О корректности многомерных задач Дарбу для волнового уравнения
В області Dε ⊂ Em₊₁ обмеженій поверхнями |x|=t+ɛ and |x|=1−t і t=0,0≤ɛ<1, доведена коректність багатовимірних задач Дарбу для хвильового рівняння, а також вивчена поведінка одержаних розв’язків при ɛ→0. It is proved that multi-dimensional Darboux problems for the wave equation are correct in the...
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| Veröffentlicht in: | Український математичний журнал |
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| Datum: | 1993 |
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Інститут математики НАН України
1993
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| Zitieren: | О корректности многомерных задач Дарбу для волнового уравнения / С.А. Алдашев // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1299–1306. — Бібліогр.: 15 назв. — рос. |
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Алдашев, С.А. 2020-02-10T09:40:15Z 2020-02-10T09:40:15Z 1993 О корректности многомерных задач Дарбу для волнового уравнения / С.А. Алдашев // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1299–1306. — Бібліогр.: 15 назв. — рос. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164605 517.956 В області Dε ⊂ Em₊₁ обмеженій поверхнями |x|=t+ɛ and |x|=1−t і t=0,0≤ɛ<1, доведена коректність багатовимірних задач Дарбу для хвильового рівняння, а також вивчена поведінка одержаних розв’язків при ɛ→0. It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain Dε ⊂ Em₊₁ bounded by the surfaces |x|=t + ɛ and |x|=1- t and the planet = 0, 0 ≤ ɛ < 1. The behavior of the solutions as ɛ→0 is studied. ru Інститут математики НАН України Український математичний журнал Короткі повідомлення О корректности многомерных задач Дарбу для волнового уравнения Correctness of multi-dimensional Darboux problems for the wave equation Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
О корректности многомерных задач Дарбу для волнового уравнения |
| spellingShingle |
О корректности многомерных задач Дарбу для волнового уравнения Алдашев, С.А. Короткі повідомлення |
| title_short |
О корректности многомерных задач Дарбу для волнового уравнения |
| title_full |
О корректности многомерных задач Дарбу для волнового уравнения |
| title_fullStr |
О корректности многомерных задач Дарбу для волнового уравнения |
| title_full_unstemmed |
О корректности многомерных задач Дарбу для волнового уравнения |
| title_sort |
о корректности многомерных задач дарбу для волнового уравнения |
| author |
Алдашев, С.А. |
| author_facet |
Алдашев, С.А. |
| topic |
Короткі повідомлення |
| topic_facet |
Короткі повідомлення |
| publishDate |
1993 |
| language |
Russian |
| container_title |
Український математичний журнал |
| publisher |
Інститут математики НАН України |
| format |
Article |
| title_alt |
Correctness of multi-dimensional Darboux problems for the wave equation |
| description |
В області Dε ⊂ Em₊₁ обмеженій поверхнями |x|=t+ɛ and |x|=1−t і t=0,0≤ɛ<1, доведена коректність багатовимірних задач Дарбу для хвильового рівняння, а також вивчена поведінка одержаних розв’язків при ɛ→0.
It is proved that multi-dimensional Darboux problems for the wave equation are correct in the domain Dε ⊂ Em₊₁ bounded by the surfaces |x|=t + ɛ and |x|=1- t and the planet = 0, 0 ≤ ɛ < 1. The behavior of the solutions as ɛ→0 is studied.
|
| issn |
1027-3190 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/164605 |
| citation_txt |
О корректности многомерных задач Дарбу для волнового уравнения / С.А. Алдашев // Український математичний журнал. — 1993. — Т. 45, № 9. — С. 1299–1306. — Бібліогр.: 15 назв. — рос. |
| work_keys_str_mv |
AT aldaševsa okorrektnostimnogomernyhzadačdarbudlâvolnovogouravneniâ AT aldaševsa correctnessofmultidimensionaldarbouxproblemsforthewaveequation |
| first_indexed |
2025-11-24T16:49:11Z |
| last_indexed |
2025-11-24T16:49:11Z |
| _version_ |
1850487097977208832 |
| fulltext |
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It is proved that the multidimensional Darboux problems for the wave equation are correct in the domain
D, C Em+I bounded by the surfaces lxl = t + £, lxl = 1- t, and the plane t = 0, 0 $ £ < I. The behavior
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3aAa4y (12), (21) rno11.1-1M K m rrer pam,H0MY ypasuemIIo
½
k - k ):. . J-k a,(s)):. f,, (S) - <?o,1('>) -t- <?o,,(S1)a,:-l µ ~ d'> l •
1; '> I '>I
(23)
C, l. + J:.2 d k
J,k (1'-) -Ck (F) + J p [ 2 11 I '> ] ~ Jr. .
