Розв’язки двоточкових і крайових задач для гіперболічних рівнянь другого порядку: Автореф. дис… канд. фіз.-мат. наук / П.В. Цинайко, Львів. держ. ун

EUeA?ANUeEEE AeA?AEAAIEE OI?AA?NEOAO

?iai? ?aaia O?aiea

OeEIAEEI IAO?I AANEEUeIAE*

OAeE 517.944

?ICA’ssCEE AeAIOOI*EIAEO ? E?AEIAEO CAAeA* AeEss

A?IA?AIE?*IEO ??AIssIUe Ae?OAIAI II?ssAeEO

01.01.02-aeeoa?aioe?aeuei? ??aiyiiy

Aaoi?aoa?ao aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa o?ceei-iaoaiaoe/ieo iaoe

Euea?a — 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a Oa?iii?euenueeiio aea?aeaaiiio iaaeaaia?/iiio
oi?aa?neoao? ?iai? Aieiaeeie?a Aiaothea.

Iaoeiaee ea??aiee aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Oiia A?eai??e Iao?iae/,

Oa?iii?euenueea aeaaeai?y ia?iaeiiai ainiiaea?noaa, i?ioani? eaoaae?e
aeui? iaoaiaoeee.

Io?oe?ei? iiiiaioe: aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Nethna?/oe Aaneeue THoeiiae/,

??aiainueeee aea?aeaaiee oaoi?/iee oi?aa?neoao, i?ioani? eaoaae?e aeui?
iaoaiaoeee;

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani?

Eaeaithe Iao?i ?aaiiae/,

Aea?aeaaiee oi?aa?neoao “Euea?anueea iie?oaoi?ea”, caa?aeoaa/ eaoaae?e
ia/enethaaeueii? iaoaiaoeee ? i?ia?aioaaiiy.

I?ia?aeia onoaiiaa: *a?i?aaoeueeee aea?aeaaiee

oi?aa?neoao ?i. TH. Oaaeueeiae/a, eaoaae?a aeeoa?aioe?aeueieo ??aiyiue,
I?i?noa?noai ina?oe Oe?a?ie, i.*a?i?aoe?.

Caoeno a?aeaoaeaoueny “21” aeiaoiy 1999 ?. i 1520 aiae. ia can?aeaii?
niaoe?ae?ciaaii? a/aii? ?aaee Ae 035.051.07 i?e Euea?anueeiio
aea?aeaaii-io oi?aa?neoao? ?i. ?.O?aiea (290001, i. Euea?a, aoe.
Oi?aa?neoao-nueea, 1).

C aeena?oaoe??th iiaeia iciaeiieoeny o a?ae?ioaoe? Euea?anueeiai
aea?aeaa-iiai oi?aa?neoaoo (i. Euea?a, aoe. Ae?aaiiaiiaa, 5).

Aaoi?aoa?ao ?ic?neaiee “17” aa?aniy 1999 ?.

A/aiee nae?aoa? Ieeeothe ss.A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Iniiaiith i?iaeaiith a oai??? ??aiyiue
iaoaia-oe/ii? o?ceee ? a?aeooeaiiy ?ica’yce?a aeeoa?aioe?aeueieo
??aiyiue c /anoeiieie iio?aeieie, ui caaeiaieueiythoue iaai? aeiaeaoeia?
oiiae, cie?aia ii/aoeia? oa e?aeia?. Iaeiae o oai??? cae/aeieo
aeeoa?ai-oe?aeueieo ??aiyiue ii?yae c ii/aoeiaith caaea/ath (caaea/ath
Eio?) ii/aee aea/aoe aaaaoioi/eia? caaea/?, ui iaei i?e?iaeia
ocaaaeueiaiiy ye o iaoaiaoe/iiio ?icoi?ii?, oae ? a ?icoi?ii? o?ce/ii?
?ioa?i?aoaoe??. Oaea caaea/a a 60-eo ?ieao aoea iinoaaeaia ? aeey
??aiyiue c /anoei-ieie iio?aeieie. Iio??aii a iaeano?

