Polynomial quasisolutions of linear second-order differential-difference equations
The second-order scalar linear difference-differential equation (LDDE) with delay $$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$ is considered. This equation is investigated with the use of the method of polynomial quasisolutions based on the presentation of an unknown function in...
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| Date: | 2008 |
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| Format: | Article |
| Language: | Ukrainian English |
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Institute of Mathematics, NAS of Ukraine
2008
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/3144 |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509182959026176 |
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| author | Ermolaeva, P. G. Cherepennikov, V. B. Єрмолаєва, П. Г. Черепенніков, В. Б. |
| author_facet | Ermolaeva, P. G. Cherepennikov, V. B. Єрмолаєва, П. Г. Черепенніков, В. Б. |
| author_sort | Ermolaeva, P. G. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:46:36Z |
| description | The second-order scalar linear difference-differential equation (LDDE) with delay
$$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$
is considered. This equation is investigated with the use of the method of polynomial
quasisolutions based on the presentation of an unknown function in the form of polynomial $x(t)=\sum_{n=0}^{N}x_n t^n.$
After
the substitution of this function into the initial equation, the residual $\Delta(t)=O(t^{N-1}),$ appears.
The exact analytic representation of this residual is obtained.
The close connection is demonstrated between the LDDE with varying coefficients and the model LDDE with constant coefficients whose
solution structure is determined by roots of a characteristic quasipolynomial. |
| first_indexed | 2026-03-24T02:37:03Z |
| format | Article |
| fulltext |
UDK 517.929
V. B. Çerepennykov, P. H. Ermolaeva
(Yn-t dynamyky system y teoryy upravlenyq SO RAN, Yrkutsk, Rossyq)
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX
DYFFERENCYAL|NO-RAZNOSTNÁX URAVNENYJ
VTOROHO PORQDKA
The second-order scalar linear difference-differential equation (LDDE) with delay
˙̇ ( ) ( ) ˙( )x t p p t x t+ +0 1 = ( ) ( ) ( )a a t x t f t0 1 1+ − +
is considered. This equation is investigated with the use of the method of polynomial quasisolutions
based on the presentation of an unknown function in the form of polynomial x t( ) = x tn
n
n
N
= 0∑ . After
the substitution of this function into the initial equation, the residual ∆ ( )t = O t N( )−1 appears. The
exact analytic representation of this residual is obtained. The close connection is demonstrated between
the LDDE with varying coefficients and the model LDDE with constant coefficients whose solution
structure is determined by roots of a characteristic quasipolynomial.
Rozhlqda[t\sq skalqrne linijne dyferencial\no-riznyceve rivnqnnq (LDRR) zahaqnoho typu
druhoho porqdku
˙̇ ( ) ( ) ˙( )x t p p t x t+ +0 1 = ( ) ( ) ( )a a t x t f t0 1 1+ − + .
V qkosti metodu doslidΩennq vykorystano metod polinomial\nyx kvazirozv’qzkiv, wo ©runtu-
[t\sq na zobraΩenni nevidomo] funkci] u vyhlqdi polinoma x t( ) = x tn
n
n
N
= 0∑ . Pry pidstanovci
ci[] funkci] u poçatkove rivnqnnq z’qvlq[t\sq vidxyl ∆ ( )t = O t N( )−1
, dlq qkoho otrymano
toçne analityçne zobraΩennq. Vidmiçeno tisnyj zv’qzok LDRR zi zminnymy koefici[ntamy z mo-
del\nym LDRR zi stalymy koefici[ntamy, struktura rozv’qzku qkoho vyznaça[t\sq korenqmy xa-
rakterystyçnoho kvazipolinoma.
1. Vvedenye. Vo mnohyx oblastqx nauky y texnyky, osobenno v takyx, kak avto-
matyka, telemexanyka, πlektroradyosvqz\, radyonavyhacyq y druhyx, v kaçestve
matematyçeskyx modelej yspol\zugtsq dyfferencyal\n¥e uravnenyq s otklo-
nqgwymsq arhumentom. Kak pravylo, takye uravnenyq qvlqgtsq nelynejn¥my.
No v sylu toho, çto lynejn¥e dyfferencyal\n¥e uravnenyq s otklonqgwymsq
arhumentom sravnytel\no lehçe poddagtsq yssledovanyqm y teoryq takyx urav-
nenyj dostatoçno razrabotana, pry reßenyy razlyçn¥x teoretyçeskyx y oso-
benno prykladn¥x zadaç nelynejn¥e uravnenyq zamenqgtsq lynejn¥my. Nay-
bolee yssledovann¥my na sehodnqßnyj den\ qvlqgtsq lynejn¥e dyfferency-
al\no-raznostn¥e uravnenyq (LDRU), kohda otklonenye (zapazd¥vanye) arhu-
menta postoqnno. Otmetym zdes\ v pervug oçered\ rabot¥ A. D. M¥ßkysa [1],
∏. Pynny [2], R. Bellmana y K. L. Kuka [3], N. V. Azbeleva, V. P. Maksymova,
L. F. Raxmatulynoj [4], V. P. Rubanyka [5].
Pry yssledovanyy LDRU v osnovnom rassmatryvagtsq dve naçal\n¥e zadaçy:
naçal\naq zadaça s naçal\noj funkcyej, kohda na naçal\nom mnoΩestve
tem yly yn¥m sposobom zadaetsq naçal\naq funkcyq, poroΩdagwaq reßenye
yskomoj zadaçy, y naçal\naq zadaça s naçal\noj toçkoj, kohda ywetsq klas-
syçeskoe reßenye, podstanovka kotoroho v ysxodnoe uravnenye obrawaet eho v
toΩdestvo.