,, " = o,, ..., o µ s<½+111) a111 ·,1
Ypam1e1rne (23) - llaCTllblA c.11y4all ypai.memrn (35.17) H3 [ 12]. K0Topoe o6paTHMO B
£ml.le (CM. TalOKe 114])
(24)
CJienou,rrem ,110. q)y11K r1.1rn. (17) >1.11Jrn erc>1. pewe1111eM Jana,111 (1), (4). ni.e v;, ( r, t).
k = I, k,,. II= 0, I, .... onpenem1er c>1. no ctl0pMyJ1e (22). B K0T0poh cp7i,, (s) IH\X0-
11-,m.:}f 113 (24).
Y ,111T1,ma>1. 01pa11ll'1e1111>1 11a 3a;1.a11111,ie qly11KqH11 1:0(r , 0). <J 1 (r, 0) 11 -reopeMy
35. 2 H3 [ l 2]. a TaK>Ke <t >op ~1y Jlbl ( 18) - (20), M0>Kl!0 n0Ka3aTb. 'ITO non yYemme
peu1e1rne u(r, 0 , 1) (17 ) np1111a1u1e>KHT K1iaccy u e C ( D 0 )nC 2(D 0 ). Ha '.HOM
3,lBepwaeTC}f /l0Ka:J<\TeJlhCTII0 reope~lbl 2 /lJl>I 3aWlYII ( l ). (4) .
CnpaBe/VllHlOCTh Teope~11,1 2 /\JI}! :J<Uta'III ( l ). (5) yciammmmaeTC>l a11aJIOf'HYH0.
3. ,UoKa33TCJlhCTBO TCOpC~I B CJIY'HlC E > 0. C 11a,iam1 paccwffpHM 3a/J.aYY ( I ),
(2). PeUJe1rne 3 1 o ti 3a/Hl'IH oy/leM IICKaTh TaK)Ke fl BHJte p >l.i \a ( 10). TOI / HI ee K03</>
ct \J11\ll e11Tbl 6y II.YT y /lOIIJler11op>1.Tb ypam1e111110 (12) . fl pH 3T0 ~t Kpae 11 1,1e YC.ll()Bfl}I (2 )
n11>1. cfly 11K1111n v;,(~, T] ) . k = f-c. 11 = 0, 1 . . .. . :J<1n1-1wyrc >1. 11 11 11ne
ISSN 0041 -6053 . .Y,p. \f(l/ /1 . )K\'f' II . • / <)93 . 111 . ./5 , N" 9
1304 C. A. AJlL(AlIJEB
(25)
TaKHM o6pa30M, 3a.naqa (1), (2) CBOAHTCjj K Ja,AatJe (12), (25).
,Ua.nee, ucn0Jib3Yjj pewettue (14) 3MatJu Kown AJijj ypaeHeHHjj (12) npu ll = E / 2
c ytJeT0M KpaeebIX ycnoeuA (25), nonytJaeM mner·pa.nbHoe ypaene1me Bonbreppa
nepeoro poaa
gk(r-) = ../z ,nk (r-) - 't~11 (E/2) _ ,:,, (S) +
11 ~ 't'£11 ~ .Ji ../2
r- S k 2
+ ~ - e/2 J •r,,(S, ) P.'( s, +se/2 ) ds
.Ji<s+e/2) s, µ s,<s+e/2) 1• £/2
K0T0poe AH<pcpepemiupoeattueM CBOAHTC51 K ypaeHeHmo BoJibTeppa BT0poro poaa
dg~ k JS k d ( Sf +se/2 )
dx = v£n(S) + v£n(S,)~kµ(s, s,) dS1, kµ(S, s,) = pµ S (s+e/2) ·
£/2 ~I I
O6parue nocneattee ypaeHeHue, pewenue 3MatJH (12), (25) H3 (14) M0)KH0 nonytJHTb
B 51BH0M BHAe (CM. TaK)Ke [14]):
vk(S ll) = ,k (Tt) + s- Tl 511 1:!,,<s, ) P.'( STl + Sf ) dr- +
n ' £II S + Tl £/2 s, µ (s + Tt)S1 ~,
+ J [P ( 2s11+S1E+(s-11)(s,-e/2)) +
£ 12 µ <s + 11)(s1 +e/2)
+ sgn(s1-11) - Ip ( 2STt+S1E-(S-Tt)(s1-E/2) )]d'l'~n dS. (26)
2 µ <s + 11)(s1 +e/2) ds1 1
CneaoeareJibHO, cpyHKQHj! (17) j!BJijjeTcj! pewe1meM 3aAa'l.H (1), (2), rAe v!(r. t),
k = 1, k,., n = 0, 1, .. . , onpeAeJijjlOTCjj H3 (26).
