Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions
We consider new eigenvalue problems with discontinuous eigenfunctions and construct computational algorithms whose accuracy is not worse than the accuracy of analogous known algorithms for problems with smooth eigenfunctions.
Збережено в:
| Дата: | 1999 |
|---|---|
| Автори: | , , , , , |
| Формат: | Стаття |
| Мова: | Українська Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
1999
|
| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/4730 |
| Теги: |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860510896080551936 |
|---|---|
| author | Deineka, V. S. Sergienko, I. V. Skopetskii, V. V. Дейнека, В. С. Сергієнко, І. В. Сконецький, В. В. |
| author_facet | Deineka, V. S. Sergienko, I. V. Skopetskii, V. V. Дейнека, В. С. Сергієнко, І. В. Сконецький, В. В. |
| author_sort | Deineka, V. S. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T21:12:54Z |
| description | We consider new eigenvalue problems with discontinuous eigenfunctions and construct computational algorithms whose accuracy is not worse than the accuracy of analogous known algorithms for problems with smooth eigenfunctions. |
| first_indexed | 2026-03-24T03:04:16Z |
| format | Article |
| fulltext |
Y,RK 519.624
B. C. ~e~neKa, I. B. CepricnKO, B. B. CKoneubKmt
(hl-r Ki6epneTrlKrt HAH Ygpahm. Kffin)
3 A ~ A q I H A BJIACHI 3HAqEHHJ;I
3 P O 3 P H B H H M H B , / I A C H H M H r
TA I X q H C E J I b H I P O 3 B ' ~ I 3 K H
We consider new eigenvalue problems with discontinuous eigen-functions. We construct computational
algorithms which are not worse in precision than similar ones known for problems with smooth r
functions.
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�9 B.C. J1E~HEKA. I. B. CEPFICHKO. B, B. CKOREI.ibKHI~, 1999
ISSN 0041-6053. YKp. ~tam. ~.'vpu.. 1999, m. 51, N e- 10 1317
1318 B.C. ~EI~IHEKA, I. B. CEPFIEHKO, B. B. CKOFIELIbKMfl
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[41. T y r
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(9)
ISSN 0041-60M, Y~p. ~uam. ~'vpn.. 1999. m. 51. N e 10
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ISSN 0041-6053. Y~p. ~am. ~.'Vlm., 1999, m. M. 1~ !0
1320 B.C. ~E~IHEKA, I. B. CEPFI~HKO, B. B. CKOFIEIXbKHI;I
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o
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~ kij ~ - - + qV(x)uVv d ~ l
fit 1 ~xj ~X i
ge
= ~ ~ V ~ pu~vda,, (13)
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III0 piBHJIHH~I (14) cnpaBe~0mae a o6~acTi D~_.
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ISSN 00,11-6053. YKp. ~am. ~.'ypn., /999. m .S I . N e 10
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= I, II , Ha BJIacHi 3HaqeHH$1 [5, 7]
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a v ArT> 0, BY = BvT>0, a v {a~jji,j=l, ={b~} i , j= 1 ,
{ "~s=" ~cp'~q)~" qV{pf (x ){p~(x)}d~+, R+V[~0, ] [~0~ ]dT ' V
,, I ~xs ~xt
ai j = f f ~ v k~. " +
Q
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pa~erpa M C E (nesi~oMi 3rlaqeaH~ Ha6anaCCHOi saacHoi qbyazttii U~ N(X) y sya.no-
sr~x "ro,~Kax o6aacri ~, ,~a 3HaqeHH~I dpyrmuii u~N(x) i Si~OSi~anX neptuHx ii
'~aCTHHHHX noxi~aax ~ y mma~zy nuKoprm-rauHs epHiT0sOi irrrcpnoasttii). TyT
v N vk k
U k ( X ) = Z �9 co i ~0 i (x) n ff . (18)
i=l
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moro polly Ha Bi~apiazy 7 pO3HCmysaaH)l o6Jmcrell f~, ~2-
Ha OCHOSi npHHtmny HiaiHazcy [5 ] HaeMo
X~ = min ImaxRV(u)} , 1=1,2 , ' . . . ,
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J
OT~e,
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ai~noaitmoro npOCTOpy Hk N, k = 1, 2, 3
~JI~l OTpHMaHH~I ORiHOK noxn60Z Ha6JIHmCHHX BJIaCHHX 3HaqCHb ra sijanosizmax
Ha6YiM)zeHax B.rlacHHX dpytIzai~{/~oRiJ1bHO BBCCTH B po3ras/I onepaTop npoezTysaH-
Ha l'I ~iHiiaHHX MHO)ZHH M v, V = I, If, Ha npocropa Hff, k = I, 2, 3, diJyHKtdll
M CE TaKHH qartOM-."
