Group analysis of boundary-value problems of mathematical physics
We obtain conditions for invariance and invariant solvability of boundary-value problems of mathematical physics.
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| Date: | 1999 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
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Institute of Mathematics, NAS of Ukraine
1999
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4593 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510740956315648 |
|---|---|
| author | Netesova, T. M. Нетесова, Т. М. Нетесова, Т. М. |
| author_facet | Netesova, T. M. Нетесова, Т. М. Нетесова, Т. М. |
| author_sort | Netesova, T. M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:09:14Z |
| description | We obtain conditions for invariance and invariant solvability of boundary-value problems of mathematical physics. |
| first_indexed | 2026-03-24T03:01:48Z |
| format | Article |
| fulltext |
Y ~ 517.946.9
T. M. HeTecona (HH-T HaTeHaTHKH HAH YKpaHHbl, KtleB)
ITYIIHOBOI~ AHAJIH3 KPAEBbIX
3Aj]Aq MATEMATHtIECKOI~ |
Invariance conditions and conditions of invariant solvability are obtained for boundary-value problems
in mathematical physics.
OTpHHaui yHosu iHBapiaHTXOCTi Ta iHnapiaHTHOi pOan'a3H0CTi Kpaflonax 3a~la,~ MaTeMaTHqH0i
B o6mapao~t cdpepe npaaomemna Teoprlri r pynnoso ro aHaJm3a ~aqbqbcpcrmHa2mHHX
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KpaCBhlX 3a~atI MaTCMaTHqCCKOIt dpH3HKH. ~ 0 CHX Hop BCTpeHalOTC~ J-lHllIb OT-
~e/Ibrlr~e rlprlHepbt Hay,4eHIrUI CBOItCTB HI-mapHaHTHOCT~ KpaeBhlX 3a~a,~ [2, 3].
~aHHaJ~ pa6oTa nocBJtmeHa Hccne/~oBamno ~onpocoB HHBapHaaTaOCTH KpaeB~x
3a~a~ HaTeHarrrqccKolt C~DH3HKH. Hpe~npHmrra nonmaxa ycTaHoB.rleHH$I KpHTepHeB
HX HHBapHaHTH0~ paapcmnHoca-a.
IIycTh B HeKOTOpOM IIpOCTpaHCTBe RN(x, u), X = (Xl, X 2 . . . . . Xn), U = (Ul, U 2 . . . .
. . . . Um), N = n + m, paccHaTpHBaerca KpaeBaa (naqa~bao-~paeBaz) aa/Iaqa ~na
CHCTeMt~I//~I(~C1DCpCHI/~aYI~H/~X ypaBHeHI~:
~ff~: F V ( x , u , u ', . . . . u (~)) = 0, v = 1,2 . . . . . k, (1)
C KpaeBhlMrl (Haqa.rlbHO-KpaeBhlHa) yc.rl0Brl~IHa
~ : ~r t ( x ,u ,u ' , . . . ) = 0, ~t = 1,2 . . . . . k, (2)
,, u(~) _ - F~C U', U , . . . , IIpOH3BO~HhlC IICpeMCHHHX U 1 , U2, . . . , /~m IIO Xl, X2, . . . , X n
go nop~Ka ~: BKJ/IOUj4TC/~HO.
PaccMaTpHBa~ ~'~ H ~ KaK MHOFOO6pa3H~ B HCKOTOpOH Hpo/~oII~KeHHOM
npoc'rpaHcrBe Rg(x, u, u ' , u " , . . . . u00) , M = n + m + (~: - 1), 6yl~cH Ha3uBaT~,
KpacBym (Ha'~a.m,HO-KpaCBy',o) 3a~a,~ ~ - ~ - (1), (2) uu~apuanmno~ ( c u ~ e m -
pu~noa), ecJm cymecTsyer rpyrma npeoSpa30BaHma G~ OTHOCHTeJIbHO KOT0p01~
mmapHanTri~ KaZ ypaBHem~a ~]~- (I), TaX H zpaeB~ac (HaaonmHo-Kpaes~e) yCnOBnJi
- (2) 3a/Iaqrl. E/II4~ICTBCHHOe p~IIICHHC U TaKOI~ ~a~a~a, ccJm OHO CylRCCTByeT,
6y~;cM Ha3blBaTb unoapuaumn~.~t (cu~t~tempul~nbt~t)pelIIeHHeM 3a/~aqH ~ - ~ , a
aa~a,~y ~ ~ p u a n m n o pazpeu~u~tozL
~a.ra,me a/Ipo OCHOBI-II~X FpyIIH G N npeo6pa3osaHrd~ HCXO~HOFO ypaBHeHaJt (CH-
CTCHI~ ypaBHeHrn~) 6ygeH o6o3Ha~laTb qepc3 G .