Tenepb aepeAAeM K Ja.na'l.e (1 ), (3). Ee peweHHe TaK)Ke M0)KH0 U0JIYtJHTb e BHAe
k --pjjAa (17), rAe v,.(r, t), k = I, k,., n = 0, I, . .. , j!BJIReTcjj pewettueM ypaetteHHjj
(12), YAOBJiereopRIOlllHM K.paeeoMy ycnoemo
( ~€ -~ ) ls~11 = vt,@, v!( S, ½) = <?!,.(s), (27)
v:,.(s) = ../2 (2s)'m---l)/2 v:n<2s), ~ $ S $ .!._
2 2
3aaatJa (12), (27) pewaercjj atta.norHtJH0 3aAatJe (12), (25), H ee pewettue Janucb1ea
ercJ1 B BHAe [14)
k ( 4s11 + e2 ) k ( e ) 5s ( s11 + s? ) k
v,.(S, Tl) = pµ 2(s + Tt)E <p£11 2 + £// µ (s + Tt)S1 vt,,(S,) ds, +
ISSN 0041-6053. YKp. J,W/11 , JK.JpH., 1993, Ill. 45, N• 9
O KOPi-'i:KTHOCTH MIIOfOMEPllblX ]i\nA'-1 nAPGY . .. nos
+ f [ p ( 2sri + ½le+(f, - ri)(f,l - ei 2) ) +
½ µ (f, + T1)(S1 +c:/2)
+ 1-sgn(f,1-ri)p ( 2f,11+f,1e- (f, - ri)(f,1 - e/2) )]d<p~,, df,. ,28)
2 µ (f, + ri)(f,, +e/2) elf,, ~
Tenepb H3Y'HIM none11.e11He 33l\a4H l npu £ ➔ 0. lh npeL1.CTa1u1e1rnjf pewemu1
(17) 3T0t:l: 3aJla4'.H BHLJ.H0, 4'.T0 LlJUI 3T0r0 LI.0CTaT0'IH0 HCCJieJl(maTb n0Be11.e1me cpyHK
QHH (26) u c11y4:ae 3Ma41-1 (1), (2) 1-1 (28) n c11y4ae 3aLl.a'm (I), (3) . I-fa (26) HeTPYJlll0
n0JIY4'.HTb, lfT0 eCJm 't:11 (f,) - e" H <p!11 (f,) - E"-1, TO v!11 (f,) - c:11- l np11 e ➔ 0 . A
H3 (28) CJ1eL1.yeT, 4TO eCJIH v!,,(f,) - i::11- 1 H <i>!,,(f,) - E" , TO -r!,,(f,) - E" npu: e ➔ 0.
113 YKa3allHblX Bb!We <:paKTOB H H3 orpamc14e1rnH Ha 3a)l.am11,1e cpyHKUHlt -re<r, 0),
vr(r, 0), cr E(r., 0), a TaK)Ke 1-13 cpopMy11 (18) - (20) BhITeKaeT cnpaBeHJIHBOCTb
Te0peMbl 3.
Tenepb nepet:l:11.eM K I\0Ka3aTeJibCTBY TeopeMhl 4. Ec11u: pewe,rne 3Ma4'.u: (1), (4)
6yJJ,eM HCKaTb 8 BHJJ,e pHJJ,a (17), TO Olla CBeJJ.eTCH K peweumo 3al\a'IH (12), (21), rJJ,e
e / 2 ~ S ~ I/ 2, e / 2 ~ Tl ~ I/ 2.
,llanee. u:cn0Jib3Y H pewe,me 3c'\D.a4'.1-1 fypca JJ.JIH ypam1ettu:H (12)
k I'- _ k I'- J ( 1 . I'- ) V\jlf,, v,.(..,, Tl) - <i>e,,(..,) - J R 2,111, ..,,, Tl ~ dT11 -
e/2 v•11
I;
- f <p~,,<s,> 0! R( S1, f; s, ri) cJri, ,
1/2 ':,)
(29)
3Malfy (12). (21) CBOAHM K yprumemno
k * 'J'2
k )'- a ( s 2 +s,c:12 ) )'-
1,, (S) = <i>e,.@ + ~ <i>e,.(..,,,) OS1 p µ S(S1 +e/2) c1..,,,, (30)
rD,e
!; I Tl +f,2 d k
f, k (I'-) 1 t (I'-) + J p [ "'> I ] 'JI 111 dT) .
