-
0@I 83B3F5 (39), (40). ' M1@2 =;:F34 H:B42 ?@D@C51J, F1@ 83B3F3 (1), (2) 52441 03F3;J0<I
=63F@6 n1-?@ E@CGB63 D 1@F64 t = 0 (! GD;G41=G D4;5F50@I 03F3;J0@?@ =63F63 C4K405G 83-
B3F5 (1), (2)).
&EC4B4;52 ! Ñ D4;5F50: =63F63. /8 N@C2:; (12), (25) Ð(27) = :F41@2 N@C2:;< (41) 5
:=;@D5I 10Ð40 037@B52 =;4B:OL54 3=52E1@15F4=654 EC4B=13D;405G:
Kt
( j )(1, s, !)! = !!m K ( j )(1, s)
An (s)
+O(!)
"
#$
%
&'
, j = 0, n2 ( 1, (48)
!i
(n1)(0, ")! = !!i
(n1)(0) #
$1
(n1)(t)
$(0)
!i
(n1)(0) + O("), i = 1, n1 ,
!n1+m+i
(n1) (t, ")! = !!n1+i
(n1) (0) #
$1
(n1)(0)
$(0)
!n1+m+i
(n1) (0) +O("), i = 1, n2 , (49)
! n1+i
(n1) (0, " )! = !" i#1 $ i
(n1)(0)
$ (0)
+O(" )
%
&
'
(
)
* , i = 1,m .
A4E4CJ 58 (34) D =5;: (48), (49) E@;:F342
y(n1)(0, ! )! = !y(n1)(0) "
# 1
(n1)(0)
# (0)
y(n1)(0) " $ n1 %&+ O(! )'( . (50)
Q=;5 D<E@;040< :=;@D5G 10Ð40, 1@ 58 (53) 037@B52 N@C2:;: 03F3;J0@?@ =63F63
! =
" 1
(n1)(0)
" (0)
# n1
$ y(n1)(0)( ) . (51)
'<G=052 73C3614C C@=13 EC@58D@B0<7 n1 + j E@CGB6@D C4K405G 83B3F5 (1), (2), D
@6C4=10@=15 1@F65 t = 0. /8 (25) Ð(27) 037@B52
! i
(n1+ j ) (0, " )! = !#
1
" j
$ 1
(n1+ j ) (0)
$ (0)
! i
(n1)(0) + O "( )( ) , i = 1,!n1, j = 1,!m+ n2 # 1,
! n1+m+ i
(n1+ j ) (0, " )! = !#
1
" j
$ 1
(n+ j1)
(0)
$ (0)
! n1+ i
(n1) (0) + O "( )( ) , i = 1,!n2, j = 1,!m+ n2 # 1, (52)
!n1+i
(n1+ j ))(0, ")! = !
"i#1
" j
$ i
(n1+ j )(0)
$(0)
+O "( )
%
&'
(
)*
, i = 1,m, j = 1,m + n2 #1.
"#/%RA&A/UQ#!/Q &VQ) !/ +QTQ)/W !+"Q'&X Y "("U/ # )"U".Z)$% #! "U!&% É 641
ISSN 1027-3190. !"#. $%&. '(#)., 2013, &. 65, * 5
R@B=13D;GG (48), (52) D N@C2:;: (34), 037@B52 H@;44 1@F0:O @9406: B;G y(n1+ j )(0, ! ),
F42 (47):
y(n1+1)(0, ! )! = !O 1
!
"
#$
%
&'
,!y(n1+2)(0, ! )! = !O 1
! 2
"
#$
%
&'
,!É !,!y(n+m( 1)(0, ! )! = !O 1
! n2+m( 1
"
#$
%
&'
.
/=7@BG 58 (51) 5 73C3614C3 C@=13 EC@58D@B0<7, 836;OF342, F1@ =50?:;GC0@ D@82:L4003G
6C34D3G 83B3F3 D @6C4=10@=15 1@F65 t = 0 52441 03F3;J0<I =63F@6 n1-?@ E@CGB63, F1@ GD;G-
41=G @B0@I 58 @=@H400@=14I 58:F342@I 83B3F5.
Замечание. *=130@D;400<I C@=1 EC@58D@B0<7 E@8D@;G41 =D4=15 6C34D:O 83B3F: (1), (2) 6
83B3F4 !@K5 = 03F3;J0<2 =63F6@2, F1@ D =D@O @F4C4BJ =;:P51 @=0@D@I B;G E@=1C@405G 3=52E-
1@15F4=657 C38;@P405I 046@1@C<7 =50?:;GC0@ D@82:L400<7 6C34D<7 83B3F = 03F3;J0<25
=63F6325.
1. 64C1)1. F. G. & 83D5=52@=15 C4K405I B5NN4C40953;J0<7 :C3D0405I @1 23;@?@ E3C3241C3 // %31. =H. Ð 1948. Ð
22 (64), [ 2. Ð #. 193 Ð204.
2. E4;4" H. I., J>,&3#)4" J. F. +4?:;GC0@4 D<C@PB4054 5 E@?C305F0<I =;@I B;G ;504I0<7 B5NN4C40953;J-
0<7 :C3D0405I = 23;<2 E3C3241C@2 // *=E475 231. 03:6. Ð 1957. Ð 12, [ 5. Ð #. 3 Ð122.
3. E%,42-3.% F. K. "=52E1@1563 C4K405I 046@1@C<7 6C34D<7 83B3F B;G 6D385;504I0<7 :C3D0405I = 23;<2 E3C3-
241C@2 EC5 =13CK4I EC@58D@B0@I // (@6;. ") ###+. Ð 1958. Ð 123, [ 4. Ð #. 583 Ð586.