Oqenmmo, wro aJ~a paccMarprmaeHoll 3a~aqH npeo6pa3oBar!H)~HH HHBapHaHT-
Hocra 6y/lyr T0SmKO TC npeo6pazosarm~ P Ha J~pa G ocnosmax rpynn cHcreHu
ypaaHern~ (I), KOTOpUe XSmnOTCS TazomaMrI ~ana nocramucnmax KpaeBuX (Ha,~am,-
H0-KpacBhlX) ycn0nHlt (2) 3atla~H. B 3T01t CBH3H I~IHOBOI~/aHaYlH3 Kpaesblx ~ a a
~)q~IP.,CTHO B ~ B T~pMP~aX o /moro OT~e,21bHO B3JITOFO IIpeo6pa3oBalta.q P H3 J~tpa
G OCHOSmax rpynn aCXO/IaOtt cHerema ypa~erma pacc~aTprlaaeMott aa/Ia~m.
J;[CHO, trro 1t KOpHr aeeae~tyeHoro sonpoca ae~3r r gpltTCpP3t HHBapHaHTHOCTH
~aoroo6pa3rta. H a n o m m ~ , wro paccHarprmaeMoe n E . MHOroo6pa3rlC (noBepx-
nocr~) ~2, aazIamaoc ypaaacmtam~
T. M. ~ O B A ~ 1999
140 I$SN 0041.6053, Yxp. ~ m . .,wylm,, 1999, m. 51, IV ~ ]
FPYl'IFIOBO~I AHAJIH3 KPAEBbIX 3A,/IAq MATEMATHqECKO~I ~H3HKH 141
�9 ~ ~= 1,2 ..... s, x=x I,x2 ..... Xn, (3)
Ha3hIBaCTC$1 HHBapHaHTHrJM MHOFOO6pa3HCM HeKOTOpOI~ FpyIIHKI G r npeo6paaoBa-
HI4fl ( n - pa3MCpHOCTb npocTpaHCTBa, r - napaMcTp rpynn~) , cc~H ~ .mo6oro
npco6paaosaaHa Tae G~ (Ta: x i = j ~ ( x , a ) , a = a l , a 2 . . . . . ar), x e ~'d, c n e ~ y e T
T a x �9 ~ff~. CornacHo KpHTepHIO rlHBaprlaHTH0CTrI MH0roo6pa3nJl/~.rlJt TOrO, Wl'O6bl
3a~aHHoe MHOroo6paaae (3) 6ranO HrmapHarrrHh~, Heo6xomrMo H ~OCTaTO~mO, ~rro-
6m ~na Bcex r0~eK 3Tor0 Mnoroo6pazHa mano0mmmcb paBeHCTBa
Xa~FO(x) = 0, a = 1,2 . . . . . r, 6 < 1 ,2 . . . . . s, (4)
r~e X a - - HHqbHHrrre3aMara~Hralt oneparop rpynwa G~ n, COOTSeTCTSymttmil npe-
o6pa3oBam o Ta.