Bp = (t, x): 0 F t F T; — Y < x < Y, i=1,2,...,p ciaeoe ?ica'ycie a?ia?aie?/iiai ??aiyiiy L[u(t,x)] = f(t,x), (t,x) I Bp , (0.1) yeee caaeiaieueiy? oiiae u(tj ,x) = jj(x), j=1,2,...,n, 0F t1F t2 F …F tnFT. (0.2) Aeyaeeiny, ui ?ica'ycie aaaaoioi/eiai? caaea/? (0.1), (0.2) acaaae? ia aoaea ?aeeiei. sse anoaiiaeaii A.E.Ioaoieeii, aeine?-aeaeaiiy oaeeo caaea/ aeiaaa? aeiaeaoeiaeo oiia, iaeeaaeaieo ia ooieoe?th f(t, x). sseui aeei?enoiaoaaoe iaoiae Oo?'? aeey aeine?aeaeaiiy caaea/? (0.1), (0.2), oi aeiaeaoeiaeie oiiaaie aeey ooieoe?? f ? ia??iaee/i?noue ca i?inoi?iaeie ci?iieie. Iaeii/anii c ?icaeoeii oai??? aaaaoioi/eiaeo caaea/ iaoiaeii Oo?'? aeine?aeaeoaaeeny e?aeia? ia??iaee/i? caaea/? utt - a2 uxx = g(x,t)+e f (x,t,u,ut ,ux), u(0,t)= u(p,t)=0, (0.3) u(x,t+T)= u(x,t), aeey a?ia?aie?/ieo ??aiyiue ae?oaiai ii?yaeeo. Ia aeaiee iiiaio iioae?eiaaii /eiaei i?aoeue, i?enay/aieo aeine?-aeaeaiith e?aeiaeo caaea/ ? e?aeiaeo ia??iaee/ieo caaea/ aeey ??cieo eean?a aee-oa?aioe?aeueieo ??aiyiue. Cai?oeii, ui e?aeia? ia??iaee/i? caaea/? aeey cae/aeieo aeeoa?aioe?aeueieo ??aiyiue a?oioiaii aea/ai? A.I.Naiieeai-eii ? eiai o/iyie ca aeiiiiiaith /enaeueii-aiae?oe/iiai iaoiaeo. Aei 80-o ?ie?a aeey ??aiyiue c /anoeiieie iio?aeieie caeaa?eueoiai aeiaaaeai-iy ?nioaaiiy ia??iaee/ieo ?ica'yce?a i?iaiaeeeinue ca aeiiiiiaith ?yae?a Oo?'?, aei oiai ae ia??iae T ? e?aeiaa oiiaa i?aeae?aeenue oae, uia iiaeia aoei aeinyaoe aaaeaiiai ?acoeueoaoo. Ia?oith na?aae ?ia?o o oeueiio ia-i?yi? aoea ?iaioa I. A. A?oai'?aa, a ye?e ?icaeyaeaeinue eiie?aoia ia-e?i?eia a?ia?aie?/ia ??aiyiiy ae?oaiai ii?yaeeo aeaeyaeo ztt-zxx=O(x,t)+ef(z). Iia? /anoeia? ?acoeueoaoe i?e ?ica'ycaii? e?aeiai? ia??iaee/ii? caaea/? aoee iaea?aeai? TH. I. Aa?acainueeei, A. I. Ai?ieii, O. A?ac?nii, I. Aaeaiaeith, I. I. Eaaeeaeainueeith, A. I. Ieo?yeiaei, E. I??aiaa?aii, A. E. Ioaoieeii, I. ?aa?iiae/ai, ?. A. ?oaeaeiaei, ?. A. Ne?eiieeii, N. E. Niaie?aei, A. O. Nieieiaei, A. I. Nieiaeiaei. Aeooaeuei?noue ?icaeoeo oai??? e?aeiaeo caaea/ ye aeey cae/aeieo aeeoa?aioe?aeueieo ??aiyiue, oae ? aeey ??aiyiue ?c /anoeiieie iio?aeieie aecia/a?oueny iio?aaaie i?aeoeee o ca'yceo c aaaeeea?noth ?? canoino-aaiiy aeey ?ica'ycaiiy aaaaoi/enaeueieo i?iaeai ? o.ae. sse oia?oa ao-ei aeacaii I. A. A?oai'?aei, iaei??th c i?e/ei, yea ca'ycaia c ?ica'ycaiiyi ia??iaee/ieo caaea/ (0.3), ? i?iaeaia iaeeo ciaiaiiee?a. Oaea i?iaeaia aeieeea i?e ?ica'ycaii? aaaaoioi/eiaeo caaea/ aeey ??aiyiiy a?ia?aie?