V nastoqwej stat\e vnymanye udelqetsq naçal\noj zadaçe dlq LDRU zapaz-
d¥vagweho typa vtoroho porqdka s naçal\noj toçkoj, kohda naçal\n¥e uslovyq
zadagtsq v naçal\noj toçke. Yzvestno, çto dlq LDRU s postoqnn¥my koπffy-
cyentamy suwestvuet beskoneçnoe mnoΩestvo analytyçeskyx reßenyj, oprede-
© V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA, 2008
140 ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 141
lqem¥x kornqmy xarakterystyçeskoho kvazypolynoma. V sluçae peremenn¥x
koπffycyentov vopros¥ razreßymosty LDRU v klasse analytyçeskyx
funkcyj na sehodnqßnyj den\ ostagtsq otkr¥t¥my.
V rabote dlq yssledovanyq reßenyj LDRU s polynomyal\n¥my koπffy-
cyentamy yspol\zuetsq metod polynomyal\n¥x kvazyreßenyj (PK-reßenyj)
[6 – 8], kotor¥j osnovan na predstavlenyy neyzvestnoj funkcyy v vyde polyno-
ma nekotoroj stepeny. Podstanovka πtoho polynoma v ysxodnoe uravnenye pry-
vodyt k nevqzke, dlq kotoroj poluçeno toçnoe analytyçeskoe predstavlenye.
2. Postanovka zadaçy. Rassmotrym skalqrnoe dyfferencyal\no-raznost-
noe uravnenye zapazd¥vagweho typa vtoroho porqdka
˙̇ ( ) ( ) ˙( )x t p p t x t+ +0 1 = ( ) ( ) ( )a a t x t f t0 1 1+ − + , t R∈ , (2.1)
hde
f t( ) = f tn
n
F
n
=
∑
0
. (2.2)
Na osnovanyy obwej teoryy dyfferencyal\n¥x uravnenyj (bez otklonenyq
arhumenta) budem yssledovat\ naçal\nug zadaçu s naçal\noj toçkoj dlq urav-
nenyq (2.1), zadav v toçke t = 0 naçal\n¥e uslovyq
x x( )0 0= , ˙( )x x0 1= . (2.3)
Kak otmeçaetsq v [7], v πtom sluçae prymenyt\ klassyçeskyj metod neoprede-
lenn¥x koπffycyentov, kohda reßenye predstavlqetsq v vyde rqda x t( ) =
= x tn
n
n=
∞∑ 0
, ne udaetsq, poskol\ku postroyt\ rekurrentnug formulu dlq op-
redelenyq neyzvestn¥x koπffycyentov xn ne predstavlqetsq vozmoΩn¥m.
Vvedem polynom
x t( ) = x tn
n
n
N
=
∑
0
, t R∈ ; N F= + 2 . (2.4)
V πtom sluçae
˙̇ ( )x t =
n
N
n
nn n x t
=
−∑ −
0
21( ) , ˙( )x t =
n
N
n
nnx t
=
−∑
0
1,
(2.5)
x t( )− 1 =
n
N
n
nx t
=
∑ −
0
1( ) =
n
N
n
nx t
=
∑
0
˜ .
Zdes\
x̃n =
i
N n
i
n i
i
n iC x
=
−
+ +∑ −
0
1( ) = x C xn
i
N n
n i
i
n i+
=
−
+ +∑
1
, (2.6)
hde Cp
q = ( )−1 q
p
qC , Cp
q =
p
q p q
!
! ( )!−
— bynomynal\n¥e koπffycyent¥.
Provedem analyz razmernostej polynomov (2.4) y (2.5) po otnoßenyg k urav-
nenyg (2.1). Proyzvodnaq ˙̇ ( )x t predstavlqetsq polynomom stepeny N – 2, po-
lynom ( ) ˙( )p p t x t0 1+ ymeet stepen\ N, a polynom ( ) ( )a a t x t0 1+ — stepen\ N +
+ 1. Tohda dlq sohlasovanyq stepenej ukazann¥x polynomov vvedem nevqzku
∆ ( )t = f t f t f tN
N
N
N
N
N
−
−
+
++ +1
1
1
1
, (2.7)
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
142 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
hde fN −1 , fN y fN +1 — nekotor¥e neyzvestn¥e koπffycyent¥.
Zapyßem sledugwee uravnenye:
˙̇ ( ) ( ) ˙( ) ( )x t p p t x t t+ + −0 1 ∆ = ( ) ( ) ( )a a t x t f t0 1 1+ − + . (2.8)
Prynqv vo vnymanye (2.2) y (2.4), opredelym funkcyg f t( ) v vyde
f t( ) =
n
F
n
nf t t
=
∑ +
0
∆ ( ) =
n
N
n
nf t
=
+
∑
0
1
. (2.9)
Zdes\ fi = fi , i = 1, F .
S uçetom (2.8) y vvedenn¥x oboznaçenyj rassmotrym naçal\nug zadaçu
˙̇ ( ) ( ) ˙( )x t p p t x t+ +0 1 = ( ) ( ) ( )a a t x t f t0 1 1+ − + , t R∈ ,
(2.10)
x x( )0 0= , ˙( )x x0 1= ,
kotoraq po otnoßenyg k naçal\noj zadaçe (2.1) – (2.3) qvlqetsq vozmuwennoj
na nevqzku ∆ ( )t .