B HatuHx HCCo-le/~0BaHHJ~X 0FpaHHqI4MC.q paccMoTpeHHeM o/monapaMeTpHuecKHx
rpyrm npeo6pa3oBaHHia ( r = 1 ). ~TO o3aaaaeT, aTO ec0m rrpeo6paaoBarm~ T a H3 r-
napaMeTpauecKo~t rpyrmta G~ npeo6paaosaHH~ COOTBeTCTByeT miqbmmTezrtMa-
0mH~ll onepaTop X a, TO upa HatUHX npezmoaoaceHHaX npeo6pazonaHam P Ha
O~HonapaMeTprrqecKott rpynma G 6y~eT COOTSeTCTBOBaTb orIepaTop
i Xp = ~i(x,u)~x, + ;(x,U)Ouj, i = 1,2 . . . . . n, j = 1,2 . . . . . m. (5)
Tor~a MHOF006pa3HJ/ ~ - (1) a ~ - (2) 5y~yT HHBapHaHTHI~I OTHOCHTeYlbHO O~HO-
ro H TOt 0 ~Ke npeo6paaoBanaa P, ecom B IIpo~0JDKeHHOM np0cTpaHCTBe R M Bbl-
nonaaiOTCa paBericTBa
Xt,̂ r FV(x, u, u', .... u(~))]~ = 0, (6)
A~
o~g (x,
u, u .... )[S~
Xp " = 0, (7)
r~e ,, ̂ " o3HaqaeT n p o ~ t o ~ e n a e onepaTopa X v ~to nopaz~Ka ~=
PaBeHCTBa (6) H (7), KOTOpBIe nl~gCTaBn.qiOT co6olt ycnosHJt HHBapHaHTHOCTH
ypaBHeHma (1)H (2) OTHOCHTe~HO rrpeo6pa3oBarma P c G, 6y~eM Haa~maT~ paaen-
cmsa~tu (yc.aosu~tu) unsapuanmnocmu KpaeBoR (HaqaYmHO-KpaeBo~I) 3a~aaH OTHO-
CHTenBHO rlpeo6pa3oBaHHJt P H3 a~pa G OCHOBHI~IX rpynn npeo6pazoBanHll CHcre-
TpaBHeHHI~I (1).
HOO6XOgHMO OTMOT~T~, '-IYO HOHnT~O mmapHaHTHOCTH KpaoBoro ycnomIa n aaga-
~ax MaTeMaTHqecKofl CIDH3HKH HMeeT cnelii4dpI4qecKyIO OCO~HHOCTb, KOTOpa.q ilpe/~-
nonaraeT ssmoomeH.e ~On0~HaTen~HOr0 yco~onHa. A HMeHHO, TpeSoBarme arma-
p.aHT~OCTH KpaeBoro ~HorooSpaaHx ~ oTaocrrreomHO npeo6pa3o~aHna P c G,
~onycKae~oro ypaSHeHHJ~H (2), B~mouaeT B ce6a eme H n ~ o s m e H H e a'pe6oBaaHa
onpo~eneHHOlt cor~aco~armocr~ nocras~eHmax KpaeBraX (a Haaasmmax) ycn0~Hi~
paccMaTpH~ae~olt 3a~a.ra. ~ r o aono0mrrresmHoe Tpe6onarrae 3a~sno~aeTc~ B TO~,
qTO ssrtayyMermmerm~ ,tocsin Hc3aBHCHM~X ncpestCHmaX B peayamTaTe IIpHMCHeHI~t
~eToao~ rpyrmosoro aHa0maa COOTSeTCTSermo ~O~'KHO y~ermmaT~ca ~ac~o nocTa-
Bnemnax ~ono0mrrrem, max ycaoBma pacc~aapmmeMoit 3a~a~m. ~ p y r ~ enoBa~m,
/~Ba (H~.rl 6once) ycnosHJl rIOCTaI~CHHOfl xpaesolt (Haqa.llbHO-XpacBOIt) 3a~aqH
~O~KHr~ TpaHcqbopM~posaT~CX B O/toO yc~oBHe B pe~aytmpoBamao~l 3a~a,~c. HanpH-
Mop, B c~y~ae HatlaYlSHO-Xpacsl~X 3a/~aq IUI~ O/IHOMCpHI~X ~BO/IIOI~HOHH~X ypaBHC-