" ':, = (II ':, µ I'-( I +n ) an I
e/2 ':, 2 'II 'II
O6paTHB (30), 6y,!J.eM HMeTb (14]
k . k f,-E/2 IJ/2 l '[ f,f+f,,e/2] k
<pe,.(SJ = f,. (S) + s + e/2 S1 Pµ f,(s1 + E/2) f,, (S1) df,1.
!;
(3 I)
TaKHM o6pa30M. pHA (17) .llBJrnercH pewe~meM 3aAalfH (1), (4). rne v~(r, t) , k =
= I, k11 • 11 = 0, I, ... , onpeAeJIHeTCH H3 (29), B Koropot:l: <p:11 (f,) uaxoD..lITCH H3 (31).
TipH 3T0M C y4'.eT0M rJIMKOCTH rpaHH'IHblX n.aHHblX. a TaK)Ke cpopMyJI (18) - (20)
tterpy.ntto noKa3aTb, lfT0 n0JIY'lemme pewem1e u e C(l5 e)()C2(De)-
Cnpa~~.UJIHBa c11e.ny10ll.la.ll 11et-1Ma.
JleMMa 2. flpu z ~ 1, µ ~ 3 <PYHK14uu P µ(z) u P~ (z) Jt011R10mcR oo3pacma
,ou~u.Mu U y008AClll80JJRIOn1 HCJ)Q8CHCnl8QJ.1
ISSN 0041 -6053 . YKp. Mam . xyp1<., 1993, m. 45. N• 9
1306 C. A. AJijlAWEB
CrrpaBe/lJlHBOCTb JieMMbl 2 Jiel' K 0 CJ1e1weT H3 mnerpaJibHhlX npeHCTaBJie1mtr
Q)YHKI.J;liH Jle)Kannpa H ee np01i3B0/lH0TT:
1 7( ~
Pµ(z) = - J (z + ✓z~-lcos cpl dcp,
7t 0
P~(z) = µPµ _1(z) + µ(µ4+l) j (z + ✓z 2 -lcos cpt-2 sin2cpd<p.
0
)la.nee, ecJm B03bMe!>1' cr1 (r, 0) = 0. 'tr(r, 0) = 'r~ (r) Y,~,,, (8), TO p ewe,me 3aAa'-IH (1) ,
8 -k - k --(4) M0)KH0 HCKan, B BUL(e u(r, , t) = v,, (r) Y,,_,., (8), k = 1. k,.. 11 = 0, l , . . . . Ton~a
cnpase,u1mB0CTb ouenKH (7) IIJrn Ka)Knoro II c .11enyeT H3 tpopMyJI1,1 (31) c y<JeT0M
JieMMbl 2.
TaKli1'1 o6pa30M, nepna>1 •iaCT b TeopeMhl 4 H0K33aHa; BT0pa.>1 lJ3CTI, 1WKa3h!BaeTC51
aHaJIOrli'Hl0.
CJieAyer 3aMeTHTb, lJT0 npH f, ➔ 0 TeopeMa 3 cormtcyeTC51 C pe3yJJbTaTaMM
reopeMhl l, jlJI.>I Teope).f 2 H 4 T3K)Ke c rrpaueJ~JIHB am1JI0rHl.J.l·lhl0 cpaKT.
B [15] /l0Ka:XlH0 cyw.eCTB0BclHHe KJiaCCHlJeCKHX pewe,rnrl. 3a/lalJH 2 npH m = 2 ,
y /WBJieTD0pRI0W.HX yCJIOBH51M er(H HCTBeJIH0CTH. HeTpy /lH0 3UMeTHTh, '-ITO 3TH yc
JI0BH>I e/llfHCTBCHH0CTH BblIT0JIIUHOTC.sl, B l.J.3CTII0CTH, H {lJl.>I 1·pa1rn'-!I-Ib!X Il.fillHblX Te0-
peMbl 2.