Opredelenye 2.1. Zadaçu (2.10) budem naz¥vat\ sohlasovannoj po razmer-
nosty polynomov otnosytel\no zadaç (2.1) – (2.3).
Celesoobraznost\ vvedenyq naçal\noj zadaçy (2.10) opredelqetsq tem, çto,
kak budet pokazano v p. 4, v πtom sluçae dlq ee yssledovanyq moΩno vospol\zo-
vat\sq metodom neopredelenn¥x koπffycyentov.
Ytak, zadaça sostoyt v ustanovlenyy uslovyj suwestvovanyq y sposobov
naxoΩdenyq neyzvestn¥x koπffycyentov fN −1 , fN y fN +1 , poroΩdagwyx
reßenye naçal\noj zadaçy (2.10) v vyde polynoma (2.4).
Opredelenye 2.2. Esly suwestvuet polynom stepeny N = F + 2
x t( ) =
n
N
n
nx t
=
∑
0
, t R∈ , (2.11)
toΩdestvenno udovletvorqgwyj naçal\noj zadaçe (2.10), to πtot poly-
nom))budem naz¥vat\ polynomyal\n¥m kvazyreßenyem (PK-reßenyem) zada-
çy (2.1) – (2.3).
Zameçanye 2.1. Poskol\ku stepen\ polynoma x t( ) ravna F + 2, πto pozvo-
lqet v¥brat\ stepen\ polynoma f t( ) v (2.2) v zavysymosty ot Ωelaemoj stepeny
polynoma x t( ), dobavlqq k f t( ) sootvetstvugwee çyslo nulev¥x çlenov.
PreΩde çem yssledovat\ vopros¥, svqzann¥e s naxoΩdenyem PK-reßenyj,
pryvedem nekotor¥e neobxodym¥e v dal\nejßem svedenyq.
3. Predvarytel\n¥e rezul\tat¥. V oboznaçenyqx zadaçy (2.1) – (2.3) ras-
smotrym snaçala naçal\nug zadaçu dlq model\noho LDRU s postoqnn¥my ko-
πffycyentamy
˙̇ ( ) ˙( )x t p x t+ 0 = a x t0 1( )− , t R∈ , x x( )0 0= , ˙( )x x0 1= , (3.1)
poskol\ku dlq πtoj zadaçy metodom ∏jlera mohut b¥t\ poluçen¥ toçn¥e çast-
n¥e reßenyq. Dejstvytel\no, polahaq x t( ) = Cekt
y podstavlqq πtu funkcyg
v ysxodnoe uravnenye, pryxodym k xarakterystyçeskomu kvazypolynomu (XK)
k p k a e k2
0 0+ = −
. (3.2)
Yssleduem strukturu kornej (3.2), vospol\zovavßys\ heometryçeskym pryemom.
Vvedem funkcyy
y k1( ) = a e k
0
− , y k2( ) = k p k2
0+ . (3.3)
Hrafyky πtyx funkcyj pry razlyçn¥x znaçenyqx koπffycyentov a0 y p0
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 143
peresekagtsq v toçkax, kotor¥e sootvetstvugt vewestvenn¥m kornqm XK. Pry
πtom vozmoΩn¥ sluçay kak kasanyq hrafykov, tak y otsutstvye toçek pereseçe-
nyq.
Rassmotrym bolee podrobno sluçay kasanyq hrafykov. V toçke kasanyq k =
= k∗ v¥polnqgtsq sledugwye uslovyq:
a) y k1( )∗ = y k2( )∗ — ravenstvo znaçenyj funkcyj;
b)
dy k
dk k k
1( )
= ∗
=
dy k
dk k k
2( )
= ∗
— ravenstvo kasatel\n¥x k hrafykam funkcyj.
Uçyt¥vaq (3.3), yz pervoho uslovyq naxodym
a0 = e k p kk∗
∗ ∗+( )2
0 , (3.4)
a yz vtoroho —
− − ∗a e k
0 = 2 0k p∗ + . (3.5)
Podstavlqq (3.4) v (3.5), ymeem
p0 = –
k k
k
∗ ∗
∗
+
+
2 2
1
. (3.6)
S uçetom πtoho v¥raΩenyq yz (3.4) poluçaem
a0 = –
k e
k
k
∗
∗
∗
+
2
1
. (3.7)
Perepyßem (3.6) y (3.7) v vyde
a k0( )∗ = −
+
∗
∗
∗k e
k
k2
1
, p k0( )∗ = − +
+
∗ ∗
∗
k k
k
2 2
1
. (3.8)
Tohda kaΩdomu dopustymomu znaçenyg k = k∗ sootvetstvugt opredelenn¥e
znaçenyq a k0( )∗ y p k0( )∗ . Ysklgçaq parametr k∗ v formulax (3.8), poluçaem
kryvug (rys. 1), kotoraq razbyvaet ploskost\ a k0( )( , p k0( )) na neskol\ko
oblastej, xarakteryzugwyx çyslo vewestvenn¥x kornej XK
1
.
Cyfroj I oboznaçen¥ oblasty znaçenyj a0 y p0 , pry kotor¥x XK (3.2)
ymeet odyn vewestvenn¥j koren\, cyfroj II — oblasty dvux vewestvenn¥x
kornej, cyfroj III oboznaçena oblast\ trex vewestvenn¥x kornej. Kryvaq,
oboznaçennaq cyfroj IV, sootvetstvuet odnomu dvukratnomu vewestvennomu
korng. Pry πtom suwestvuet oblast\ znaçenyj a0 y p0, pry kotor¥x XK ne
ymeet vewestvenn¥x kornej. Oboznaçym ee cyfroj V.
