Generalized procedure of separation of variables and reduction of nonlinear wave equations
We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries.
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| Datum: | 2009 |
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| Sprache: | Ukrainisch Englisch |
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Institute of Mathematics, NAS of Ukraine
2009
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| author | Barannyk, T. A. Barannyk, A. F. Yuryk, I. I. Баранник, Т. А. Баранник, А. Ф. Юрик, І. І. |
| author_facet | Barannyk, T. A. Barannyk, A. F. Yuryk, I. I. Баранник, Т. А. Баранник, А. Ф. Юрик, І. І. |
| author_sort | Barannyk, T. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:44:40Z |
| description | We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries. |
| first_indexed | 2026-03-24T02:35:34Z |
| format | Article |
| fulltext |
UDK 517.5
V. F. Babenko
(Dnepropetr. nac. un-t, Yn-t prykl. matematyky y mexanyky NAN Ukrayn¥, Doneck),
N. V. Parfynovyç (Dnepropetr. nac. un-t)
NERAVENSTVA TYPA BERNÍTEJNA
DLQ SPLAJNOV DEFEKTA 2
New exact inequalities of a Bernstein type for periodic polynomial splines of order r and defect 2 are
obtained.
Otrymano novi toçni nerivnosti typu Bernßtejna dlq periodyçnyx polinomial\nyx splajniv po-
rqdku r defektu 2.
Vo mnohyx sluçaqx dlq tryhonometryçeskyx polynomov y splajnov mynymal\-
noho defekta yzvestn¥ toçn¥e neravenstva typa Bernßtejna, kotor¥e yhragt
vaΩnug rol\ vo mnohyx voprosax teoryy pryblyΩenyq (obzor y yzloΩenye mno-
hyx yzvestn¥x toçn¥x neravenstv, a takΩe byblyohrafyg moΩno najty, napry-
mer, v [1, 2]). M¥ ustanovym nekotor¥e toçn¥e neravenstva typa Bernßtejna
dlq splajnov defekta 2.
Pust\ Lp , 1 ≤ p ≤ ∞, — prostranstva 2π-peryodyçeskyx funkcyj f : R →
→ R s sootvetstvugwymy normamy ⋅ Lp
= ⋅ p , C — prostranstvo neprer¥v-
n¥x 2π-peryodyçeskyx funkcyj.
Çerez S n r2
2
, , r = 1, 2, … , oboznaçym prostranstvo 2π-peryodyçeskyx poly-
nomyal\n¥x splajnov porqdka r defekta 2 s uzlamy v toçkax tk =
2k
n
π
,
k ∈Z , t.7e. mnoΩestvo 2π-peryodyçeskyx funkcyj s, udovletvorqgwyx sle-
dugwym uslovyqm:
1) s ymeet neprer¥vn¥e proyzvodn¥e do porqdka r – 2 vklgçytel\no;
2) dlq kaΩdoho k ∈Z najdetsq alhebrayçeskyj polynom p xk ( ) stepeny r
takoj, çto s x( ) = p xk ( ) dlq x ∈7 ( , )t tk k+1 .
Otmetym, çto s r( )−1
moΩet ymet\ razr¥v¥ (pervoho roda) v toçkax tk ,
k ∈Z , y v πtyx toçkax polahaem
s t s t s tr
k
r
k
r
k
( ) ( ) ( )( ) ( )− − −( ) = + + −
1 1 11
2
0 0 .
Nayluçßym pryblyΩenyem funkcyy f podprostranstvom konstant v prost-
ranstve Lp , 1 ≤ p ≤ ∞, naz¥vaetsq velyçyna
E f f cp
c p
( ) inf= −
∈R
.
Funkcyy Bernully opredelqgtsq sledugwym obrazom (sm. [3, s. 72]). B x1( ) —
2π-peryodyçeskaq funkcyq, kotoraq na 0 2, π[ ) zadaetsq tak:
B x1( ) =
π
π
π
−
∈
=
x
x
x
2
0 2
0 0
, ( , ),
, .
Esly r > 1, to B xr ( ) est\ (r – 1)-j 2π-peryodyçeskyj yntehral s nulev¥m
srednym znaçenyem na peryode ot B x1( ) .
