The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation
The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established.
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| Datum: | 1993 |
|---|---|
| Hauptverfasser: | , , , |
| Format: | Artikel |
| Sprache: | Ukrainisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
1993
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/5925 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512147601096704 |
|---|---|
| author | Mitropolskiy, Yu. A. Khoma, L. G. Митропольський, Ю. О. Хома, Л. Г. |
| author_facet | Mitropolskiy, Yu. A. Khoma, L. G. Митропольський, Ю. О. Хома, Л. Г. |
| author_sort | Mitropolskiy, Yu. A. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-19T09:21:13Z |
| description | The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established. |
| first_indexed | 2026-03-24T03:24:10Z |
| format | Article |
| fulltext |
Y,LJ;K517.946
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JI. r. XoMa, acn. (IH-T MaTeMaTHKH AH YKpaiHH, KHiB)
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The problem on solvability of a mixed problem for a linear hyperbolic partial-differential equation of
second order is studied. The minimal necessary and sufficient conditions are found for the existence of a
unique classic solution of this problem.
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u(0, t) = u(1t, t) = 0, 0 s ts T,
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u X, - <p X • dt - "' X • - X - 1t .
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u2(x, 0) = 'lf(x) - <p'(x) :!! <p2(x), u3(x, 0) = u(x, 0) = <p(x) !!! <p3(x),
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2
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2 0
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0 j= I
ISSN 0041-6053 . YKp. Mam. )KYP"-· 1993, m. 45 , N" 9
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+ 't, 't) d't + f 2, a 1/x + t- 't, 't) u/x + t - 't, 't) d't + f J (x + t- 't, 't) d't,
t-1t+x }= I 0
(ll)
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u2(x, t) = cp2(x - t) + f I, a2/x - t + 't, 't) u1(x - t + 't, 't)d't + J i(x - t + 't, 't)d't,
0 j=I 0
I t
u(x, t) = cp(x) + - J {u 1(x, 0) + ui(x, 0)}d0.
2 0
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· 0 I
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ISSN 0041-6053. YKp. MOIi! . AY/J"·· 1993, Ill. 45 , N" 9
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<p(x) e C1t,1t)• 'Jl(X) e ci10,1t]• (24)
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ax O X 0
(25)
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(27)
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x-t 1:I 0
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2 0
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ISSN 0041-6053 . Y,qJ. Mam~ ;,cyp1t., 1993, m. 45, N' 9
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HHX au?(x, t)/ox , i = I , 2, 3. OcKiJibKH BHK0HYIOTbCR YM0BH (18), (19) i (28), TO
oaep)KanaCHcTeMaMae eam1ttltpo3s ' J130K ou?(x,t) /cJxe C(X). i= 1,2,3. Aua
nori'lno, AH(pepenuiIOIO'IH piBHOCTi (9) ((27)) no t, oaep)KyeM0 CHCTeMy iHTerpa
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= I , 2, U 3 = 0, i BpaJf0ByI0'IH JieMy [3 , c. 101], nepeK0HyeM0Cb, ll.\0 cpyHKI(ij{
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3JJ.ii!:CHIOIO'IH m1anori'IHi MipKyBaHIUI ll.Jl.ll CHCTeM irnerpaJibHHX piBHHHb (10) i
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u0(x,t) , (x,t) E ~ = {(x,1) : 1 '.S: x s; 1t-t, 0 '.S: 1 '.S: rt/2},
u (x, t) = ul(x,t), (29)
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TioKa)KeMO, ll.\0 BiH e KJiaCH'{flHM p03B' R3K0M Miwattoi: 3all.a'li (1) - (3) B np.llM0Ky
THHKY Tir1.
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no6yaosaua cpynKttiH u(x, t) Ja,n0B0JJbHRe nepwy Il0'laTK0BY YM0BY u(x, 0) = <p(x).
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(12), MaeMo, mo u(0, t) = u(rt, t) = 0 ll.JIR ncix t e [0, rt /2).
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MYJIH (29) HaJie)KHTb KJiaCy C2 (I1r), RKll.\0 BHK0HyI0TbCH yM0BH (12) - (19) . .U,JR
u1,0I·0 noTpi6no nepeBip1n-1-1 uenepepsnicn, p03B ' jf3Ky u(x, t) i ltoro noximmx AO
apyr·oro nopRAKY BKJII0'IH0 11a xapaKTepHCTHKax X = I j X = 1t - I.