Vvedem sledugwye opredelenyq.
Opredelenye 3.1. Reßenyq dyfferencyal\no-raznostnoho uravnenyq (3.1),
poroΩdaem¥e vewestvenn¥my kornqmy xarakterystyçeskoho kvazypolyno-
ma (3.2) v oblastqx I – IV, budem naz¥vat\ domynantn¥my reßenyqmy.
Opredelenye 3.2. Reßenyq dyfferencyal\no-raznostnoho uravnenyq (3.1)
v oblasty V, poroΩdaem¥e kompleksn¥m kornem xarakterystyçeskoho kvazy-
polynoma (3.2) s mynymal\noj mnymoj çast\g, budem naz¥vat\ predomynant-
n¥my reßenyqmy.
Celesoobraznost\ vvedenyq πtyx opredelenyj y yzuçenye takyx reßenyj
svqzano s tem, çto v pryloΩenyqx realyzugtsq, kak pravylo, ymenno πty reße-
nyq. Krome toho, pry stremlenyy zapazd¥vanyq k nulg, reßenyq, opredelen-
n¥e takym obrazom, stremqtsq k reßenyqm yzuçaem¥x uravnenyj, v sluçae, koh-
1
Na rys. 1 v skobkax ukazan¥ znaky vewestvenn¥x kornej XK.
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
144 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
da zapazd¥vanye ravno nulg.
Rys. 1
4. Osnovn¥e rezul\tat¥. Vernemsq k naçal\noj zadaçe (2.10).
Podstavlqq (2.4), (2.5) y (2.9) v (2.10) y pryravnyvaq koπffycyent¥ pry ody-
nakov¥x stepenqx t, ymeem
( )n nxn− 1 =
a x p x f n
a x p n i x f n N
i
i n i i n i n
0 0 0 1 0
0
1
2 1 2
2
1 3
˜ , ,
˜ ( ) , ,
− + =
− − −[ ] + ≤ ≤
=
− − − − −∑
(4.1)
0 = a xi n i
i
˜ − −
=
∑ 2
0
1
–
i n N
i n ip n i x
= − −
− −∑ − −
1
1
11( ) + fn−2, N n N+ ≤ ≤ +1 2 ,
0 = a x fN N1 1˜ + + , n N= + 3.
Dlq reßenyq postavlennoj v p. 2 zadaçy v¥razym neyzvestn¥e koπffycyen-
t¥ xn , n = 1, N , polynomyal\noho kvazyreßenyq (2.11) çerez neyzvestn¥e ko-
πffycyent¥ fN −1 , fN y fN +1 polynoma (2.9) y najdem uslovyq, pry kotor¥x
poslednye mohut b¥t\ opredelen¥.
S uçetom (2.5) y (2.6) dlq s = 0 1, N − posledovatel\no yz (4.1) pryxodym k
cepoçke ravenstv
a xNN N + fN +1 = 0,
a xN N N−1, + a xN N N− − −1 1 1, + fN = 0,
a xN N N−2, + a xN N N− − −2 1 1, + a xN N N− − −2 2 2, + fN −1 = 0,
(4.2)
………………………………………………………………………
a xN s N N− , + a xN s N N− − −, 1 1 + … + a xN s N s N s− − −, + fN s+1– = 0,
……………………………………………………………………
hde
aNN = a1;
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 145
aN N− −1 1, = a1,
aN N−1, = a0 + a CN1
1 – Np1,
aN N− −2 2, = a1,
aN N− −2 1, = a0 + a CN1 1
1
− – N p−( )1 1,
aN N−2, = a C a C NpN N0
1
1
2
0+ − ;
……………………………………………………………
aN s N s− −, = a1,
aN s N s− − +, 1 = a a C N s p0 1 1
1
11+ − − +( ) ,
aN s N s− − +, 2 = a C a C N s pN s N s0 2
1
1 2
2
02− + − ++ − − +( ) ,
……………………………………………………………
aN s N s k− − +, =
i
i N s k
k ia C
=
− +
− +∑
0
1
1 – N s k N s k k− +( ) − + −( )1 3δ , 3 ≤ ≤k s ,
symvol Kronekera
δ p
n =
1
0
, ,
, .
n p
n p
=
≠
Lemma 4.1 [8]. Obwyj πlement posledovatel\nosty xn n
N{ } =1, poroΩdae-
moj sootnoßenyqmy (4.2), opredelqetsq formuloj
xN s− =
i
s
N s N s i N s iK f
=
− − + + − +∑
0
1, , (4.3)
hde
KN s N s− −, = −
− −
1
aN s N s,
,
KN s N s− −, = −
− − =
−
− − + − + −∑1
1a
a K
N s N s i
s r
N s N s i N s i N r
,
, , , s r> .
Vernemsq k formulam (4.1). Perepyßem pervug formulu s uçetom (2.5) v
vyde
2 2x = a x p x f0 0 0 1 0˜ − + = a x x x xN
N0 0 1 2 1( ( ) )− + −…+ − – p x f0 1 0+ .