© V. F. BABENKO, N. V. PARFYNOVYÇ, 2009
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7 995
996 V. F. BABENKO, N. V. PARFYNOVYÇ
PoloΩym ψn r x, ( ) = −
2π
n
B nx
r r ( ) (zametym, çto ψn r x, ( ) ∈ S n r2
2
, ).
DokaΩem sledugwye utverΩdenyq.
Teorema 1. Pust\ n , r ∈N , r ≥ 2. Dlq lgboho splajna s S n r∈ 2
2
, y lgbo-
ho j ∈Z ymegt mesto neravenstva
sup ( )( )
t t t
r
j j
s t
< < +1
≤
E s
E
t
n r t t t
n
j j
( )
( )
sup ( )
,
,
∞
∞ < < +
′
ψ
ψ
1
1 , (1)
sup ( )( )
t t t
r
j j
s t
< <
−
+1
1 ≤
E s
E
t
n r t t t
n
j j
( )
( )
sup ( )
,
,
∞
∞ < < +ψ
ψ
1
1 , (2)
ω s t tr
j j
( ), ( , )−
+( )1
1 ≤
E s
E
t t
n r
n j j
( )
( )
, ( , )
,
,
∞
∞
+( )
ψ
ω ψ 1 1 , (3)
hde ω( , )f M = sup ( ) ( )
,′ ′′∈
′ − ′′
t t M
f t f t — kolebanye funkcyy f na mnoΩestve
M ⊂ R .
Sledstvye 1. Pust\ n, r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
,
V
0
2π
s r( )−
1 ≤
E s
E
t
n r
n
( )
( )
( )
,
,
∞
∞
[ ]
ψ
ψ
π
V
0
2
1 , (4)
s r( )−
∞
1 ≤
E s
E
t
n r
n
( )
( )
( )
,
,
∞
∞ ∞ψ
ψ 1 . (5)
Teorema 2. Pust\ n, r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
, y lgboho j ∈Z
V
t
t
r
j
j
s
+
−
1
2( ) ≤
E s
E n r t
t
n
j
j
( )
( ),
,
∞
∞
+
[ ]
ψ
ψV
1
2 (6)
y, sledovatel\no,
V
0
2π
s r( )−
2 ≤
E s
E n r
n
( )
( ),
,
∞
∞
[ ]
ψ
ψ
π
V
2
0
2 . (7)
Teorema 3. Pust\ n , r ∈N , r ≥ 2. Dlq lgboho s S n r∈ 2
2
, 6 y lgboho p ∈
∈ [ ∞)1,
s r
p
( )−1 ≤
E s
E
t
n r
n p
( )
( )
( )
,
,
∞
∞ψ
ψ 1 . (8)
Zameçanye. Oçevydno, çto neravenstva (1) – (8) obrawagtsq v ravenstva
dlq funkcyy s n r= ψ , y, sledovatel\no, qvlqgtsq toçn¥my.
Dokazatel\stvo teorem¥ 1. Pust\ s S n r∈ 2
2
, . PoloΩym
ϕ
ψ
ψ( )
( )
( )
( )
,
,t
E s
E
t
n r
n r= ∞
∞
.
V sylu lynejnosty funkcyj ϕ( )r−1
y s r( )−1
na ( , )t tj j+1 , j = 0 1, n − , sootno-
ßenyq (1) y (3) budut sledovat\ yz sootnoßenyq (2). Poπtomu dlq dokaza-
tel\stva teorem¥ dostatoçno ustanovyt\ sootnoßenye (2), t.7e. dokazat\, çto
dlq lgboho j = 0 1, n −
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
NERAVENSTVA TYPA BERNÍTEJNA DLQ SPLAJNOV DEFEKTA 2 997
sup ( ) sup ( )( ) ( )
t t t
r
t t t
r
j j j j
s t t
< <
−
< <
−
+ +
≤
1 1
1 1ϕ . (9)
PredpoloΩym, çto najdetsq j0 takoe, çto na yntervale ( , )t tj j0 0 1+ sootnoße-
nye (9) ne v¥polnqetsq, t.7e. ymeet mesto po krajnej mere odno yz neravenstv
s t tr
j
r
j
( ) ( )( ) ( )− −+ >1 1
0 0
0 ϕ ,
s t tr
j
r
j
( ) ( )( ) ( )− −
+− >1 1
10 0
0 ϕ .