/iiai oeio. Caoaaaeeii, ui eeoa i?e eiie?ao-iiio aeai?? /enea a (?aoe?iiaeueiiio), ia??iaeo T=1 ? a?aeiia?aeieo e?aeiaeo oiiaao x=0 ? x=1 I. A. A?oai'?ao aaeaeiny aeiaanoe oai?aio ?nio-aaiiy ? ?aeeiino? ?ica'yceo caaea/? (0.3). Oaeee i?aeo?ae i?e aeiaaaeai-i? ?nioaaiiy ia??iaee/ieo ?ica'yce?a ??aiyiue ?c /anoeiieie iio?aeieie aeei?enoiaoaaany aaaaoueia iaoaiaoeeaie (A. I. Ea?iii, A. I. Ieo?yeiaei, A. E. Ioaoieeii, A. I. Iie?uoeii, I. ?aa?iiae/ai, A. O. Nieieiaei, I. A. Nieiaeiaei), aei oiai ae ?acoeueoaoe iaea?aeoaaeenue eiaeai ?ac o niaoe?aeueii aeae?eaieo i?inoi?ao ooieoe?e. O 1984 ?ioe? /anueeeie iaoaiaoeeaie I. Aaeaiaee ? I. Ooaae?e a ?iaio? aaeaeiny eeaneo?eoaaoe i?inoi?e ?ica'yce?a e?aei-ai? ia??iaee/ii? caaea/? (0.3). A?eueoa aueiai, oi/i? ?ica'ycee e?i?e-ii? caaea/? (0.3) (e=0) ciaeaeai? ca aeiiiiiaith i?inoi? iiaeeo?eaoe?? oi?ioee Aeaeaiaa?a, yea aeicaieeea oieeiooe ae?ac?a, a yeeo iio??aii noioaaoe iane?i/aii? ?yaee. Ia?aaaaith ?ic?iaeaiiai aiae?oe/iiai ia-oiaeo, yeee aeei?enoiao?oueny a i?inoi?? iaia?a?aii aeeoa?aioe?eiaaieo ooieoe?e, e??i iaaecae/aeii? i?inoioe, ? ? a?aenooi?noue oiiae ia cia-/aiiy ooieoe?e, ye? noiyoue a i?aa?e /anoei? ??aiyiiy utt - uxx =f(x,t,u,ut, ux, e), o iaaeiaeo oi/eao ?ioa?aaeo [0, p]. Iio??aii cao-aaaeeoe, ui a aiioiaai?e aeua ?iaio? I. Aaeaiaee ? I. Ooaae?e ia aea-/aeany i?iaeaia aeieeiaiiy i?inoi??a ?nioaaiiy ?ica'yce?a. Ae??oaiith oaeeo ieoaiue, a oaeiae a?aeooeaiith aiae?oe/ieo ?ica'yce?a caaea/? (0.3), i?enay/ai? ?iaioe TH. I. Ieo?iiieuenueeiai ? A. I. Oiie, A. I. Oiie ? ss. A. Iao??anueeiai, I. A. Oiie. O ieo /anoeiai ?icaeiaii iiaee iaoeiaee iai?yi o oai??? /enaeueii-aiae?oe/ieo ia-oiae?a ye aeey oaeeueiaeo cae/aeieo aeeoa?aioe?aeueieo ??aiyiue ae?oaiai ii?yaeeo, oae ? aeey oaeeueiaeo ??aiyiue a?ia?aie?/iiai oeio c /anoei-ieie iio?aeieie. Aeei?enoiaoth/e iaoiaeeeo aeine?aeaeaiiy e?aeiaeo ia??iaee/ieo caaea/ (0.3) aeuacaaaeaieo aaoi??a, iaie a aeena?oaoe?ei?e ?iaio? i?iaaaeaii aeine?aeaeaiiy e?i?eii? aeaiooi/eiai? caaea/? utt - uxx = g(x, t), (x,t) I R2, (0.4) u(x,0) = u(x, p)=0, x I R; e?i?eii? e?aeiai? caaea/? utt - uxx = g(x, t), x I R, 0

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