,[l,JIR A0Bell.eHlljf 11enepepBHOCTi cpyuKui'i u(x, t) na np51Mitl: X = t niacTaBHMO
3Ua'leHIU! t = X B CHCTeMH (9) i (10). BpaxoByl0'IH n0'taTK0Bi YM0BH (6), neprny
YM0BY IlOI'0A)KeHHjf (13) i €;\HHiCTb p0313 ' jf3KY CHCTet-m (9). o.nep)KyeM0
u? (x, x) = uf (x, x), u~ (_,;, x) = ukr. x). ,A.\ , x) = u 1 (x, x) , (30)
T06To cpyHKUi51 u( x, 1) nenepepsua ua xapaKTep11cnu1i x = t. AnaJIOri'IHO n epeKo
nyeM0C51, mo cpynKqi51 u(x, t) 1-1enepepBna 11a xapaKTep1-1crnqi x = 1t - t.
BHKOpHCTOBylO'IH piBIIOCTi (30) i t1)0PMYJJH nepexoay [l, C. 13) Bil\ Miwano'i :k'\Aa-
ISSN 0041 -6053 . Y,:p. stam . .1Kyp1t., /993, m. 45 , N'' 9
ICl-lYBAHlHI KJIACH1-1.HOfO P03B '.sl3KY MllllAHOI 3A.llA '-II . .. 1237
'-li (1) - (3) 11.0 3.41a'-li (4) - (6), Ma€MO, mo '-lacnmtti noximri nepworo nop.HAKY BiA
cpyHKu,ii u(x, t) € HenepepBHHMH Ha xapaKTepHCTHKax X = t i X = 1t - t.
)],JI.lf o6rpyHTYBaHH.lf HenepepBHOCTi 11.pyrux '-laCTHHHHX noxi,llHHX cpyttKu,ii
u(x, t) na xapaKTepHCTHKax x = t i x = 1t - t 6yAyTb BHKOpHCTani piBHOCTi
a2u"'(x,t)
a12
a2um (x,t)
ax2 =
(31)
1-( u1(x,t) - u21(x,t) ) =
ax 2
2-( cJu1(x ,t) _ ouz' (x,t))
2 dx dx '
(32)
m = 0, 1, 2.
Ili.llCTaBl-lMO cnoY.aTKY 3HaY.emI.ll t = X B CHCTeMH ,llJI.lf 3HaXO,ll.)Kelm.lf noxi,D,HHX
au;' (x, t)I dX, i = 1, 2, 3, m = 0, 1. BpaxoBylO'-IH IlO'l.aTKOBi YMOBH (6), Il03Ha'-leHIUI
Koecpiu,i€HTiB (8) , nepwy YMOBY noro11.)KeHH.lf (14), €AHHiCTb p03B' .lf3KY CHCTeMH iu-
Terpa.,tbHHX piBH.llHb fl]l.ll 3HaXOll,)KeHH.lf noxiAHHX au? (x , t) I dx, i = 1, 2, 3, i piBHO
CTi (32), Ma€MO, mo JJ.pyra noxi11.ua no x Bill cpynK1~ii u(x, t) € nenepepanoIO cpyu
Kl.l,i€IO ua np.HMill: X = t. Ananori'l.HHMH MipKyBaHH.lfMH MO)KHa noKa3aTli, ll.(0 JJ.pyra
noximrn no x niA cpynKu,ii u(x, t) € uenepepaumo q>ynKUi€IO na np.HMiil x = 1t - t.