4. K1712>L1. G. G., H4ϴ-,"4: M. F. "=52E1@15F4=654 241@B< D 14@C55 04;504I0<7 6@;4H305I. Ð %.: )3: -
63, 1974. Ð 503 =.
5. H4=3)"1 D. @., N1<1. G. O. (5NN4C40953;J0<4 :C3D0405G = 23;<2 E3C3241C@2 5 C4;36=395@00<4 6@;4H3-
05G. Ð %.: )3:63, 1979. Ð 154 =.
6. I$%)%243. H. I. "=52E1@15F4=654 241@B< D 14@C55 =50?:;GC0@ D@82:L400<7 5014?C@-B5NN4C40953;J0<7
=5=142 // /==;4B@D305G E@ 5014?C@-B5NN4C40953;J0<2 :C3D0405G2. Ð SC:084: /;52, 1962. Ð A. 2. Ð #. 21 Ð39.
7. J1$1. P. F. 'D4B4054 D @HL:O 14@C5O =50?:;GC0<7 D@82:L405I. Ð %.: )3:63, 1981. Ð 399 =.
8. K(&(<1. E. @. *?;@D@I E@?C30=;@I D =24K300<7 =50?:;GC0@ D@82:L400<7 83B3F37 B;G ?5E4CH@;5F4=657 :C3D-
0405I D1@C@?@ E@CGB63 // (@6;. ") ###+. Ð 1977. Ð 235, [ 5. Ð #. 997 Ð1000.
9. E4;4" H. I., J>,&3#)4" J. F. & 03F3;J0@2 =63F64 B;G 04;504I0<7 B5NN4C40953;J0<7 :C3D0405I, =@B4CP3-
L57 23;<I E3C3241C // (@6;. ") ###+. Ð 1960. Ð 132, [ 6. Ð #. 1242 Ð1245.
10. B%,/$1. B. F., G(#7%L/2 Q. G. "=52E1@15F4=6@4 E@D4B4054 C4K405I ;504I0<7 =50?:;GC0@ D@82:L400<7
@HL57 C38B4;400<7 6C34D<7 83B3F, 524OL57 03F3;J0<I =63F@6 // *6C. 231. P:C0. Ð 2003. Ð 55, [ 11. Ñ
#. 1496 Ð1508.
11. B%,/$1. B. F., G(#7%L/2 Q. G. "=52E1@15F4=654 @94065 C4K405G =50?:;GC0@ D@82:L400@I 6C34D@I 83B3F5 =
03F3;J0<2 =63F6@2 B;G ;504I0<7 B5NN4C40953;J0<7 :C3D0405I // (5NN4C409. :C3D0405G. Ð 2004. Ð 40, [ 4. Ð
#. 597 Ð607.
R@;:F40@ 08.07.12
|
| id | nasplib_isofts_kiev_ua-123456789-165321 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1027-3190 |
| language | Russian |
| last_indexed | 2025-11-25T23:47:37Z |
| publishDate | 2013 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Касымов, К.А. Нургабыл, Д.Н. Уаисов, А.Б. 2020-02-13T09:08:54Z 2020-02-13T09:08:54Z 2013 Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных / К.А. Касымов, Д.Н. Нургабыл, А.Б. Уаисов // Український математичний журнал. — 2013. — Т. 65, № 5. — С. 629–641. — Бібліогр.: 11 назв. — рос. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/165321 917.926 Встановлено асимптотичні оцінки розв'язків сингулярно збурених крайових задач з початковими стрибками. We establish asymptotic estimates for the solutions of singularly perturbed boundary-value problems with initial jumps. ru Інститут математики НАН України Український математичний журнал Статті Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных Asymptotic estimates for the solutions of boundary-value problems with initial jump for linear differential equations with small parameter in the coefficients of derivatives Article published earlier |
| spellingShingle | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных Касымов, К.А. Нургабыл, Д.Н. Уаисов, А.Б. Статті |
| title | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| title_alt | Asymptotic estimates for the solutions of boundary-value problems with initial jump for linear differential equations with small parameter in the coefficients of derivatives |
| title_full | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| title_fullStr | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| title_full_unstemmed | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| title_short | Асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| title_sort | асимптотические оценки решения краевой задачи с начальным скачком для линейных дифференциальных уравнений с малым параметром при производных |
| topic | Статті |
| topic_facet | Статті |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/165321 |
| work_keys_str_mv | AT kasymovka asimptotičeskieocenkirešeniâkraevoizadačisnačalʹnymskačkomdlâlineinyhdifferencialʹnyhuravneniismalymparametrompriproizvodnyh AT nurgabyldn asimptotičeskieocenkirešeniâkraevoizadačisnačalʹnymskačkomdlâlineinyhdifferencialʹnyhuravneniismalymparametrompriproizvodnyh AT uaisovab asimptotičeskieocenkirešeniâkraevoizadačisnačalʹnymskačkomdlâlineinyhdifferencialʹnyhuravneniismalymparametrompriproizvodnyh AT kasymovka asymptoticestimatesforthesolutionsofboundaryvalueproblemswithinitialjumpforlineardifferentialequationswithsmallparameterinthecoefficientsofderivatives AT nurgabyldn asymptoticestimatesforthesolutionsofboundaryvalueproblemswithinitialjumpforlineardifferentialequationswithsmallparameterinthecoefficientsofderivatives AT uaisovab asymptoticestimatesforthesolutionsofboundaryvalueproblemswithinitialjumpforlineardifferentialequationswithsmallparameterinthecoefficientsofderivatives |