Otsgda
−a x0 0 = ( )− −a p x0 0 1 + ( )− −a x0 22 – a x0 3 + … + ( )−1 0
N
Na x + f0
yly
x0 = 1 0
0
1+
p
a
x + − +
1 2
0
2a
x + x3 – x4 + … + ( )− +1 1N
Nx –
f
a
0
0
. (4.4)
Oboznaçym
V1 = 1 0
0
+ p
a
, V2 = − +1 2
0a
,
V3 1= , V4 1= − , … , Vn
n= − +( )1 1
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
146 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
y perepyßem (4.4) tak:
x0 = V x V x V x V x V x
f
aN N1 1 2 2 3 3 4 4
0
0
+ + + +…+ − . (4.5)
Na osnovanyy lemm¥ 4.1 v¥razym koπffycyent¥ xn çerez koπffycyent¥
fi , i = 1 1, N + , sledugwym obrazom:
x0 =
i
N
i iK f
=
+∑
0
0 1, =
k
N
N k N kf K
=
+ − −∑
0
1 0, ,
x1 =
i
N
i iK f
=
−
+ +∑
0
1
1 1 2, =
k
N
N k N kf K
=
−
+ − −∑
0
1
1 1, ,
x2 =
i
N
i iK f
=
−
+ +∑
0
2
2 2 3, =
k
N
N k N kf K
=
−
+ − −∑
0
2
1 2, ,
………………………………………………………
xm =
k
N m
m i m m iK f
=
−
+ + +∑
0
1, =
k
N m
N k m N kf K
=
−
+ − −∑
0
1 , ,
………………………………………………………
xN = K fNN N +1.
Podstavym najdenn¥e takym obrazom koπffycyent¥ xn v formulu (4.5):
x0 = V f K
k
N
N k N k1
0
1
1 1
=
−
+ − −∑ , + V f K
k
N
N k N k2
0
2
1 2
=
−
+ − −∑ , + …
…+ V f Km
k
N m
N k m N k
=
−
+ − −∑
0
1 , + … + V K fN NN N +1 –
f
a
0
0
.
Hruppyruq v πtom ravenstve slahaem¥e pry odynakov¥x koπffycyentax fN i+ ,
ymeem
x0 =
i
N k
i i N k N k
k
N
V K f
=
−
− + −
=
−
∑∑
1
1
0
1
, –
f
a
0
0
. (4.6)
Oboznaçym
K N k0, − =
i
N k
i i N kV K
=
−
−∑
1
,
y perepyßem (4.6) v vyde
x0 =
k
N
N k N kK f
=
−
− + −∑
1
1
0 1, –
f
a
0
0
. (4.7)
Poluçenn¥e rezul\tat¥ pozvolqgt sformulyrovat\ sledugwug teoremu.
Teorema 4.1. Pust\ zadana naçal\naq zadaça (2.1) – (2.3).
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 147
Tohda πta zadaça ymeet edynstvennoe PK-reßenye v vyde x t( ) =
n
N
n
nx t=∑ 0
,
N = F + 2, s nevqzkoj ∆ ( )t = O t N( )−1
, esly opredelytel\
D =
K K K
K K K
K K K
N N N
N N N
N N N
0 0 1 0 2
1 1 1 1 2
0 0 1 0 2
, , ,
, , ,
, , ,
− −
− −
− −
ne raven nulg.
Dokazatel\stvo. S uçetom (2.3) rassmotrym formulu (4.3) pry s = N y
s = N – 1:
x0 =
k
N
N k N kK f
=
− + −∑
0
0 1, , x1 =
k
N
N k N kK f
=
−
− + −∑
0
1
1 1, . (4.8)
Zapyßem ravenstva (4.7) y (4.8) v vyde lynejnoj system¥ otnosytel\no neyz-
vestn¥x koπffycyentov fN i+ −1 , i = 0 2, :
K fN N0 1, + + K fN N0 1, − + K fN N0 2 1, − − = W1,
K fN N1 1, + + K fN N1 1, − + K fN N1 2 1, − − = W2 , (4.9)
K fN N0 1, + + K fN N0 1, − + K fN N0 2 1, − − = W3 ,
hde
W1 = x0 –
k
N
N k N kK f
=
− + −∑
3
0 1, ,
W2 = x1 –
k
N
N k N kK f
=
−
− + −∑
3
1
1 1, ,
W3 = x0 –
k
N
N k N kK f
=
−
− + −∑
3
1
0 1, +
f
a
0
0
.
Vvedem sledugwye oboznaçenyq:
D1 =
W K K
W K K
W K K
N N
N N
N N
1 0 1 0 2
2 1 1 1 2
3 0 1 0 2
, ,
, ,
, ,
− −
− −
− −
,
D2 =
K W K
K W K
K W K
N N
N N
N N
0 1 0 2
1 2 1 2
0 3 0 2
, ,
, ,
, ,
−
−
−
,
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
148 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
D3 =
K K W
K K W
K K W
N N
N N
N N
0 0 1 1
1 1 1 2
0 0 1 3
, ,
, ,
, ,
−
−
−
.
Poskol\ku po uslovyg teorem¥ opredelytel\ system¥ (4.9) otlyçen ot nulq,
reßaq πtu systemu metodom Kramera, naxodym koπffycyent¥ fN −1 , fN y
fN +1:
fN −1 =
D
D
3 , fN =
D
D
2 , fN +1 =
D
D
1
.
Tohda yz cepoçky ravenstv (4.2) posledovatel\no opredelqgtsq koπffycyent¥
PK-reßenyq xn y, sledovatel\no, samo PK-reßenye v vyde (2.11).
Najdenn¥e koπffycyent¥ fN −1 , fN , fN +1 pozvolqgt zapysat\ v qvnom
vyde v¥raΩenye dlq nevqzky
∆ ( )t =
i
N i
N if t
=
− +
− +∑
0
2
1
1 =
D
D
t N3 1− +
D
D
t N2 +
D
D
t N2 1+
.
Teorema dokazana.
