Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action
We use the scheme of the classic least-squares method for the construction of an approximate pseudosolution of a linear ill-posed boundary-value problem with pulse action for a system of ordinary differential equations in the critical case. The pseudosolution obtained is represented in the form of p...
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| Date: | 2010 |
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| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2010
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860508892664954880 |
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| author | Chuiko, S. M. Чуйко, С. М. |
| author_facet | Chuiko, S. M. Чуйко, С. М. |
| author_sort | Chuiko, S. M. |
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| datestamp_date | 2020-03-18T19:39:51Z |
| description | We use the scheme of the classic least-squares method for the construction of an approximate pseudosolution of a linear ill-posed boundary-value problem with pulse action for a system of ordinary differential equations in the critical case. The pseudosolution obtained is represented in the form of partial sums of a generalized Fourier series. |
| first_indexed | 2026-03-24T02:32:26Z |
| format | Article |
| fulltext |
UDK 517.9
S. M. Çujko (Slavqn. ped. un-t)
METOD NAYMEN|ÍYX KVADRATOV
V TEORYY NEKORREKTNO POSTAVLENNÁX
LYNEJNÁX KRAEVÁX ZADAÇ
S YMPUL|SNÁM VOZDEJSTVYEM
The classical least-squares method is used for the construction of an approximate pseudosolution of ill-
posed linear boundary-value problem with pulse influence for a system of ordinary differential equations
in the critical case. This pseudosolution is presented in the form of partial sums of the generalized
Fourier series.
Sxemu klasyçnoho metodu najmenßyx kvadrativ vykorystano dlq pobudovy nablyΩenoho psev-
dorozv’qzku linijno] nekorektno postavleno] krajovo] zadaçi z impul\snym vplyvom dlq systemy
zvyçajnyx dyferencial\nyx rivnqn\ u krytyçnomu vypadku u vyhlqdi çastkovyx sum uzahal\ne-
noho rqdu Fur’[.
1. Postanovka zadaçy. Yssleduem zadaçu o naxoΩdenyy reßenyj [1 – 5]
z t z t z tn( ) ( ), , ( )( ) ( )= …( )col 1 , z C a bj
i I
( )( ) , \⋅ ∈ [ ] { }{ }1 τ , j = 1, 2, … , n,
system¥ ob¥knovenn¥x dyfferencyal\n¥x uravnenyj s pereklgçenyqmy
dz
dt
A t z f t= +( ) ( ) , t i≠ τ , i = 1, 2, … , p, (1)
udovletvorqgwyx kraevomu uslovyg
Lz( )⋅ = α , α ∈ Rm . (2)
Zdes\ A t( ) = a tij ( )( ) – ( )n n× -matryca y f t( ) = col f t( )( )1( , … , f tn( )( )) — vek-
tor-stolbec
aij ( )⋅ , f C a bi
I
( )( ) , \⋅ ∈ [ ] { }{ }1 τν , i, j = 1, 2, … , n, ν = 1, 2, … , p,
Lz( )⋅ — lynejn¥j ohranyçenn¥j vektorn¥j funkcyonal vyda
Lz zi
i
p
( ) ( )⋅ = ⋅
=
∑ �
0
,
pryçem
� i i i i i
mz C C R( ) : , ,⋅ × … × →+ +τ τ τ τ1 1 ,
i = 0, 1, 2, … , p – 1, τ0 = a ,
� p p p
mz C b C b R( ) : , ,⋅ [ ] × … × [ ] →τ τ