Tenep niACTaBHMO 3Ha'-leHH.lf t = X B CHCTeMH All.lf 3tiaX0,D,)KeHH.lf noxiAHHX
a u;"(x, t)/ dx, i = 1, 2, 3, m = 0, 2. BpaxoByIO'-IH llO'-laTKOBi )'MOBH (6), n03Ha'-leHH.lf
Koe<pi1.1,i€HTiB (8), nepwy YMOBY IlOI'OA)KeHH.lf (14) , €AHHiCTb po3B'JJ3KY CHCTeMH iu-
TerpanbHHX piBH.llHb ,llJUI 3tiaX0l).)KeHH.lf noximrnx a u?(x, t)/ dt , i = 1, 2, 3, i piBHOC
Ti (31), Ma€MO, mo 1.1,pyra '-laCTHHHa noximrn no t Bill. cpynK1.1,ii u(x, t) € t1enepepa
no10 cpy11K1.1,i€IO 1-m xapaKTepuc-rm,i x = t. ITo ananorii nepeKony€MOCj{, mo 11.pyra
noxi11.Ha no t sin. <pyHKl.l,il u( x, t) € HenepepBHOIO <PYHKl.l,i€IO Ha np.SIMilt X = 1t - t .
OT)Ke,MH no6y.llyBaJIH €D.HHH!t KJiaCH'l.HHlt po3B' j{30K u(x, t), BH3tla'Iem-1fi cpop-
MYJIOIO (29) , Miwauoi 3aAaY.i (1)-(3) s npj{MOKyrnttKy ITT 1. To,D,i ttaM si,D,OMi 3Ha-
'-lemij{ cpynK1.1,i1 Ha npj{Mifi t = 7t / 2. ITpu8MaIO'-IH
( 7t ) * du(x,1t/2) *( )
U X, l = <p (X), at = \jf X
3a IlO'-laTKOBi yMOBH, 01.1,ep)Ky€MO Miwaizy 3al).a'-ly J.l,Jlj{ npj{MOKYTHHKa Ilr2 = {(x, t):
0 $ x $ 1t, 1t I 2 $ t $ 7t} B)Ke po3rJij{Ayaanoro rnny i T. A- rro a1-1anorii 3 np.HMOKYT
HHKOM Ilr1 • Miwaim 3a.ll,a'-la (1) - (3) Ma€ €AHHHfi KJJaCH'-IHHfi p038'j130K B ycix o6na-
CT.llX Ilrt TaKHX, mo LJk Ilrk = ITT. TeopeMa I 11.oue,.1.ena.
TeopeMa 1 Aa€ MO)KJII-IBiCTb ccpopMyJIIOBaTH neo6xiJ{Hi Ta L{OCTan1i YMOBH icny
saumt) €A1-n1ocTi Knacw-moro po3B' j{3KY Miwanoi 3aJ{a'li BHAY
I ? ? a-u a-u
~ - -;7° + q(x) II = 0, (33)
ut ux
u(0, t) = 11(1t, t) = 0, 0 $ t $ T,
0 ( ou(x,0)
u(x, ) = <p x), dt = \jf(x), 0 $ x $ 1t,
.llKi OAep)KaHi illWHM MeTC)H0M u po6ori [3 ].
(34)
(35)
Hac;iiooK 1. /) :1R i c 11yoaw-1R ma €0 11Hocmi J.:.1U1 c u•111mo p o 3<3' R3K)' Miwa,wi'
3aaa'li (33) -- (35) 0 06,wcmi TT T 11eo6xia1w i aocmamf/t:,0 , 111 06 cf_>)'IIKl{ii' cp(x),
\jf(X), q(x) 3aaooo ;11:,11J1; 1u )'MO/lll 1101.00;KC/l/l}I (12), (13) , (15) i )'MOllll
(j)" (0) = tp" (7t) = 0. (36)
ISSN 004/ -6053 . Y•p . .\/(1111. A _)'fllt ., 1993. Ill . ./5 . N'' 9
1238 JO. O,MliTPOITOflbCbK.Hit, n. r. XOMA
q(x) e: Cio,itJ · (37)
j(o6eiJeuusi. TTpH JJ,OBeJJ,eHHi BHXOJJ,HTHMeMO 3 TOro, w;o 3aJJ,alfy (33)-(35)
MO)Kffa po3rJJ.llJJ,aTH .llK MiwaHy 3al(alfy (1)-(3), KOJIH f(x. t) = 0, b(x. t) = 0,
c(x, t) = o. k(x, t) = q(x). 3po3yMiJIO, w;o 3 yMOBH (37) BHilJIHBa€ BKJJIOlfeHH.ll
q(x) e; c°·1(TTT)· TaKHM \JHHOM, BHKOHYIOTbCJI sci YMOBH TeOpeMH 1 JJ.Jl51 MiwaHoi
3Ma'!i (33)-(35), a 3Ha\JHTb, BOHa Ma€ €I~HHHfi KJJaCH\JHH!t p038'5130K u(x, t) B
o6JiaCTi TIT.