Pust\, dlq opredelennosty, s tr
j
( )( )− +1
0
0 > ϕ( )( )r
jt−1
0
. Tohda pry podxodq-
wem 0 < λ < 1 ymeem
λ ϕs t tr
j
r
j
( ) ( )( ) ( )− −+ =1 1
0 0
0 . (10)
PoloΩym δ ϕ( ) ( )t t= – α – λ βs t( ) −( ) , hde α y β — konstant¥ nayluçßeho
ravnomernoho pryblyΩenyq dlq funkcyj ϕ( )t y s t( ) sootvetstvenno.
Otmetym, çto δ( )r−1
na kaΩdom yz yntervalov ( , )t tj j+1 moΩet menqt\ znak
ne bolee odnoho raza, krome toho, peremena znaka u δ( )r−1
vozmoΩna pry pere-
xode arhumenta çerez toçku t j . V sylu (10) δ( )r−1
ne menqet znak na (t j0 ,
t j0 1+ ) y, znaçyt, δ( )r−1
ymeet na peryode ne bolee 2n – 1 peremen znaka. S
druhoj storon¥, tak kak E( )ϕ ∞ > E s( )λ ∞ , δ( )t ymeet po krajnej mere odnu
peremenu znaka meΩdu lgb¥my dvumq sosednymy toçkamy πkstremuma funkcyy
ϕ( )t – α. Znaçyt, δ( )t ymeet na peryode ne menee 2n peremen znaka. Tohda v
sylu teorem¥ Rollq u δ( )r−1
budet takΩe ne menee 2n peremen znaka. Polu-
çennoe protyvoreçye dokaz¥vaet, çto dlq lgboho j = 0 1, n −
sup ( ) sup ( )( ) ( )
t t t
r
t t t
r
j j j j
s t t
< <
−
< <
−
+ +
≤
1 1
1 1ϕ .
Takym obrazom, sootnoßenye (2) ustanovleno.
Teorema 1 dokazana.
V sylu lynejnosty funkcyj s r( )−1
y ψn,1 na kaΩdom yntervale ( , )t tj j+1
y
2π
n
-peryodyçnosty funkcyy ψn,1 yz sootnoßenyj (2) y (3) sledugt ut-
verΩdenyq (4) y (5).
Dokazatel\stvo teorem¥ 2. Pust\ snaçala promeΩutok t tj j, +[ ]1 takov,
çto s r( )−1
ymeet nul\ v ( , )t tj j+1 . Poskol\ku v sylu (5)
s r( )−
∞
1 ≤ ϕ( )r−
∞
1
y v sylu (1)
s tr( )( ) ≤ ϕ( )( )r t =
E s
E n r
( )
( ),
∞
∞ψ
, t ∈ ( , )t tj j+1 ,
dlq kaΩdoho x ≥ 0
mes t t t t xj j
r∈ ≥{ }+
−( , ) : ( )( )
1
1ϕ ≥ mes t t t s t xj j
r∈ ≥{ }+
−( , ) : ( )( )
1
1 . (11)
Yz (11) neposredstvenno sleduet, çto pry vsex p ∈ ∞[ )1,
s t dt t dtr p
t
t
r p
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ (12)
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
998 V. F. BABENKO, N. V. PARFYNOVYÇ
y, v çastnosty,
s t dt t dtr
t
t
r
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ ,
tak çto
V V
t
t
r
t
t
r
j
j
j
j
s
+ +
− − ≤
1 1
2 2( ) ( )ϕ . (13)
Pust\ teper\ promeΩutok t tj j, +[ ]1 takov, çto s r( )−1
ne obrawaetsq v nul\
na yntervale ( , )t tj j+1 . V πtom sluçae s r
t tj j
( )
,
−
+
2
1
— monotonnaq funkcyq y,
sledovatel\no,
sup ( ) max ( ) , (( ) ( ) ( )
t t t
r r
j
r
j j
s t s t s
≤ ≤
− − −
+
=
1
2 2 2 tt j+{ }1) .
Pust\, dlq opredelennosty,
sup ( ) ( )( ) ( )
t t t
r r
j
j j
s t s t
≤ ≤
− −
+
=
1
2 2
.
Otmetym takΩe, çto
ϕ ϕ ϕ( ) ( ) ( )( ) ( )r r
j
r
jt t−
∞
− −
+= =2 2 2
1 .