— lynejn¥e ohranyçenn¥e funkcyonal¥. Pust\ W t0( ) — normal\naq
(W a0( ) = In ) fundamental\naq matryca system¥ (1) na otrezke a; τ1[ ], a W t1( )
— fundamental\naq matryca system¥ (1) na otrezke τ τ1 2;[ ] , kotoraq udov-
letvorqet uslovyg W0 1( )τ = W1 1( )τ . Suwestvovanye normal\noj (W0 1( )τ =
= W1 1( )τ ) fundamental\noj matryc¥ system¥ (1) na otrezke τ τ1 2;[ ] sleduet
yz nev¥roΩdennosty fundamental\n¥x matryc system¥ (1) na otrezkax a; τ1[ ]
© S. M. ÇUJKO, 2010
690 ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
METOD NAYMEN|ÍYX KVADRATOV V TEORYY NEKORREKTNO POSTAVLENNÁX … 691
y τ τ1 2;[ ] . Takym obrazom, normal\naq ( X a0( ) = In ) fundamental\naq matryca
X t0( ) odnorodnoj çasty system¥ (1) predstavyma v vyde
X t
W t t a W a I
W t t
n
0
0 1 0
1 1 2
( )
( ), ; , ( ) ,
( ), ;
=
∈[ ] =
∈
τ
τ τ[[ ] =
……… ………… …………………
∈
, ( ) ( ),
( ), ;
W W
W t tp p
0 1 1 1τ τ
τ bb W Wp p p[ ] =
−, ( ) ( ).τ τ1 1
Matryca X t0( ) neprer¥vna na otrezke a b;[ ] y udovletvorqet odnorodnoj
çasty system¥ (1); pry πtom obwee reßenye odnorodnoj çasty system¥ (1)
ob¥knovenn¥x dyfferencyal\n¥x uravnenyj s pereklgçenyqmy predstavymo v
vyde
z t c X t c( , ) ( )= 0 , c Rn∈ .
Lemma. Pry uslovyy PQ ≠ 0 odnorodnaq çast\ zadaçy (1), (2) ymeet se-
mejstvo reßenyj [4] z t c( , ) = X t c( ) , c Rn∈ , predstavymoe normyrovannoj
X i( )τ
0
0+( = 1) fundamental\noj matrycej
X t
p
X t P I t a
p
X t P
Q
Q( )
( ) , ; ,
( )
( )
( )
=
∈[ [1
1
0
0
0
1
0
0
1
� τ
��
�
I t
p
X t P I tQ
p
, ; ,
( ) ,( )
∈[ [
……………… ……………
∈
τ τ1 2
0
0
1
ττ p b; ,[ ]
P
P
P
P
Q
Q
Q
Q
p
=
……
( )
( )
( )
0
1
,
�I
I
I
I
n
n
n
= …
,
odnorodnoj çasty zadaçy (1), (2). Zdes\
Q X X Xp= ⋅ ⋅ … ⋅[ ]� � �0 0 1 0 0( ) ( ) ( )
— postoqnnaq matryca razmera m n p× +( )( )1 ,
P R N QQ
n p: ( )( )+ →1
— n p( )+( 1 × n p( )+ )1 -mernaq matryca-ortoproektor [2], PQ
i( ) , i = 0, 1, … , p,
— n n p× +( )( )1 -mern¥e bloky ortoproektora PQ ,
X P I pi Q
i
0 0 00
0( ) ( )τ + =� , max ( ) ( )rank rankP I P IQ
i
Q
i� �= 0 .
Pry uslovyy PQ∗ ≠ 0 zadaça (1), (2) razreßyma dlq tex y tol\ko tex neod-
norodnostej
col f t f tn1( ), , ( )( )…( ) , f C a bi
j I
( )( ) , \⋅ ∈ [ ] { }{ }1 τ , i = 1, 2, … , n,
y α ∈ Rm , kotor¥e udovletvorqgt uslovyg
P K f sQ∗ − [ ] ⋅{ } =α L ( ) ( ) 0 , (3)
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
692 S. M. ÇUJKO
hde PQ∗ — (m × m)-mernaq matryca-ortoproektor: R N Qm → ∗( ) .
Sleduq tradycyonnoj klassyfykacyy kraev¥x zadaç [2], sluçaj PQ∗ ≠ 0
nazovem krytyçeskym. Reßenye zadaçy (1), (2) v πtom sluçae predstavymo s po-
mow\g obobwennoho operatora Hryna [4]
G f s t
X t K f s t t a
X
( ); ( )
( ) ( ) ( ), ; ,
α
γ τ
[ ] =
+ [ ] ∈[ [0 0 1
0(( ) ( ) ( ), ; ,t K f s t t
X
γ τ τ1 1 2
0
+ [ ] ∈[ [
……………………… …………
(( ) ( ) ( ), ; ,t K f s t t bp pγ τ+ [ ] ∈[ ]
(4)
hde
col ( , , , )γ γ γ0 1 … p = Q K f s+ − [ ] ⋅{ }α L ( ) ( ) ,
K f s t( ) ( )[ ] = X t X s f s ds
a
t
0 0
1( ) ( ) ( )−∫
— operator Hryna zadaçy Koßy dlq system¥ (1) s pereklgçenyqmy.