- - 0, u �9 M v �9 (19)
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JIeMa 2. ]la,~ KO~HOi" claynK~ii" U �9 M v V = I, If, icHye i npumo~ty eOunutl
e,le~tenm l'lu �9 Hk #, k = l, 2, 3, maKua, u4o zaOosoabnae cnissiOnoutenn~ (19).
YlKu4o u(x) a KoY,~itl 3 oS/zacme~ [21 , E4 2 ~tae oS~w~eHi nenepepsHi ~acmunni
noMOHi ( k + l)-zo nop.~9~.% mo cnpase&ausa oLliHKa
I$SN 0041-6053. Y~p. ~tam. #,~pn.. 1999, in. 51. A r~ I0
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ll._rl.llw~ < chk -. f(O)' (20)
ae c const > 0, h ~ naaSiabvaais Ooeorun cmopin mpuKymnurie -i posaumma = e r
oaaacmi'~i k~cmeninb'noaiuOadeMCEinpu k= I f(O) -cosO (O
-i no.aoeuna ee.au,~unu na~Fi.abu~ozo 3 rymie e r ) ; npu k = 2, 3 f ( O ) = sin O ( 0
- i 6e.au~ut.ta na~,~tet-tutozo 3 rymi6 e r ) .
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fV(x) = L vu, u ~ M v,
i n06Y/IyBaTri Kpafloay 3a, aa,qy 3 po3pHBHHM pO3B'.q3KOM U(X) Ha y TaKHM tlHHOM:
LVu =fV(x) ," x e ~ 2 , u e M v, f v e L2 (~ ), (21)
/~e KpafloBa yMOBa Ta yMoarl Crlp~DKeHH~I BrlaHaqalOTbCJl Bil~noBi/~HO cIIiBBi~HomeH-
H.,qMH (2 ) - - (4 ) qH (6)-(8).
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cOBaHoMy f r e~aH)iM pOaB'aaKOM aa~aqi (21) i e~aand~ eKcTpeMamno qbymctdoHana
eHeprii
FV(v) - a V ( v , v ) - 2 ( f v , v ) , v ~ H v, (22)
Ha orpHManoMy panitue eHepre~qHO~/npOCTOpi H v .
Hexafl u(x) ~ pO3B'513OK KpafloBoi 3a41a~Ii (21), a uN(x) e n N - - ii 6KHHHI~
Ha6JIH~eHrlII y3ara~bHeHHfl po3a'~13OK. To~ai cnpaBe]mHBi piBHOCTi
o~V(u, v : ) " ( fv , Vff), (23)
a V ( u : , v f f ) = ( fv ,v ;v ) . :Vvff'e H : . (24)
Bi~iMmo'~n (24) BiA (23), oAep~xyeMo .
aV ( u - uff , vff ) = 0 V vff e H : . (25)
Orz~e, rlu = u~(x).~ H~.