Ustanovym neravenstvo
sup ( ) ( ) ( )( ) ( ) ( )
t t t
r r
j
r
j
j j
s t s t t
≤ ≤
− − −
+
= ≤
1
2 2 2ϕ , (14)
otkuda y budet sledovat\ sootnoßenye (13) dlq rassmatryvaemoho sluçaq.
PredpoloΩym, çto vopreky (14)
sup ( ) ( ) ( )( ) ( ) ( )
t t t
r r
j
r
j
j j
s t s t t
≤ ≤
− − −
+
= >
1
2 2 2ϕ .
Tohda najdetsq λ, 0 < λ < 1, takoe, çto λs tr
j
( )( )−2 = ϕ( )( )r
jt−2
y, sledova-
tel\no, λs tr
j
( )( )−
+
2
1 ≤ ϕ( )( )r
jt−
+
2
1 .
Pust\ δ( )t = ϕ( )t – α – λ βs t( ) −( ) , hde α y β — konstant¥ nayluçßeho
ravnomernoho pryblyΩenyq funkcyj ϕ( )t y s t( ) sootvetstvenno.
Poskol\ku δ( )r−2 ∈ S n2 2
2
, y δ( )( )r
jt−2 = 0, δ( )r−2
na promeΩutke t tj j, +[ ]1
moΩet ymet\ ne bolee trex peremen znaka. Tohda na peryode u δ( )r−2
budet ne
bolee 2n – 1 peremen znaka.
S druhoj storon¥, kak y pry dokazatel\stve teorem¥71, ubeΩdaemsq, çto
δ( )t ymeet na peryode ne menee 2n peremen znaka. No tohda v sylu teorem¥
Rollq δ( )( )r t−2
menqet znak na peryode ne menee 2n raz.
Poluçennoe protyvoreçye dokaz¥vaet neravenstvo (14). Yz (14) s uçetom mo-
notonnosty s r( )−2
na t tj j, +[ ]1 sleduet, çto
V V
t
t
r
t
t
r
j
j
j
j
s
+ +
− − ≤
1 1
2 2( ) ( )ϕ
takΩe dlq yntervalov, v kotor¥x s r( )−1
ne ymeet nulej.
Teorema 2 dokazana.
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
NERAVENSTVA TYPA BERNÍTEJNA DLQ SPLAJNOV DEFEKTA 2 999
Dokazatel\stvo teorem¥ 3. Pry dokazatel\stve teorem¥72 ustanovleno,
çto esly s r( )−1
ymeet nul\ v ( , )t tj j+1 , to ymeet mesto neravenstvo (12). Poka-
Ωem, çto πto neravenstvo ymeet mesto y dlq takyx yntervalov, v kotor¥x s r( )−1
ne ymeet nulej. Dlq πtoho pokaΩem, çto dlq vsex x ∈ 0 2, /π n[ ] v¥polnqetsq
neravenstvo
r s t dt r t dtr
x
r
x
( ) ( ), ,− −( ) ≤ ( )∫ ∫1
0
1
0
ϕ , x ∈ 0 2, /π n[ ] , (15)
hde r f t( , ) — nevozrastagwaq perestanovka (sm., naprymer, [3], hl. 3) suΩenyq
funkcyy f na t tj j, +[ ]1 .
Pust\ ∆( )x = r t dtrx
ϕ( ),−( )∫ 1
0
– r s t dtrx ( ),−( )∫ 1
0
. V rassmatryvaemom sluçae
obe funkcyy r trϕ( ),−( )1
y r s tr( ),−( )1
lynejn¥ na 0 2, /π n[ ] , tak, çto yx raz-
nost\ lybo ne menqet znak na yntervale ( , / )0 2π n y( tohda v sylu (6) dlq
lgboho t ∈ ( , / )0 2π n budet r s tr( ),−( )1 ≤ r trϕ( ),−( ))1
, lybo menqet znak rovno
odyn raz v toçke x0 ∈ ( , / )0 2π n , pryçem s „+” na „–”.
V pervom sluçae neravenstvo (15) oçevydno. Vo vtorom sluçae, tak kak
′∆ ( )x = r xrϕ( ),−( )1 – r s xr( ),−( )1
, ∆( )x vozrastaet na yntervale ( , )0 0x y ub¥-
vaet na ( , / )x n0 2π . Krome toho, ∆( )0 = 0 y ∆( / )2π n ≥ 0 (poslednee neravenst-
vo ymeet mesto v sylu (6)). Takym obrazom, raznost\ ∆( )x neotrycatel\na na
0 2, /π n[ ] , çto πkvyvalentno (15).