HHa MaTeMaTrraecKolt ~ b n ~ o t (rrpH WrOM HeKOMOe pemeHHe u = u(x, t ) - - qbyHKtma
IIpocTpaI-ICTBeHHOI~ FIeI~MOHHOI~ X H BI~MCI-IH t) CTaB$1T~R/]~Ba Kpacs~tX H O/~HO Ha-
qa3I~HOC yCnOBHC. B pe3ym~TaTe pe ;ayra l~ npH~eM K KpaeBl,IM 3a~a~aM ~ 06UK-
HOBeHIIOrO /~llqbqbepeHl/~a~soro ypaBHeHH~l, S KOTOplax Ha IICKOMy~o ~ ) y ~
ISSN 0041-6053. Yxp. ~.aun. affptt, 1999, m. 51,1~1
142 T.M. HETECOBA
HaKJIa/I~IRaIoTC~[ TOJISKO ~Ba yC/IOBH$L OI'IHCaHHOr TaKHM o6pa3oM TpC6OBaHHC yMe-
HbmCH~ ~Hc~a nocTaB~cm~x ~OHOJIHHTC/IbHI~IX yC~OBHR Kpae~ofl 3aRaqH 6y~CM
Ha3HBaTb c8o~cm~934 w~apuawnnoa pei~ymcuu KpacB~X (a HaqaJIbHHX) yCIIOBHH 3a-
~aqH, a casm KpacB~e yc~ostta ~ un6apuawnno peDyt~upye~a~va.
rlptme~eHHUe paccyx~aeuHa MOmHO CdpOpMysmpOBar~ B BH~C c~c~3notuero y m c -
p ~ e H a m
Teope~a. ]/n.a moeo ~mo6~ rpaeoa~ (na~a,u, n o - r p a e e ~ ) za~a~,a ~tame~tamu-
~ecrotl ~ u ~ K u (1), (2) 6~za u n o a p u a u m ~ pazpemu:~to~ omRocumemmo neromo-
pozo npeo@azooanu~ P, neoSxoSu~w u Oocmamo~no, wnoS~:
a) npeo@a3o~anue P npunaS/~e~aao ,~Opy ocnoon~x epynn G npeo@a3ooa-
null cucme~4b~ ypaonenu~ ( 1 ) - - P c G;
6) o ~ , D o t ~ mo~re pacc~ampuoae~toz] o6~acmu obmoAn,~UCb yC,~OOUJ~ unoa-
puawnnocmu (6) u (7);
o) rpaeeb~e u (~a~a~b~b~e) yc,wou,~ ( 2 ) saOar 6b~nu unoapuaum~o pec3y~u-
pye~b~U.
3a,~e,~anue. HOCKOabKy rpynnomott aHa~ma OCHOaaH Ha JIOma/lbHOl~ Teop~H
FpyrlH /IH rlpr TO RCHO, qTO TCXHHKa r MOYKCT 61dTb IIpHMCHHMa
/IHmb K 3a~aqaM, ~oIIycKaIOIIIHM JIOKaJIbHOO paCCMOTpeHHr (HalIpHMop, 3a~aqa
KOUIH, Fypca).
B Kaqccq3e npti~epa paccMoTpHM Haqaylbno-Kpae~yIo aa~aqy, a~n~ott ty~c~ Ma-
TOMaTHqCCK0i~ M0]]OJI~IO II~I~CCCOB TOIHI0-MaCCOIICpCH0Ca B cTpaTH~t~HIJ~IpOBaHH0~
BO~Ott cpe~e [4]:
[f(ux)Ux] x - u t = O, O ~ x < **, t > O; ( 8 )
u(x,0) = 0, x > 0;
: l f(Ux)Ux Ix=0 = 0, t > 0; (9)
[ l i m u ( x , t ) = 0, lim [ f (ux )u x] = O, t > O.
i.x "~ x --~ oo
K a a ycrauosae tmz tmeap.atrr~ott pa3peum~ocam HOCTaBJIOHHOI~ 3a/IaqH n p e ~ e
scero a e o 6 x o ~ o onpe~e0mTb ~ p o ocnoBmax rpyrm G rrpeo6pa3oaa~m~ HCXO/IHO-
ro ypa~HeHHa (8) 3a~awa.