PredpoloΩym, çto uslovye (3) ne v¥polneno, pry πtom zadaça (1), (2) qv-
lqetsq nekorrektno postavlennoj [6]; pry postroenyy reßenyj πtoj zadaçy v
vyde [2]
z t X t K f s t( , ) ( ) ( ) ( )ξ ξ= + [ ] , ξ ∈ Rn ,
poluçaem n-parametryçeskoe semejstvo lynejno nezavysym¥x psevdoreßenyj
z t c X t c G f s t+ = + [ ]( , ) ( ) ( ); ( )α , c Rn∈ .
Postroennoe takym obrazom psevdoreßenye z t c+ ( , ) udovletvorqet dyfferen-
cyal\noj systeme (1) y mynymyzyruet normu nevqzky ∆0 = �z+ ⋅ −( ) α v
kraevom uslovyy (2). Cel\g dannoj stat\y qvlqetsq postroenye pryblyΩen-
n¥x psevdoreßenyj kraevoj zadaçy (1), (2) v vyde çastyçn¥x summ obobwennoho
rqda Fur\e [7 – 9]
z t c
t c t a
t c t†( , )
( ) , ; ,
( ) , ; ,
=
⋅ ∈[ ]
⋅ ∈[ ]
ϕ τ
ϕ τ τ
0 1
1 1 2
………… …………
⋅ ∈[ ]
ϕ τ( ) , ; ,t c t bp p
c
c
c
c
R
p
p k=
……
∈ +
0
1 1( ) .
Zdes\ ϕ1( )t , ϕ2( )t , … , ϕk t( ) , …M— systema lynejno nezavysym¥x vektor-funk-
cyj,
ϕ ϕ ϕ ϕ( ) ( ) ( ) ( )t t t tk= …( )1 2
— ( )n k× -mernaq matryca. Po analohyy s metodom naymen\ßyx kvadratov [7,
8] potrebuem mynymyzacyy norm¥ nevqzky, dyfferencyal\noj system¥ (1) y
kraevoho uslovyq (2)
∆1( , )ϕ c = A t z f t
dz
dt
L
( ) ( )†
†
+ −
2
2
+ Lz
Rm
†( ) min⋅ − →α
2
,
hde [10, s. 123]
A t z f t
dz
dt
L
( ) ( )†
†
+ −
2
2
=
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
METOD NAYMEN|ÍYX KVADRATOV V TEORYY NEKORREKTNO POSTAVLENNÁX … 693
= A t z f t
dz
dt
A t z f t
dz
da
b
( ) ( ) ( ) ( )†
†
†
†
+ −
+ −∫
∗
tt
dt
,
Lz
Rm
†( )⋅ − α
2
= L Lz z† †( ) ( )⋅ −( ) ⋅ −( )∗
α α .
Pry fyksyrovannoj matryce ϕ( )t mynymum funkcyy ∆1( , )ϕ c suwestvuet,
tak kak neprer¥vnaq poloΩytel\naq funkcyq ∆1( , )ϕ c dostyhaet mynymuma.
Neobxodymoe uslovye mynymyzacyy funkcyy ∆1( , )ϕ c pryvodyt k uravnenyg
A t t t A t t c t c f( ) ( ) ( ) ( ) ( ) ( )ϕ ϕ ϕ ϕ[ ] − ′[ ]{ } [ ] − ′[ ] +∗ (( )t dt
a
b
[ ]{ }∫ +
+M L Lϕ ϕ α( ) ( )⋅[ ]{ } ⋅[ ] − [ ]{ } =∗ c 0 .