5. Prymer¥. Prymer 5.1. Rassmotrym naçal\nug zadaçu dlq LDRU s pos-
toqnn¥my koπffycyentamy
˙̇ ( ) ˙( )x t x t+ 4 = − −x t( )1 , t R∈ ,
(5.1)
x( )0 = x0 = 1, ˙( )x 0 = x1 = 0.
V oboznaçenyqx zadaçy (3.1) zdes\ p0 = 4, a0 = – 1. ∏tym znaçenyqm na rys. 1
sootvetstvuet toçka, leΩawaq v oblasty suwestvovanyq dvux vewestvenn¥x
kornej XK (3.2): k1 = – 0,43134 y k2 = – 1,22241. Sootvetstvenno, domynantn¥e
reßenyq uravnenyq (5.1) opys¥vagtsq formuloj
x t( ) = C e t
1
0 43134− , + C e t
2
1 22241− ,
,
hde C1 y C2 — nekotor¥e konstant¥. Sledovatel\no, dlq naxoΩdenyq kons-
tant C1 y C2 , udovletvorqgwyx naçal\noj zadaçe (5.1), dostatoçno zadanyq
dvux naçal\n¥x uslovyj. S uçetom zadann¥x v (5.1) uslovyj naxodym
C1 = 1,54526, C2 = – 0,54526.
Tohda domynantnoe reßenye zadaçy (5.1) prymet vyd
x t( ) = 1,54526e t−0 43134, – 0,54526e t−1 22241,
. (5.2)
Predstavym (5.2) v vyde rqda Maklorena
x t( ) = 1 – 0 26363 2, t + 0 14532 3, t – 4 85006 10 2 4, × − t +
+ 1 22101 10 2 5, × − t – 2 51298 10 3 6, × − t + 4 40405 10 4 7, × − t + … . (5.3)
V πtom sluçae v sylu opredelenyq 2.1 naçal\naq zadaça
˙̇ ( ) ˙( )x t x t+ 4 = − −x t( )1 + ∆N t( ) , t R∈ , x( )0 = x0 = 1,
˙( )x 0 = x1 = 0,
hde ∆N t( ) = f tN
N
−
−
1
1 + f tN
N
, budet sohlasovannoj po razmernosty polynomov
otnosytel\no zadaçy (5.1).
Budem yskat\ PK-reßenyq v vyde
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 149
x tN ( ) =
n
N
n
nx t
=
∑
0
. (5.4)
V rezul\tate rasçetov dlq N = 5, 6, 7 b¥ly najden¥ sledugwye PK-reße-
nyq y sootvetstvugwye ym nevqzky:
x t5( ) = 1 – 0 26278 2, t + 0 14514 3, t – 0 04962 4, t + 0 01501 5, t ,
∆5( )t = 0 17553 4, t + 0 01501 5, t ;
x t6( ) = 1 – 0 26292 2, t + 0 14511 3, t – 0 04844 4, t + 0 01248 5, t –
– 3 08463 10 3 6, × − t ,
∆6( )t = – 0 04304 5, t – 3 08463 10 3 6, × − t ;
x t7( ) = 1 – 0 26356 2, t + 0 1453 3, t – 0 04846 4, t – 0 0122 5, t –
– 2 56881 10 3 6, × − t + 5 40328 10 4 7, × − t ,
∆7( )t = 8 77808 10 3 6, × − t + 5 40328 10 4 7, × − t .
Sravnyvaq x t5( ), x t6( ) y x t7( ) s x t( ) v (5.3), pryxodym k v¥vodu, çto s uvely-
çenyem stepeny polynoma koπffycyent¥ PK-reßenyj pryblyΩagtsq k koπf-
fycyentam rqda Maklorena dlq x t( ), a samy PK-reßenyq posledovatel\no
prytqhyvagtsq k domynantnomu reßenyg, çto nahlqdno yllgstryruetsq hra-
fykamy na rys. 2.
Rys. 2
Prymer 5.2. Yssleduem sledugwug naçal\nug zadaçu:
˙̇ ( )x t + ( ) ˙( )4 + t x t = ( ) ( )− + −1 1t x t , t R∈ ,
(5.5)
x( )0 = x0 = 1, ˙( )x 0 = x1 = 0.
Poskol\ku znaçenyq a0 y p0 model\noho uravnenyq (5.1) πtoj zadaçy oprede-
lqgt toçku, sootvetstvugwug oblasty suwestvovanyq dvux vewestvenn¥x kor-
nej, yspol\zuem dva zadann¥x naçal\n¥x uslovyq.
Zapyßem zadaçu, sohlasovannug po razmernosty polynomov:
˙̇ ( )x t + ( ) ˙( )4 + t x t = ( ) ( )− + −1 1t x t + ∆N t( ),
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
150 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
(5.6)
x( )0 = x0 = 1, ˙( )x 0 = x1 = 0,
hde
∆N t( ) = f tN
N
−
−
1
1 – f tN
N + f tN
N
+
+
1
1
.