II,O TaKi, II~O Hexall B H N iCHy~T~ ~Ba e~eMerm[ urn'N; u k
~,,(,,_ ~N ,,~")=0, o:(u-.~'",o"~)=o. /,t k ,
To~ai, , n6Hpam~m v N = u I 'N - ulk I 'N, o'rpHMyeMo cynepe,-IUiCTl,. OT~Ke, i cny -
sanHz i r esleMeHTa 17 u e H~ r /Iose/Ieab.
l'lcpefl~teMo ~o oTpHMaHHZ ottimcn (20). Hexafl u - - po3pnBmm ~aCaqHHa poa-
n'aZOK ~p.aaoagi ' aattaqi (21). IIoKa3KeMO, t~O ~ a 6y/xg-aKoi V (X) e H v cnpaBea-
mtsa piBaicn, [4]
F~(v)-F"(u) = I Io -u l l~ , , ~ M v, V v ~ H r . (26)
/3i~CHO, oczism~rt
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TO
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3Haqenlt,q a poaplIBHHMH B~IaCHI.IMtl dpyl-IKtd~lMH [7]. HpoBc/leHi o6nrtc~qmBam, ni eK-
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1. 3apy~un B.C. HHxeHepm,m MeTOllbl petuemt8 m/tan "letumnpoaolttmcTrL- M. :~guepr0aTOM-
HaltaT. 1983. -- 326 C.
2. l.U.at, tKoa 10. I-I.. f'tmun E.A. Kom'aK'rm,fla rem|oo6MeH. -- M. ; J'l. :~nepvoa'roMt~altaT, 1987. -
265 c.
3. Koa3oga f/ . /I . Me'rolu,~ pemetma He~mne|lnt,~x aa/taq ren.nonpo~ottuOCl~. - KrmB: HayK. ltyMKa,
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4. Cepzuemr H. B, Ctr B. B., ~eiineKa B. C. Ma'reMartlttecKoe MO/leJIapoBaill, le H I.leCJle/to-
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7. ~eiineKa B. C.. Cepzuetttr H. B.. C~onet,Kuii B. B. Ma'teMarrtqeCKrte MO/IeJIH H M~:'l'OllJd paeqe'ra
aa/laq c pa:~pbmmaMn petuemiaMm - KneB: HayK. ttymKa. 1995. -262 r
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nic~/toonpa,0otmtnm - 19.01.98
ISSN 0041-6053. Yh'p. ~tam. ~.'Vlm., 1999. m. 51. N ~ I0
|
| id | umjimathkievua-article-4730 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | Ukrainian English |
| last_indexed | 2026-03-24T03:04:16Z |
| publishDate | 1999 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/db/1b870fc525dccc5f91bcb34d22fe21db.pdf |
| spelling | umjimathkievua-article-47302020-03-18T21:12:54Z Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions Задачі на власні значення з розривними власними функціями та їх чисельні розв'язки Deineka, V. S. Sergienko, I. V. Skopetskii, V. V. Дейнека, В. С. Сергієнко, І. В. Сконецький, В. В. We consider new eigenvalue problems with discontinuous eigenfunctions and construct computational algorithms whose accuracy is not worse than the accuracy of analogous known algorithms for problems with smooth eigenfunctions. Розглянуто нові задачі на власні значення з розривними власними функціями. Побудонано обчислювальні алгоритми, які за точністю не гірші за аналогічні, відомі для задач з гладкими власними функціями. Institute of Mathematics, NAS of Ukraine 1999-10-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/4730 Ukrains’kyi Matematychnyi Zhurnal; Vol. 51 No. 10 (1999); 1317–1323 Український математичний журнал; Том 51 № 10 (1999); 1317–1323 1027-3190 uk en https://umj.imath.kiev.ua/index.php/umj/article/view/4730/6161 https://umj.imath.kiev.ua/index.php/umj/article/view/4730/6162 Copyright (c) 1999 Deineka V. S.; Sergienko I. V.; Skopetskii V. V. |
| spellingShingle | Deineka, V. S. Sergienko, I. V. Skopetskii, V. V. Дейнека, В. С. Сергієнко, І. В. Сконецький, В. В. Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title | Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title_alt | Задачі на власні значення з розривними власними функціями та їх чисельні розв'язки |
| title_full | Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title_fullStr | Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title_full_unstemmed | Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title_short | Eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| title_sort | eigenvalue problems with discontinuous eigenfunctions and their numerical solutions |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/4730 |
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