I/IH(IDHHHTO3HMaJIbPJ:d~ onepaTop ~pynn~ G 6yReM HCKaTB B Bl4~e
X = ~ x + not + ;0u, (10)
r~e ~, rl . ~ - - qSyHKIItIH OT X, t,'U; ~X) ~t H ~u - - rlpOH3BO]~fla$1 rlO COOTBeTCTBylo-
~ett nepe~enHott.
Bse~aa o6omm.~etma u x = p. u t = q, uxz = r, uxt = s. u . = l. ~a-mtne~ ~TOpOe npo-
aOJL~Kerme ncxo~oro onepaTopa:
r~e npo~onxerm~e K0~l~dpHRaeHTta a a I~ JmJL~OrCa qbyammmda x, t, u, p H q,
a p, ff H X - - q b y H m m ~ nepeMeHmax x, t, u, p, q, r, s, I.
Tor~a. r~exo~aa r~a ycJ~omtR mmapHaHTHOerH (6), no.uy~ae~ onpe/zeJmmmee ypae-
HCKI4e]~JLq HCKOMHXKOOp]DIHaT ~, 1] 14 ~ onepaTopa(10)
X2{[f(P)P]x -q}l~ - (~ + 2f~,, + p 2 ; u u + ;u - 2r~x - 2 ~ x - P ~ --
- qr ln) tPf ' + f ] - 4, - q;u + q~x + qrl, + ( ~ + p ; . - qqt)[rf" + r p f " ] = 0.
I Ioayqetmoe ypaenetme pemae~ca ~ r o ~ o ~ noc~e~o~aTeJmnoro paculermerma
c o w . e c h o c o n e p a n a m ~ 14cr~moaetm~ 14 ~clxl>epemmpoeauv~, wro n pe~y~sTaxe
rrp~omaT x CaCTe~e HeTp~eHa~H~X ~qb~pepeHtma.~Hux ypaBHeHHg, o6mee pe-
metme KOT0pOg 14 oIIp~e.2IRU'W HCKOMM0 KO~]XI~HI~eHTH onepaTopa X. A 14biermo,
ISSN 0041-6053, Y~p. ~un. ucytm, 1999, m. 51. N~I
l'~Yl'lHOBOl~ AHAJIH3 KPAEBHX 3A/~H MATEMATHHECKOfl OH3HKH 143
O~a4x + ( a - 1 )a I , ~l 0~2a4 t + 0~2-1 = = a 2, ~ = a a 4 u + ( a - 1 ) a 3.
2
CooTaeTcrsymmae aTUM KOaClaqbHmteH'raM HHqbHHHTeaaManbHUe onepaTop~ n
onpc~e~moT Jt/Ipo G OCHOBHIxlX Fpyl'IIl npeo6paaosanHfl ypaBHeHng (8):
PI: ~ = x + a , ~=t , ~ = u ;
P2: .~=x, t '= t+o~ 2, ~'=u;
P3: ~ = x , ? = t , ~'= u + a ;
P4 : .~=17.x, T=a2t , ~ '=au.
BTopoe ycYtoBae HHSapHaHTHOCTH (7)KpaeBrax 3a~aq rIOaBOJIJteT ar.r~e~aa-s rt3
a'roro sz~pa rpyrnay npeo6paaoaaHHlt P4, OTHOCnTem,HO KOTOpOit KpaeBue ycno-
BHJ~ (9) paccMaTpHaaeMofl 3a/iaqri rmBapHaaTHra H, B TO ~Ke BpeMz, aHBaprmnTaO
pe~ytmpye~a,I. ~e~tc'rBHTem,HO, rpynna npeo6paaoBannlt P4 aazse ' r c s rpynno~
Macttrra6max npeo6pa3osaHrIfl c nHBapHaHTamf:
X U
r l = ~ t a V = ~ . (11)
B~apaa,B petuerme paccMarprmaeMo~ aa~a'~n U = u(x, t) ,aepea rmBapnanTu (11)
B BHae U(X, t) = xff-V(rl), .nerKo Brt~e~, v ro nepBoe ri r p e ~ e aa ycnoBrI~ (9) aa~a-
'au rpancqbopMupy~oTCS B O~HO yc.noane ~m~ qby~KUrIH V(rl), a rIMeUHo,
lim V('q) = 0.