3ayooJK.eUIIJI. BCTaHOBJJeHi s JieMi 4.1 [3. C. 66) rpaHHlfHi YMOBH /(0, t) =
= /(re, t) = 0 JJ,Jl.ll q:>yHKl\il f(x , t) pa30M 3 BJJaCTHBiCTIO 11 uenepepBHOCTi JJ.O3BOJJ.ll
l0Tb 3p06HTH KOpHCHi BHCHOBKH BiJJ.HOCHO Ilp0JJ.0B)KeHOI q:>yHKUil ](z, 't). Ilo
nepwe,
j(z, 't) [ C(.IR 1 X [0, T]). (38)
TTo-JJ.pyre, Tenep yMosaM (18), (19) MO)KHa HMarn: i11wo1 eKsisaneHTHOi cf>opMH.
TcopcMa 2. a11.11 ic1tyeam1.11 ma €0UHOcmi KllGCU'tHOlO po38' Jl3KY Millla1toi" 3GOG'ti
(1)- (3) e 0611acmi nT ueo6xioHo i oocmamubO, 11406 cpy1tK4ii" <p(x), 'lf(X), b(x, t),
c(x, t), k(x, t), 3aiJoeo11b11.111111 yMoeu nowo;1wm.11 (12) - (14) i y;11oeu (15), (16),
cpy1tK4i.11 f(x, t) 3(1ooeo11bHJ111a y;11oey (17) i zpaHU'tHY y;11oey
/(0, t) = /(re, t) = 0, t e [0, T], (39)
p KOJKHUU 3 iHmezpa11ie p - (x, t), p+(x, t), JIKi (3U3Ha'ta/OmbCJI 3liOHO 3 cpopMyllaMU
(18), (19), HG/leJKGB XO'ta 6 OOHOMY 3 KllGCie C 1•0(TTT) a6o c0· 1(TT7-).
,[loseJJ,eHHJI TeOpeMH 2 auanori'!He JJ,OBeJJ.eHHIO TeOpeMH 2 po6onf [3).
B nopiBH.llHHi 3 TeOpeMOIO 1 TeopeMa 2 JJ.O3BOJJ51€ cnpocTHTH npoueJJ.ypy nepesip
KH YMOB po3B'.ll3HOCTi Miwanoi 3al(a'Ii (1) - (3) . 3aMiCTb nepesipKH 'IOTHpbOX BKJIIO
lfeHh, .llKi o6'€JJ.HaHi y TBepJJ.)KeHHJIX (18), (19), JJ.OCTaTHbO 3rim10 3 TeOpeMOIO 2 ne
peKOHaTI-fC.ll B cnpaBeJJ.JIHBOCTi TiJJbKH JI.BOX 3 HHX - no OJJ,HOMY Ha KO.>KHY 3 q:>yHK
uin. p-(x, t), p+(x, t). JJ,OJJ,aBllIH YMOBY (39) • .llKa nerKO nepesip.ll€TbCX.
Ha 3aKiH'IeHI-J51 3anponony€MO Jl,JI.ll MiwaHOl 3aJl,a'Ii (]) - (3) JI.Ba THilH Jl,OCTaTHix
YMOB p038' 513HOCTi, 6iJihllI .>KOpcTKHX, Hi.>K (17) - (19), ane BHpa)KeHHX B 6iJibllI
3BH'IHHX TepMinax 8 nopiBHXHHi 3 (18), (19).
Hac.11iiJoK 2. JIKU{O cpy1tKt4ii" b(x, t) , c(x, t). k(x, t) , cp(x), 'lf( x) 3aiJoeo11b
u.1110mb yMoeu meopeMu l , a cpy1tK4i .11 f ( x, t) 3aooeO/lbflJ1€ yJ1toey (39) i pwey
f(x, t) e c1·°(nT), mo iCHy€ €0LIHIIU KllGCU'tHUU JJ038' Jl30K Millla1toi" 3aOa'ti (1) - (3) (3
06/lacmi TIT.