Zdes\
A t t
A t t O O
O A t t O
O O A t
( ) ( )
( ) ( )
( ) ( )
(
ϕ
ϕ
ϕ[ ] =
…
…
… … … …
… )) ( )ϕ t
,
f t
f t
f t
f t
( )
( )
( )
( )
[ ] =
…
— n p k p( ) ( )+ × +( )1 1 -mernaq matryca y n p( )+( )1 -mern¥j vektor-stolbec,
ϕ ϕ ϕ ϕ( ) ( ), ( ), , ( )t t t t[ ] = …[ ]diag , ′[ ] = ′ ′ … ′[ ]ϕ ϕ ϕ ϕ( ) ( ), ( ), , ( )t t t t
— n p k p( ) ( )+ × +( )1 1 -mern¥e matryc¥,
Lϕ ϕ ϕ ϕ( ) ( ), ( ), , ( )⋅[ ] = ⋅ ⋅ … ⋅[ ]diag � � �0 1 p , α α α α[ ] = …[ ]col
— m p k p( ) ( )+ × +( )1 1 -mernaq matryca.
Oboznaçaq k p k p( ) ( )+ × +( )1 1 -mern¥e matryc¥ Hrama
Γ ϕ ϕ ϕ ϕ( ) ( ) ( ) ( ) ( ) ( )⋅[ ] = [ ] − ′[ ]{ } [ ] −∗∫ A t t t A t t
a
b
′′[ ]{ }ϕ ( )t dt ,
Γ L L Lϕ ϕ ϕ( ) ( ) ( )⋅[ ] = ⋅[ ]{ } ⋅[ ]{ }∗ ,
dlq naxoΩdenyq pryblyΩennoho psevdoreßenyq
z t c t c†( , ) ( )= [ ]ϕ
kraevoj zadaçy (1), (2) pryxodym k uravnenyg
Γ Γϕ ϕ( ) ( )⋅[ ] + ⋅[ ]{ }{ }L c = Lϕ α( )⋅[ ]{ }[ ]∗ – A t t t f t dt
a
b
( ) ( ) ( ) ( )ϕ ϕ[ ] − ′[ ]{ } [ ]∗∫ ,
odnoznaçno razreßymomu otnosytel\no vektora c R p k∈ +( )1
pry uslovyy nev¥-
roΩdennosty matryc¥
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
694 S. M. ÇUJKO
Γ Γϕ ϕ( ) ( )⋅[ ] + ⋅[ ]{ }L
y opredelqgwemu psevdoreßenye
z t c t†( , ) ( ) ( ) ( )∗ −= [ ] ⋅[ ] + ⋅[ ]{ }{ }ϕ ϕ ϕΓ Γ L 1 ×
× Lϕ α ϕ ϕ( ) ( ) ( ) ( ) ( )⋅[ ] [ ] − [ ] − ′[ ]{ } [ ]∗ ∗A t t t f t dt
a
b
∫∫
. (5)
V sylu edynstvennosty poluçennoe psevdoreßenye obespeçyvaet mynymum
funkcyy ∆1( , )ϕ c y zavysyt ot v¥bora matryc¥ ϕ( )t .
Teorema. Pry uslovyy PQ ≠ 0 , PQ∗ ≠ 0 zadaça (1), (2) razreßyma dlq
tex y tol\ko tex neodnorodnostej
f t f t f tn( ) ( ), , ( )( ) ( )= …( )col 1 , f C a bi
j I
( )( ) , \⋅ ∈ [ ] { }{ }1 τ , i = 1, 2, … , n,
y α ∈ Rm , kotor¥e udovletvorqgt uslovyg (3). Esly v sluçae PQ ≠ 0 ,
PQ∗ ≠ 0 uslovye (3) ne v¥polneno, to dlq lgboho natural\noho çysla k y
fyksyrovannoj matryc¥ ϕ( )t pry uslovyy
det ( ) ( )Γ Γϕ ϕ⋅[ ] + ⋅[ ]{ }{ } ≠L 0 (6)
formula (5) opredelqet nayluçßee sredy funkcyj vyda z t c†( , ) = ϕ( )t c[ ] ,
c R p k∈ +( )1 , psevdoreßenye zadaçy (1), (2), mynymyzyrugwee nevqzku ∆1( , )ϕ c .
