Задачі з вільними границями для еліптичних та параболічних рівнянь: Автореф. дис… д-ра фіз.-мат. наук / М.О. Бородін, НАН України. Ін-т приклад. мат

22

Iaoe?iiaeueia Aeaaeai?y Iaoe Oe?a?ie

?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee

Ai?iae?i Ieoaeei Ieaen?eiae/

OAeE 517.946

Caaea/? c a?eueieie a?aieoeyie aeey ae?ioe/ieo oa ia?aaie?/ieo ??aiyiue

01.01.02 — aeeoa?aioe?aeuei? ??aiyiiy

Aaoi?aoa?ao

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

aeieoi?a o?ceei-iaoaiaoe/ieo iaoe

Aeiiaoeuee-1999

Aeena?oaoe??th ? ?oeiien

?iaioo aeeiiaii a Aeiiaoeueeiio aea?aeaaiiio oi?aa?neoao? I?i?noa?noaa
ina?oe Oe?a?ie

Io?oe?ei? iiiiaioe:

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani? Ai?en Aaneeueiae/ Aacae?e,
caa?aeoaa/ a?aeae?eii ??aiyiue iaoaiaoe/ii? o?ceee ?III IAI Oe?a?ie

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani? Na?a?e Iaaeiae/ Eaa?aithe,
caa?aeoaa/ eaoaae?ith aeeoa?aioe?aeueieo ??aiyiue Euea?anueeiai
aea?aeaaiiai oi?aa?neoaoo

aeieoi? o?ceei-iaoaiaoe/ieo iaoe, i?ioani? Ieeiea Ea?aiaoiae/
Ea?aiaoyioe, caa?aeoaa/ eaoaae?ith aeeoa?aioe?aeueieo oa ?ioaa?aeueieo
??aiyiue ?inoianueeiai aea?aeaaiiai oi?aa?neoaoo

I?ia?aeia onoaiiaa:

?inoeooo iaoaiaoeee IAI Oe?a?ie, i.Ee?a, a?aeae?e aeeoa?aioe?aeueieo
??aiyiue

Caoeno a?aeaoaeaoueny “ 22 “ aeiaoiy 1999. i 15 aiaeei? ia
can?aeaii? niaoe?ae?ciaaii? ?aaee Ae 11.193.01 aeey caoenoo
aeena?oaoe?e ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a o?ceei-iaoaiaoe/ieo
iaoe i?e ?inoeooo? i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie ca
aae?anith: 340114, Aeiiaoeuee, aoe. ?ice Ethenaiao?a, 74.

C aeena?oaoe??th iiaeii iciaeiieoenue o a?ae?ioaoe? ?inoeoooo
i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie ca aae?anith: 340114,
Aeiiaoeuee, aoe. ?ice Ethenaiao?a, 74.

Aaoi?aoa?ao ?ic?neaiee 15 aa?aniy 1999?.

A/aiee nae?aoa? niaoe?ae?ciaaii?

a/aii? ?aaee
Eiaaeaanueeee I. A.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Iniiaiei ia’?eoii iaoiai aeine?aeaeaiiy yaeythoueny
iae?i?ei? caaea/? iaoaiaoe/ii? o?ceee ?c a?eueieie a?aieoeyie. Aiie
i?aaenoaaeythoue iaoaiaoe/i? iiaeae? i?ioean?a, aeey yeeo oa?aeoa?iith
iniaeea?noth ? iayai?noue ??cieo ca nai?ie oa?aeoa?enoeeaie oac, ui
?icae?eythoueny iaa?aeiiith a?eueiith iiaa?oiath. Oae? i?ioeane
a?aeaoaathoueny a aeayeeo no/anieo iaoaeo?a?eieo oaoiieia?yo (
iaia?a?aiee ?iceea noae?, aeaeo?ioeaeiaee ia?aieaa), i?e ooai?aii? oa
aaiethoe?? iiey?ii? e?eae, i?e ae?iuoaaii? iiiie?enoae?a, a oai???
i?oaeiino?, a?ae?aae?oe?, a oai??? ai??iiy oa aeayeeo ?ioeo iaeanoyo
iaoee oa oaoi?ee. Iayai?noue a?eueii? a?aieoe? ?iaeoue aeacai?
iaoaiaoe/i? iiaeae? nooo?ai iae?i?eieie oa iniaeeai aaaeeeie aeey
aeine?aeaeaiiy.Oiaoi c iaeiiai aieo ie ia?ii ci?noiaiee iaoaiaoe/iee
ia’?eo, a c ae?oaiai aieo — ia’?eo, yeee ia? /eneaii? canoinoaaiiy.
Aoaeue yea ei?aeoii iinoaaeaia caaea/a, ui ia? ia iao? iien ae?eniino?,
iiaeiia caaeiaieueiyoe oae? aeiiae: 1) caaea/a ia? ?ica’ycie; 2)
?ica’ycie ?aeeiee; 3) ?ica’ycie no?eeee. Aeena?oaoe?y i?enay/aia
aeine?aeaeaiith eeane/ii? ?ica’yciino? aeayeeo aaaaoiaei??ieo caaea/ ?c
a?eueieie a?aieoeyie aeey ae?ioe/ieo oa ia?aaie?/ieo ??aiyiue.