,[loseaeHH.ll 6a3y€TbC.ll Ha BHBe,ll,eHHi YMOB TeOpeMH 1 3 aanoro HaCJJiAKY.
Hac.11iiJoK 3. JIK1qo cpy1tK4ii" b(x, t), c(x, t), k(x, t), cp(x), 'lf(x) 3ai)oeo11b
H.1110mb YMOBU meopeMU 2, a cpyHK4i.11 f (x, t) 3G0060llbH.fl€ YMOBY (39) i YMOBY
f(x, t) e CO,l(TTT), mo iCHy€ €0UHUU K/lQCU'tHUU JJ038' Jl30K .;,,tiutaHOi° 3(10G'ti (1) - (3) 6
06Mcmi TIT.
,[loseJJ,eHH.ll 6a3y€TbC.ll Ha BHBeJJ.eHHi YMOB TeOpeMH 2 3 ,ll,aHoro HaCJiiJJ.KY.
1. Mump0110/lbCKUi1 IO. A .. Xoua r. n. 06 c,q><peKTHBHOCTH npHMeHeHHJI aCHMllT0TH'!eCKHX
MeT0A0B K KB33HB0JIH0BblM ypaBHCHHJIM rHnep6oJIH'!eCKoro THna. - Kues, 1989. - 32 C. -
(IlpenpHHT / AH YCCP. IiH-T MaTeMaTHKH; N" 89.15).
2. A60IIUHJ< 8 . 3 ., Mb1LUKUC A. /i. 0 CMClliaHH0ll 33Aaqe AJIJI JIHHCllH0I! rHnep6oJIH'!eCKOll CHCTCMbl
Ha nnocKOCTH // Y'!. 3an. flaTB. yH-Ta. - 1958. - 20, N" 3. -C. 87- 104.
3. l/ep,uunw, B. A. O6ocHOBaHHe MeTona <I>ypbe B CMelliaHH0ll 33Aa'!e A JIJI ypas11e1rnll B '13CTHblX
npoH3B0LIHblX. - M .: li3LI- BO MocK. yn-Ta, 1991. - 112 C.
4. llempoao,ua /1. r. fleKI.IHH o6 ypasHCHHJIX C '13CTllblMH npoH3B0AHblMH. - M .: <I>H3MaTrH3, 1961.
-400c.
Onep)l(a110 20.01.93
ISSN 0041-6053. YKp. uam. J1<.yp1< .• 1993, m. 45, N" 9
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| id | umjimathkievua-article-5925 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:24:10Z |
| publishDate | 1993 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/a1/cd6f7b489ab9b68598e1267c97b47ca1.pdf |
| spelling | umjimathkievua-article-59252020-03-19T09:21:13Z The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation Існування класичного розв’язку мішаної задачі для лінійного гіперболічного рівняння другого порядку Mitropolskiy, Yu. A. Khoma, L. G. Митропольський, Ю. О. Хома, Л. Г. The paper deals with the problem of solvability of the mixed problem for a linear second-order hyperbolic partial differential equation. The minimal necessary and sufficient conditions for the existence of a unique classical solution to this problem are established. Вивчаються питання розв’язності однієї мішаної задачі для лінійного гіперболічного рівняння в частинних похідних другого порядку. Встановлені мінімальні необхідні та достатні умови існування єдиного класичного розв’язку даної задачі. Institute of Mathematics, NAS of Ukraine 1993-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/5925 Ukrains’kyi Matematychnyi Zhurnal; Vol. 45 No. 9 (1993); 1232–1238 Український математичний журнал; Том 45 № 9 (1993); 1232–1238 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/5925/8533 https://umj.imath.kiev.ua/index.php/umj/article/view/5925/8534 Copyright (c) 1993 Mitropolskiy Yu. A.; Khoma L. G. |
| spellingShingle | Mitropolskiy, Yu. A. Khoma, L. G. Митропольський, Ю. О. Хома, Л. Г. The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title | The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title_alt | Існування класичного розв’язку мішаної задачі для лінійного гіперболічного рівняння другого порядку |
| title_full | The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title_fullStr | The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title_full_unstemmed | The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title_short | The existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| title_sort | existence of a classical solution to the mixed problem for a linear second-order hyperbolic equation |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/5925 |
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