Prymer 1. V kaçestve yllgstracyy k lemme postroym pryblyΩennoe psev-
doreßenye
z C I( ) ; \⋅ ∈ −[ ] [ ]{ }1
11 1 τ , τ1 0= ,
nekorrektno postavlennoj kraevoj zadaçy
dz
dt
= 1, z( )− =1 0 , z z( ) ( )0 1 1− = . (7)
Matryca
Q =
1 0
0 0
opredelqet proektor¥
P PQ Q= =
∗
0 0
0 1
.
Poskol\ku PQ∗ ≠ 0 , ymeet mesto krytyçeskyj sluçaj, pry πtom uslovye (3) ne
v¥polneno. Tradycyonnoe odnoparametryçeskoe semejstvo psevdoreßenyj
z t c X t c G f s t+ = + [ ]( , ) ( ) ( ); ( )α , c R∈ 1 ,
predstavymoe matrycej
X t
t
t
( )
, ; ,
, ; ,
=
∈ −[ [
∈[ ]
0 1 0
1 0 1
y operatorom Hryna
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
METOD NAYMEN|ÍYX KVADRATOV V TEORYY NEKORREKTNO POSTAVLENNÁX … 695
G f s t K f s t t( ); ( ) ( ) ( )α[ ] = [ ] = + 1, t ∈ −[ ]1 1; ,
ymeet nevqzku ∆0 4= . Dlq postroenyq psevdoreßenyq z t c†( , )∗
zadaçy (7) po
formule (5) vospol\zuemsq systemoj funkcyj
ϕ1
1
2
( )t = , ϕ π2 2( ) sint t= , ϕ π3 2( ) cost t= ,
ϕ π4 4( ) sint t= , ϕ π5 4( ) cost t= .
Matryc¥ Hrama, opredelqem¥e πtoj systemoj, ymegt vyd
Γ ϕ( )⋅[ ] = diag 0 4 4 16 16 0 4 4 16 162 2 2 2 2 2 2 2, , , , , , , , ,π π π π π π π π ,
Γ Lϕ( )⋅[ ]{ } =
1 0 2 0 2 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
2 0 2 0 2 0 0 0 0 0
0 0 00 0 0 0 0 0 0 0
2 0 2 0 2 0 0 0 0 0
0 0 0 0 0
1
2
0
1
2
0
1
2
0 0 0 0 0 0 0 0 0 0
0 00 0 0 0
1
2
0 1 0 1
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0
1
2
0 1 0 1
.
Uslovye (6) pry πtom v¥polneno:
det ( ) ( )Γ Γϕ ϕ π⋅[ ] + ⋅[ ]{ }{ } = ≠L 8388608 016 .
Poluçennoe po formule (5) psevdoreßenye z t c†( , )1
∗ ≡ 0, t ∈ −[ ]1 1; , ymeet ne-
vqzku ∆1 1( , )ϕ c∗ = 3, men\ßug nevqzky ∆0 4= tradycyonnoho psevdore-
ßenyq.
PredpoloΩym dalee, çto v sluçae PQ ≠ 0 , PQ∗ ≠ 0 trebovanye (3) v¥pol-
neno; pry πtom zadaça (1), (2) razreßyma, odnako v¥çyslenye operatora Hryna
K f s t( ) ( )[ ] zadaçy Koßy z a c( ) = , c Rn∈ , yly obobwennoho operatora Hryna
G f s t( ); ( )α[ ] kraevoj zadaçy (1), (2) nevozmoΩno v πlementarn¥x funkcyqx. V
πtom sluçae estestvenno voznykaet zadaça o nayluçßem pryblyΩenyy z t( , )ϕ k
reßenyg kraevoj zadaçy (1), (2), mynymyzyrugwem nevqzku
∆2
2
2
( , ) ( ) ( , ) ( )
( , )
ϕ ϕ
ϕ
c A t z t f t
dz t
dt L
= + − + Lz
Rm( , ) min⋅ − →ϕ α 2
pry fyksyrovannoj matryce ϕ( )t . ∏to pryblyΩenye ymeet vyd
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
696 S. M. ÇUJKO
z t( , )ϕ = ϕ ϕ ϕ( ) ( ) ( )t[ ] ⋅[ ] + ⋅{ }{ }−Γ Γ L 1 ×
× Lϕ α ϕ ϕ( ) ( ) ( ) ( ) ( )⋅[ ] [ ] − [ ] − ′[ ]{ } [ ]∗ ∗A t t t f t dt
a
b
∫∫
. (8)
Takym obrazom, pryxodym k sledstvyg yz dokazannoj teorem¥.