Na osnovanyy rezul\tatov, yzloΩenn¥x v p. 3, b¥ly v¥çyslen¥ PK-reßenyq
zadaçy (5.6) dlq N = 7, 8, 9 y sootvetstvugwye nevqzky ∆N t( ) :
x t7( ) = 1 – 0 25643 2, t + 0 23648 3, t – 0 03915 4, t – 0 03308 5, t +
+ 0 01469 6, t + 2 86412 10 3 7, × − t ,
∆7 ( )t = 0 22402 6, t + 0 02827 7, t – 2 86412 10 3 8, × − t ;
x t8( ) = 1 – 0 25643 2, t + 0 23675 3, t – 0 03987 4, t – 0 03261 5, t + 0 01559 6, t +
+ 6 15026 10 4 7, × − t – 1 68075 10 3 8, × − t ,
∆8 ( )t = 3 56365 10 4 7, × − t – 0 02919 8, t + 1 68075 10 3 9, × − t ;
x t9( ) = 1 – 0 25645 2, t + 0 23675 3, t – 0 03987 4, t – 0 03261 5, t + 0 01559 6, t +
+ 6 16343 10 4 7, × − t – 1 68389 10 3 8, × − t – 2 019 10 6 9, × − t ,
∆9 ( )t = – 0 02922 8, t + 1 64553 10 3 9, × − t – 2 019 10 6 10, × − t .
Na rys. 3 pryveden¥ hrafyky PK-reßenyj dlq N = 7 y N = 8 , poskol\ku kry-
v¥e x t8( ) y x t9( ) v masßtabax hrafyka praktyçesky nerazlyçym¥. Zdes\ s
uvelyçenyem stepeny polynoma nablgdaetsq tendencyq vzaymnoho prytqΩenyq
PK-reßenyj.
Rys. 3
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
POLYNOMYAL|NÁE KVAZYREÍENYQ LYNEJNÁX … 151
Prymer 5.3. Yssleduem LDRU
˙̇ ( ) ( ) ˙( )x t t x t+ + −1 1 = ( , ) ( )− + −0 5 1t x t , t R∈ . (5.7)
Vvedem model\noe uravnenye
˙̇ ( ) ˙( )x t x t+ − 1 = − −0 5 1, ( )x t , t R∈ . (5.8)
V πtom sluçae p0 = 1, a0 = – 0,5. Na rys. 1 πtym znaçenyqm sootvetstvuet toç-
ka, leΩawaq v oblasty suwestvovanyq kompleksn¥x kornej k = – 0,20931 ±
± 0,58908 i. Sohlasno opredelenyg 3.2 predomynantn¥e reßenyq uravnenyq
ymegt vyd
x t( ) = e C t C tt− +( )0 20931
1 20 58908 0 58908, cos( , ) sin( , ) ,
hde C1 y C2 — nekotor¥e konstant¥. Sledovatel\no, pry postanovke naçal\-
noj zadaçy dlq uravnenyq (5.7) dostatoçno dvux naçal\n¥x uslovyj, t. e.
˙̇ ( ) ( ) ˙( )x t t x t+ + −1 1 = ( , ) ( )− + −0 5 1t x t , t R∈ ,
(5.9)
x( )0 = x0 = 1, ˙( )x 0 = x1 = 0.
Zapyßem zadaçu, sohlasovannug po razmernosty polynomov otnosytel\no (5.9),
v vyde
˙̇ ( ) ( ) ˙( )x t t x t+ + −1 1 = ( , ) ( )− + −0 5 1t x t + ∆N t( ), t R∈ ,
x( )0 = x0 = 1, ˙( )x 0 = x1 = 0,
hde s uçetom (5.4)
∆N t( ) = f tN
N
−
−
1
1 + f tN
N + f tN
N
+
+
1
1
.
V rezul\tate v¥çyslenyj, provedenn¥x dlq zadaçy (5.9), b¥ly poluçen¥ for-
mul¥
x t7( ) = 1 – 0 18536 2, t + 0 14308 3, t + 0 04408 4, t – 0 03473 5, t –
– 4 90839 10 3 6, × − t + 3 998 10 3 7, × − t ,
∆7 ( )t = −0 096 6, t + 0 062 7, t – 3 998 10 3 8, × − t ;
x t8( ) = 1 – 0 18531 2, t + 0 14288 3, t + 0 04435 4, t – 0 03468 5, t –
– 5 57386 10 3 6, × − t + 4 99161 10 3 7, × − t + 9 71001 10 4 8, × − t ,
∆8 ( )t = 0 054 7, t + 0 011 8, t – 9 710 10 4 9, × − t ;
x t9( ) = 1 – 0 18531 2, t + 0 14283 3, t + 0 04454 4, t – 0 03492 5, t –
– 5 56427 10 3 6, × − t + 5 35343 10 3 7, × − t +
+ 4 66723 10 4 8, × − t – 4 13722 10 4 9, × − t ,
∆9 ( )t = 0 015 8, t – 8 120 10 3 9, × − t + 4 137 10 4 10, × − t .
Na rys. 4 pryveden¥ PK-reßenyq y sootvetstvugwye πtym PK-reßenyqm nevqz-
ky. Y v πtom sluçae najdenn¥e PK-reßenyq s uvelyçenyem stepeny polynoma
ymegt tendencyg vzaymnoho prytqΩenyq pry umen\ßenyy nevqzky. Tohda pod
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
152 V. B. ÇEREPENNYKOV, P. H. ERMOLAEVA
ε-prytqΩymost\g PK-reßenyj budem ponymat\ sledugwee.
Opredelenye 5.1. Pod ε -prytqΩymost\g PK-reßenyj na nekotorom ot-
rezke t t0 1,[ ] budem ponymat\ svojstvo vzaymnoho prytqΩenyq posledovatel\-
nosty PK-reßenyj, poroΩdaem¥x uvelyçenyem stepeny N polynoma PK-reße-
nyq, v sm¥sle, çto suwestvuet takoe N∗, pry kotorom dlq vsex N ≥ N∗ y
zadannoho ε
x t x tN i N i+ + −−( ) ( )1 < ε, i = 1, 2, … k, ∀ ∈[ ]t t t0 1, .