3a.Me'lauue. OrMeTI-IM, '-ITO B H0Ka3aHllhlX TeopeMax npHHaLVIe)KJI0CTb 3a)laJ-I
llblX cpyHKqHR K yKa.JaHHbIM KJiaccaM cymecrnemia . ITpu 1-1apyweHim 3THX YCJI013Htl
B [6] noKa3aH0, lJT0 ecJm £ = 0, TO 3a{lal.J.a 1 M0)KeT HMeTb 6ec•rnc11e1moe MH0)KeCT-
130 pewem-1rl., a /lJI.sJ :kVJ.a'IH 2 ur 11 111 np11 r ➔ 0 HMeI0T oco6em1ocT1, nop>1L{Ka ,-- 2 ,
a B c1iy•me £ > 0 pernen11e .JaWl '-1 1, 2 M0)KeT ue cyw.ecTBonari,.
I. Prolter M. H New boundary va lue problems for the wave eq uation and equat ions of mixed type//
J. Rath. Mech . and Anal. - 1954. - 3, N" 4. - P. 435-446.
2. Wand Guand Wind. The Goursat problems in space// Sci . Rec. New scr. - 1957. - l , N" 5. -
P. 7-10.
3 . Mux,11111 C. T. l'vlH01 ·m1epm,e c11m-y11.Hpm,1e 1111Terpai1L111,1e ypaB11eHH.H . - M.: (])u3~1:nT113, 1962. -
254 C.
--1- . A.1oaweo C. A. 0 11eK0Top1,1x Kpacm,1 x 3a11a'-lax 1111.11 ~1Hor0Mcp11oro no,1110Boro ypan11e 11 1-1.11 //
/lOK,1 . AH CCCP. - 1982. - 265, N° 6. - c. 1289-1292.
5. Tong Kwand Chang. On a lxrnndary value problem for the wave equation// Sci. Rec. New ser. -
1957. - I , N2 5. - P. 1- 2.
6. A ,1oa111eo C. A . KpaeB1,1e '.k'lila'-11-1 rvrn M11oro~1epm,x 1·m1cp(x>J1H'-!eCKHX ypa1111e 11H l-1 11 ypan11em1l-1
c~1e1u a11uo1--o -runa: /lHc .. . . 11-pa ,p113AtaT. 11ayK. - K11es, 1990. - 188 c.
7. Xe Kall '-lep . 0 HCC/lHHCTBCIIIIOCTl1 pc1uc1rnll HCKor op1,1x Kpaenux 3a/l a'-I 11 .1 1.11 BUpO)KLlaKllllHXC.11
ypaBHetmtl // )loKJJ. All CCCP. - 1983. - 272, N° 5. - C. 1066-1068.
8. A.1oa111eo C. A. 0 11eKOTopux 110KaJ1LllhlX H 11e110KanLmn Kpaen1,1x 3a;ia•rnx Ll'1.II no111101,or:o
ypaBHCHIUl // }J,11ct)(pcpc111). y paBHCIIH.11. - 1983. - 19. 1'2 1. - C. 3-8.
9. Popiva1101• N. /. , Schneider M. The Darboux-problem in R3 for a Class of Degenerated Hyperbolic
Equation// )loKJJ. Eo;w AKa11emn1 11ayK. - I 988. - 41 , N° 11. - C. 7-9.
10. Ewiao3e A . B. Ypan11c1111.H orn111a1moronma. - M.: H3/l-BO AH CCCP, 1959. - 164 c .
1 I. Copso11 E . T . On the Rie mann-Green functi on // J . Rath. Mech. and Anal. - 1958. - J. -
P. 324-348.
12. CaMk·o C. r .. Ku ,16ac A. A ., Mapu•,eo 0. 1-f. l-f11Tcr·pa111,1 1-1 11po1-130011111,1c npo611on> 11op.llHKa H Hc
KOTopue HX np1111o)Kc11>1.11. -M>1 11cK: llay Ka H Texm1Ka, 19R7 . - 688 c.
13. Eei111l.M((/l r. , 3poeilu A. BhlCllll-lC rpa!ICUCll!lt'-ITl'I-ILIC lPYIIKIWH'. B 3-x T. - M .: H ayKa, 1973. - T. I.
- 294 c .
14 . Jlu B. 0 Ht:KOTOp1,1x npoc-rpaHCTDCIUIIJX 3ULla'-!ax nma f y pca: J111cc . . Ka 1w . ,t>H3.-MaT. 11ayK. -
Am1a-A1a, 1969. - 81 c.
15. Xe Km, '-lep. 3a;1a'-IH )lap6y - npoTrcpa /lJl.11 ;m r 1cp 11010 BOJIIIOllOfO ypaBIIC III-I.>!. - Bm1m1BOCTOK,
1990. - 25 c. - \ ITpe 11pmrr / JlBO All CCCP. HnMJ.
noJ1y<1e110 23. Ill . 91
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