Sledstvye. Esly v sluçae PQ ≠ 0 , PQ∗ ≠ 0 trebovanye (3) v¥polneno, to
dlq lgboho fyksyrovannoho çysla k y fyksyrovannoj matryc¥ ϕ( )t pry uslo-
vyy (6) nayluçßee pryblyΩenye z t( , )ϕ k reßenyg kraevoj zadaçy (1), (2),
mynymyzyrugwee nevqzku ∆2( , )ϕ c , ymeet vyd (8).
Zameçanye. Dokazannoe sledstvye spravedlyvo y v nekrytyçeskom sluçae,
kohda PQ ≠ 0 , PQ∗ ≠ 0 , tak kak trebovanye (3) v¥polnqetsq dlq lgb¥x neod-
norodnostej kraevoj zadaçy (1), (2).
Prymer 2. V kaçestve yllgstracyy k sledstvyg postroym nayluçßee pry-
blyΩenye
z C I( ) ;⋅ ∈ −
{ }{ }1
1
1
2
1
2
τ , τ1 0= ,
k reßenyg korrektno postavlennoj kraevoj zadaçy
dz
dt
= 1, z −
=
1
2
0 , z z
1
2
1
2
1
− −
= . (9)
Dlq postroenyq nayluçßeho pryblyΩenyq k reßenyg zadaçy (9) vospol\-
zuemsq systemoj funkcyj
ϕ1 1( )t = , ϕ2( )t t= , ϕ3
2( )t t= .
Matryc¥ Hrama, opredelqem¥e πtoj systemoj, ymegt vyd
Γ ϕ( )⋅[ ] =
0 0 0 0 0 0
0 2 0 0 0 0
0 0
3
8
0 0 0
0 0 0 0 0 0
0 0 0 0 2 0
0 0 0 0 00
3
8
,
Γ Lϕ( )⋅[ ]{ } =
−
− −
−
2 1
1
2
0 0 0
1
1
2
1
4
0 0 0
1
2
1
4
1
8
0 0 0
0 0 0 1
11
2
1
4
0 0 0
1
2
1
4
1
8
0 0 0
1
4
1
8
1
16
.
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
METOD NAYMEN|ÍYX KVADRATOV V TEORYY NEKORREKTNO POSTAVLENNÁX … 697
Uslovye (6) pry πtom v¥polneno
det ( ) ( )Γ Γϕ ϕ⋅[ ] + ⋅[ ]{ }{ }L =
2 1
1
2
0 0 0
1
5
2
1
4
0 0 0
1
2
1
4
67
24
0 0 0
0 0 0 1
1
2
1
4
0 0 0
1
2
−
− −
−
99
4
1
8
0 0 0
1
4
1
8
131
48
=
512
9
0≠ .
Poluçennoe po formule (8) nayluçßee pryblyΩenye z t( , )ϕ k reßenyg krae-
voj zadaçy (9) ymeet vyd
z t
t t
t t
( , )
, ; ,
, ;
ϕ =
+ ∈ −
− ∈
1
2
1
2
0
1
2
0
1
2
,,
y qvlqetsq toçn¥m reßenyem zadaçy (9).
1. Samojlenko A. M., Perestgk N.A. Dyfferencyal\n¥e uravnenyq s ympul\sn¥m vozdejst-
vyem. – Kyev: Vywa ßk., 1987. – 287 s.