11--)**
Taxr~M o6pa3oM, uenrmeaHaJ~ KpaeBa.a aa~aqa (8), (9) rmaap~arrrno paapetuaMa rt
OThICKaI-IHC e e HHBapHatlTHOFO (B ~aHHOM cnyqae aBTOMO,/~eJIbHOFO) petUenHa CBO-
/XHTCa K pemeHmo cne/w~omefl 3aRaqa ~ O6taKHOaeHHoro/~qbqbepem~azmHoro
ypa~Herma:
[ f (V ' )V ' ] + 1 / 2 ( r l V ' - V)= 0, (12)
f (V')Vln=o= Q, lim V(ri) = 0, lim [ f ( V ' ) V q = 0. (13)
11-+** 1]-.-***
TeXHHKa rpyrmOBOrO aHa~43a oKa3raBaeTcJ~ yCTlemHO npHMeHHMO~ TaK~Ke npH
paCCMOTpgHHH CrI~T~HaJIbHOFO K~Tacca 3a~aq, B KOTOpHX pemeHHe U n o p o ~ a e T c ~
RelllGTBHeM HeKOTOpOFO (MFHOBOHHO HJIH IIOCTO~qHHO ~I~ffTByK)III{eFO) HCTOqHHKa. B
TaKHX 3a~aqax Ha HCKOMOe pemeHHe U HaK3I~aIOTCYl/~OTIOJIHHTCJIbHbIO yCJ~OBHJL
o6ycnosnermhte 3aKOHaMH coxpaHeHHJL PaccMoTprrg MaTeMaTH'4ecKy~O Mo/~e.rn, rrpo-
~ccca BO3HHKHOBeHHR 3~IeKTpOMarHHTHOFO TIOYIJt B dpeppoMar~HTHOfl cpenr ( j = (~E,
D = ~E, B = bH ~/n, n > 1 ) nor ~eltCTBHeM II~OCKOFO HYlH ToqeqHOFO HCTOqHHKa
~YleKTpOMarHHTHOfl 3HepFHH HHTeHCHBHOGTH Wu(t ) [5]:
1 t,,~H. ~ b~HO-") /nHt = 0, k = 1, 2, 3; (14)
I'~ e / e - n
n ( e , 0 ) = 0, n ( o . , t) = 0, (Okno)e = 0; ( i5 )
,~ n+l t ** t
bo, fHTO,_~dO + O_.k~ ~dt fHgdO- ~Wu(t)dt. (16)
n + l ~ o 0 0 0 0
,~][po OCHOBH/~X l"pyllH G npeo6pa~osaHHit ypaBHeH~ (14) npe~eramaaeTCg npeo6-
pa~olmmla~a:
: w ~=at, "g=H;
1'2: "O -" a-b/2o, ~ = t, H f o J - / ;
P3: w /'=t, ~=H.
ISSN 0041-6053. Yrp. ~tam. ~yp.., 1999, m. 51,1~1
144 T.M. HETECOBA
Ecm~ cmwravb, qTO HHTeHCHBHOCTb HC'rOqHtIKa W u ( t ) - - cTeneHHa~I qbyHKI~HZ,
T . e . W = ( t ) = pwt/'-l, TO paccMaTptmaeMa~ 3a~a~a (14)-(16) 6y~eT nHBapHanTnO
pa3pemHMott oTuocIcrem,HO rpynnu npeo6pa3osam~tt
c! -be 2
P = ClP 1 + c2P2: 0 = i X 2 0 , [=o~Ctt, H=t~C2H, (17)
r]]c c 1 H c 2- npOn3SO.rlbH~r nOCTO~HH~C.