Uçyt¥vaq (15) y predloΩenye 3.2.5 yz [3], vydym, çto (12) ymeet mesto takΩe
dlq yntervalov ( , )t tj j+1 , v kotor¥x s r( )−1
ne ymeet nulej.
Ytak, dlq lgboho p ∈ ∞[ )1, y lgboho j = 0 1, n −
s t dt t dtr p
t
t
r p
t
t
j
j
j
j
( ) ( )( ) ( )− −
+ +
∫ ∫≤1 1
1 1
ϕ ,
sledovatel\no,
s s t dtr
p
r p
t
t
j
n
p
j
j
( ) ( )
/
( )− −
=
−
=
+
∫∑1 1
0
1
1
1
≤≤
−
=
− +
∫∑ ϕ( )
/
( )r p
t
t
j
n
p
t dt
j
j
1
0
1
1
1
=
= ϕ ϕ
π
( )
/
( )( )r p
p
r
p
t dt− −∫
=1
0
2 1
1 .
Takym obrazom, yz posledneho neravenstva, s uçetom opredelenyq funkcyy ϕ,
poluçaem neravenstvo (8) dlq vsex p ∈ ∞[ )1, .
Teorema 3 dokazana.
1. Kornejçuk N. P., Babenko V. F., Lyhun A. A. ∏kstremal\n¥e svojstva polynomov y splajnov.
– Kyev: Nauk. dumka, 1992. – 304 s.
2. Babenko V. F., Kornejçuk N. P., Kofanov V. A., Pyçuhov S. A. Neravenstva dlq proyzvodn¥x
y yx pryloΩenyq. – Kyev: Nauk. dumka, 2003. – 591 s.
3. Kornejçuk N. P. Toçn¥e konstant¥ v teoryy pryblyΩenyq. – M.: Nauka, 1987. – 424 s.
Poluçeno 30.05.08,
posle dorabotky — 30.04.09
ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 7
|
| id | umjimathkievua-article-3065 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T02:35:34Z |
| publishDate | 2009 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/08/7a7666ac7d805bfacd0694b4eda88308.pdf |
| spelling | umjimathkievua-article-30652020-03-18T19:44:40Z Generalized procedure of separation of variables and reduction of nonlinear wave equations Узагальнена процедура відокремлення змінних і редукція нелінійних хвильових рівнянь Barannyk, T. A. Barannyk, A. F. Yuryk, I. I. Баранник, Т. А. Баранник, А. Ф. Юрик, І. І. We propose a generalized procedure of separation of variables for nonlinear wave equations and construct broad classes of exact solutions of these equations that cannot be obtained by the classical Lie method and the method of conditional symmetries. Предложена обобщенная процедура разделения переменных для нелинейных волновых уравнений. Построены широкие классы точных решений этих уравнений, которые невозможно получить с помощью метода С. Ли и метода условных симметрий. Institute of Mathematics, NAS of Ukraine 2009-07-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3065 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 7 (2009); 892-905 Український математичний журнал; Том 61 № 7 (2009); 892-905 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/3065/2879 https://umj.imath.kiev.ua/index.php/umj/article/view/3065/2880 Copyright (c) 2009 Barannyk T. A.; Barannyk A. F.; Yuryk I. I. |
| spellingShingle | Barannyk, T. A. Barannyk, A. F. Yuryk, I. I. Баранник, Т. А. Баранник, А. Ф. Юрик, І. І. Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title | Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title_alt | Узагальнена процедура відокремлення змінних і редукція нелінійних хвильових рівнянь |
| title_full | Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title_fullStr | Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title_full_unstemmed | Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title_short | Generalized procedure of separation of variables and reduction of nonlinear wave equations |
| title_sort | generalized procedure of separation of variables and reduction of nonlinear wave equations |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3065 |
| work_keys_str_mv | AT barannykta generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT barannykaf generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT yurykii generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT barannikta generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT barannikaf generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT ûrikíí generalizedprocedureofseparationofvariablesandreductionofnonlinearwaveequations AT barannykta uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ AT barannykaf uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ AT yurykii uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ AT barannikta uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ AT barannikaf uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ AT ûrikíí uzagalʹnenaproceduravídokremlennâzmínnihíredukcíânelíníjnihhvilʹovihrívnânʹ |