2. Boichuk A. A., Samoilenko A. M. Generalized inverse operators and Fredholm boundary-value
problems. – Utrecht; Boston: VSP, 2004. – XIV + 317 p.
3. Sčhwabik S. Differential equations with interface conditions// Čas. pestov
∨
. mat. – 1980. –
Roč.105. – P. 391 – 410.
4. Bojçuk A. A., Çujko S. M. Obobwenn¥j operator Hryna ympul\snoj kraevoj zadaçy s pere-
klgçenyqmy // Nelinijni kolyvannq. – 2007. – 10, # 1. – S. 51 – 65.
5. Çujko S. M. Operator Hryna kraevoj zadaçy s ympul\sn¥m vozdejstvyem // Dyfferenc.
uravnenyq. – 2001. – 37, # 8. – S. 1132 – 1135.
6. Tyxonov A. N. O reßenyy nekorrektno postavlenn¥x zadaç y o metode rehulqryzacyy //
Dokl. AN SSSR. – 1963. – 151, # 3. – S. 501 – 504.
7. Kr¥lov N. M. Yzbrann¥e trud¥. – Kyev: Yzd-vo AN USSR, 1961. – T. 1. – 268 s.
8. Axyezer N. Y. Lekcyy po teoryy approksymacyy. – M.: Nauka, 1965. – 408 s.
9. Çujko S. M. Metod najmenßyx kvadrativ v teori] nekorektno postavlenyx krajovyx zadaç
// Visn. Ky]v. nac. un-tu im. T. Íevçenka. – 2007. – # 7. – S. 51 – 53.
10. Kantorovyç L. V., Akylov H. P. Funkcyonal\n¥j analyz. – M.: Nauka, 1977. – 744 s.
Poluçeno 27.03.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2010, t. 62, # 5
|
| id | umjimathkievua-article-2899 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:32:26Z |
| publishDate | 2010 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/2a/79e306db9829ac56bbabc439a9228a2a.pdf |
| spelling | umjimathkievua-article-28992020-03-18T19:39:51Z Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action Метод наименьших квадратов в теории некорректно поставленных линейных краевых задач с импульсным воздействием Chuiko, S. M. Чуйко, С. М. We use the scheme of the classic least-squares method for the construction of an approximate pseudosolution of a linear ill-posed boundary-value problem with pulse action for a system of ordinary differential equations in the critical case. The pseudosolution obtained is represented in the form of partial sums of a generalized Fourier series. Схему класичного мечоду найменших квадратів використано для побудови наближеного псев-дорозв'язку лінійної некоректно поставленої крайової задачі з імпульсним впливом для системи звичайних диференціальних рівнянь у кри тичному випадку у вигляді часткових сум узагальненого ряду Фур'є. Institute of Mathematics, NAS of Ukraine 2010-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/2899 Ukrains’kyi Matematychnyi Zhurnal; Vol. 62 No. 5 (2010); 690–697 Український математичний журнал; Том 62 № 5 (2010); 690–697 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/2899/2549 https://umj.imath.kiev.ua/index.php/umj/article/view/2899/2550 Copyright (c) 2010 Chuiko S. M. |
| spellingShingle | Chuiko, S. M. Чуйко, С. М. Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title | Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title_alt | Метод наименьших квадратов в теории некорректно поставленных линейных краевых задач с импульсным воздействием |
| title_full | Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title_fullStr | Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title_full_unstemmed | Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title_short | Least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| title_sort | least-squares method in the theory of ill-posed linear boundary-value problems with pulse action |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/2899 |
| work_keys_str_mv | AT chuikosm leastsquaresmethodinthetheoryofillposedlinearboundaryvalueproblemswithpulseaction AT čujkosm leastsquaresmethodinthetheoryofillposedlinearboundaryvalueproblemswithpulseaction AT chuikosm metodnaimenʹšihkvadratovvteoriinekorrektnopostavlennyhlinejnyhkraevyhzadačsimpulʹsnymvozdejstviem AT čujkosm metodnaimenʹšihkvadratovvteoriinekorrektnopostavlennyhlinejnyhkraevyhzadačsimpulʹsnymvozdejstviem |