~cttcTstrreJmno, BI~a3HB pcmeHHe H(O, t) qcpr HI-mapHanTu V H Z I
H(0, t) = tmv(TI), 11 = 0 2 / t l, m = c2 /c 1, l = 1 - b c 2 / c r (18)
H HO]ICTaBHB B HCXO]~IOC ypaBHCHHr H yCJIOBH.q (15), (16), nony, mM peI1ylJ~IpoBalIHyIo
3a~aqy
/
v(oo) = o, = 0 , (20)
\ :TI=0
:|_b.fVn 2d + • =w:. (2,)
~Po 03
H3 paseHcrs (20) c~e~yer, ,fro HaqaJmHo-zpaeBa~ 3a~aqa (14)--(16) 6yA~r vraBa-
pHa~rrno pa3pemm~oft OTHOCnTeJZbHO rpynnu Macurra6aux npeo6pa3oBaHHR (17)
npH yc~oamt p = 2m + 1 + (k + 1 ) l / 2 , o ~ y ~ I a c 1 = c 2 H pemCHHe ee CaO~TCZ K
pcmcemo tpaesott aa~amt (19) - (2I).
BO3MO)F~0CT~ nptIMeUeHH~ TeX~m~ n MeTOAOB rpynnosoro aHa.rIH3a rlpH p~IIIe-
Hm~ zpacmax aajzaq ~o Hac'rosmero Spe~CUH OKOH'4aTCJZI~HO CmC He OUCHellbl. Haps-
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annapaToM, ]~atouiabl Jn~ub MaTeMaT~qeCKOe pemeHHC 3a~a~ MaTCMaTIIqCCKOI~ QbH-
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I. Osc~uroe ./I. 8. rpynnoaott aHa~n3 n~qbqbepeHuna~bHUX ypaaHeHHlt. -- M.: Hayza, 1978. -
400c.
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HOlt cpe~u. - 1972. - B~n. I0. - C. 70 - 84.
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ct~aeM~xe qaffrHqHO-HHBapHaHTHMMH petuenHaMtt ypaBneHti~l HaBbe -- CToKca II TaM ~Ke. --
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l'lo~yqe.o 18.04.97
I$SN 0041-6053. Yrp, ~ m . ~.'vps., 1999 , m. 51, N el
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| id | umjimathkievua-article-4593 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | rus English |
| last_indexed | 2026-03-24T03:01:48Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a3/39277c2311582a2edfca632fd19496a3.pdf |
| spelling | umjimathkievua-article-45932020-03-18T21:09:14Z Group analysis of boundary-value problems of mathematical physics Групповой анализ краевых задач математической физики Netesova, T. M. Нетесова, Т. М. Нетесова, Т. М. We obtain conditions for invariance and invariant solvability of boundary-value problems of mathematical physics. Отримані умови інваріантності та інваріантної розв'язності крайових задач математичної фізики. Institute of Mathematics, NAS of Ukraine 1999-01-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4593 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 1 (1999); 140–144 Український математичний журнал; Том 51 № 1 (1999); 140–144 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/4593/5890 https://umj.imath.kiev.ua/index.php/umj/article/view/4593/5891 Copyright (c) 1999 Netesova T. M. |
| spellingShingle | Netesova, T. M. Нетесова, Т. М. Нетесова, Т. М. Group analysis of boundary-value problems of mathematical physics |
| title | Group analysis of boundary-value problems of mathematical physics |
| title_alt | Групповой анализ краевых задач математической физики |
| title_full | Group analysis of boundary-value problems of mathematical physics |
| title_fullStr | Group analysis of boundary-value problems of mathematical physics |
| title_full_unstemmed | Group analysis of boundary-value problems of mathematical physics |
| title_short | Group analysis of boundary-value problems of mathematical physics |
| title_sort | group analysis of boundary-value problems of mathematical physics |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4593 |
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