V zaklgçenye otmetym, çto najdenn¥e PK-reßenyq mohut posluΩyt\ osno-
voj dlq reßenyq LDRU metodom ßahov, esly v kaçestve naçal\noj funkcyy na
naçal\nom mnoΩestve zadat\ PK-reßenye. Pry πtom reßenyq v toçkax st¥kov-
Rys. 4
ky budut ymet\ neprer¥vn¥e posledovatel\n¥e proyzvodn¥e porqdka N , hde
N — stepen\ polynoma PK-reßenyq.
1. M¥ßkys A. D. Lynejn¥e dyfferencyal\n¥e uravnenyq s zapazd¥vagwym arhumentom. –
M.; L.: Hostexyzdat, 1951. – 352 s.
2. Pynny ∏. Ob¥knovenn¥e dyfferencyal\no-raznostn¥e uravnenyq. – M.: Yzd-vo ynostr.
lyt., 1961. – 248 s.
3. Bellman R., Kuk K. Dyfferencyal\no-raznostn¥e uravnenyq. – M.: Myr, 1967. – 548 s.
4. Azbelev N. V., Maksymov V. P., Raxmatullyna L. F. Vvedenye v teoryg funkcyonal\no-
dyfferencyal\n¥x uravnenyj. – M.: Nauka, 1991. – 280 s.
5. Rubanyk V. P. Kolebanyq kvazylynejn¥x system s zapazd¥vanyem. – M.: Nauka, 1969. –
287 s.
6. Cherepennikov V. B. Analytic solutions of some functional differential equations linear systems //
Nonlinear Analysis: Theory, Methods and Appl. – 1997. – 30, # 5. – P. 2641 – 2651.
7. Çerepennykov V. B. Polynomyal\n¥e kvazyreßenyq lynejn¥x system dyfferencyal\no-
raznostn¥x uravnenyj // Yzv. vuzov. Matematyka. – 1999. – # 10. – S. 49 – 58.
8. Cherepennikov V. B., Ermolaeva P. G. Polynomial quasisolutions of linear differential difference
equations // Opusc. Math. – 2006. – 26, # 3. – P. 47 – 57.
Poluçeno 15.10.07
ISSN 1027-3190. Ukr. mat. Ωurn., 2008, t. 60, # 1
|
| id | umjimathkievua-article-3144 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:37:03Z |
| publishDate | 2008 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/6d/9d513bd14f59d2a4ba9a0cb59e8f0e6d.pdf |
| spelling | umjimathkievua-article-31442020-03-18T19:46:36Z Polynomial quasisolutions of linear second-order differential-difference equations Полиномиальные квазирешения линейных дифференциально-разностных уравнений второго порядка Ermolaeva, P. G. Cherepennikov, V. B. Єрмолаєва, П. Г. Черепенніков, В. Б. The second-order scalar linear difference-differential equation (LDDE) with delay $$\ddot{x}(t) + (p_0+p_1t)\dot{x}(t) = (a_0 +a_1t)x(t-1)+f(t)$$ is considered. This equation is investigated with the use of the method of polynomial quasisolutions based on the presentation of an unknown function in the form of polynomial $x(t)=\sum_{n=0}^{N}x_n t^n.$ After the substitution of this function into the initial equation, the residual $\Delta(t)=O(t^{N-1}),$ appears. The exact analytic representation of this residual is obtained. The close connection is demonstrated between the LDDE with varying coefficients and the model LDDE with constant coefficients whose solution structure is determined by roots of a characteristic quasipolynomial. Розглядається скалярне лінійне диференціально-різницеве рівняння (ЛДРР) загаяного типу другого порядку $$\ddot{x}(t) + (p_0 + p_1t)\dot{x}(t) = (a_0 + a_1t)x(t-1) + f(t)$$ В якості методу дослідження використано метод поліноміальних квазірозв'язків, що ґрунтується на зображенні невідомої функції у вигляді полінома $x(t)=\sum_{n=0}^{N}x_n t^n.$ При підстановці цієї функції у початкове рівняння з'являється відхил $\Delta(t)=O(t^{N-1})$, для якого отримано точне аналітичне зображення. Відмічено тісний зв'язок ЛДРР зі змінними коефіцієнтами з модельним ЛДРР зі сталими коефіцієнтами, структура розв'язку якого визначається коренями характеристичного квазіполінома. Institute of Mathematics, NAS of Ukraine 2008-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3144 Ukrains’kyi Matematychnyi Zhurnal; Vol. 60 No. 1 (2008); 140–152 Український математичний журнал; Том 60 № 1 (2008); 140–152 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3144/3036 https://umj.imath.kiev.ua/index.php/umj/article/view/3144/3037 Copyright (c) 2008 Ermolaeva P. G.; Cherepennikov V. B. |
| spellingShingle | Ermolaeva, P. G. Cherepennikov, V. B. Єрмолаєва, П. Г. Черепенніков, В. Б. Polynomial quasisolutions of linear second-order differential-difference equations |
| title | Polynomial quasisolutions of linear second-order differential-difference equations |
| title_alt | Полиномиальные квазирешения линейных дифференциально-разностных уравнений второго порядка |
| title_full | Polynomial quasisolutions of linear second-order differential-difference equations |
| title_fullStr | Polynomial quasisolutions of linear second-order differential-difference equations |
| title_full_unstemmed | Polynomial quasisolutions of linear second-order differential-difference equations |
| title_short | Polynomial quasisolutions of linear second-order differential-difference equations |
| title_sort | polynomial quasisolutions of linear second-order differential-difference equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3144 |
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