Na?aae caaea/ ?c a?eueieie a?aieoeyie oeaio?aeueia i?noea iin?aea?
i?iaeaia Noaoaia. Aiia aoea noi?ioeueiaaia noi ?ie?a iacaae, eiee
aano??enueeee o?cee E.Noaoai ni?iaoaaa iiaoaeoaaoe iaoaiaoe/io iiaeaeue
ooai?aiiy oa aaiethoe?? e?eae o na?oiaiio ieaai?. Nii/aoeo iniiai?
coneeey aoee ni?yiiaai? ia aea/aiiy iaeiiaei??ieo caaea/, a oaeiae
caaea/ ?c oeee?iae?e/iith oa noa?e/iith neiao???th. A ?iaioao
E.?.?oa?iooaeia, Aeae.Aeoaeana, E.O?eea, I.I?ei?/a??i, A.I.Iae??iaiiaa
oa ?i. iaeiiaei??i? caaea/? aoee aea/ai? c aeinoaoiueith iiaiioith.
Iniiaiee iaoiae aeine?aeaeaiiy — ?aaeoeoe?y aei ?ioaa?aeueieo ??aiyiue
aieueoa???anueeiai oeio.

Ii/aoie aea/aiiy aaaaoiaei??ii? caaea/? Noaoaia aoei iieeaaeaii a
?iaioao I.A.Ie?eiee oa N.A.Eaiaiiiinonueei?. Aoea canoiniaaia
eiioeaioe?y ocaaaeueiaiiai ?ica’yceo, ui aeicaieeei aeiaanoe oai?aio
?nioaaiiy oa ?aeeiino? neaaeiai ?ica’yceo. Ia ii/aoeo n?iaeanyoeo ?ie?a
a ?iaioao E.Aaeiee? oa A.Aethai aoa cai?iiiiiaaiee niin?a ?aaeoeoe??
aeayeeo caaea/ ?c a?eueieie a?aieoeyie aei aa??aoe?eieo ia??aiinoae. Oe?
?iaioe aeaee iiooaeiee iiooiao ?icaeoeo oai??? aa??aoe?eieo ia??aiinoae
oa aeicaieeee A.O?eaeiaio, Ae.E?iaea?ea?a?o, E.I??aiaa?ao aeiaanoe
?nioaaiiy aeiaaeueiiai eeane/iiai ?ica’yceo a iaeiioaci?e
ianoaoe?iia?i?e caaea/? Noaoaia. Oae? ?acoeueoaoe aoee iaea?aeai?
caaaeyee ooiaeaiaioaeueiei aeine?aeaeaiiyi E.Eaoa?aee? i?i aeaaee?noue
a?eueieo a?aieoeue.

Aeaioacia caaea/a oaeiae aeicaiey? aa??aoe?eio iinoaiiaeo, aea ia
a?aei?io a?ae iaeiioacii? caaea/? ooo aaeaeiny aeiaanoe eeoa
iaia?a?ai?noue ?ica’yceo. Iai?ee?ioe? naieaeanyoeo ?ie?a A.I.Iae??iaiia
cai?iiiioaaa ?ioee i?aeo?ae aei aea/aiiy aeaioacii? caaea/?. Aa?aoe
niaoe?aeueio ?aaoey?ecaoe?th oa ci?ii? I?cana, a?i aeia?a ?nioaaiiy
eeane/iiai ?ica’yceo o iaeiio ca /anii. A?aei?oeii oaeiae ?iaioe
?.?.Oaicaae, yeee, aeei?enoiaoth/e ?acoeueoaoe Aeae.Iaoa oa TH.Iica?a,
aeia?a aiaeia?/ia oaa?aeaeaiiy. A.A.Aacae?e cai?iiiioaaa ?ioee iaoiae
aeine?aeaeaiiy eeane/ii? ?ica’yciino? o iaeiio ca /anii, yeee
nie?a?oueny ia oai?aio Oaoaea?a. Oaeei aea iaoiaeii eiio aaeaeiny
aeiaanoe eeane/io ?ica’yci?noue o ?yae? ?ioeo caaea/ oeio Noaoaia.
Aiaeia?/i? ?acoeueoaoe iaea?aeai? ?.A.?aaeeaae/ai.

O 1987 ?ioe? aoea iioae?eiaaia ?iaioa ?.Ii/aooi, a ye?e aea/aeany
iiiaii??ia caaea/a Noaoaia o oae caai?e aioaeuei?ei?e iinoaiiaoe?.
Aeiaaaeaii e?ioeoeaa?noue a?eueii? a?aieoe?. A 1996 ?ioe? iioae?eiaaia
?iaioa Aeae.Aoaianiioeina oa E.Eaoa?aee?, a ye?e iieacaii, ui
?acoeueoaoe iiia?aaeiuei? ?iaioe i?e aeeiiaii? aeayeeo oiia iiaeia
iineeeoe.

A?aei?oeii, ui aaeeea cia/aiiy iaea iaeyaeiaa ?iaioa ?. ?. Aeaieethea,
i?enay/aia i?iaeai? Noaoaia, yea iioae?eiaaia a 1985 ?ioe?.

?ioee eean caaea/ ?c a?eueieie a?aieoeyie, yeee aea/a?oueny a
aeena?oaoe??, aeieea? i?e aea/aii? oaeeueiaeo oa eaa?oaoe?eieo oa/?e
??aeeie, a iaoaiaoe/i?e oai??? ai??iiy. Oe? caaea/? a?ae??ciythoueny
a?ae caaea/? Noaoaia oei, ui aiie iae?i?ei? ia o?eueee /a?ac iayai?noue
a?eueii? a?aieoe?, a e /a?ac iae?i?ei?noue a?aie/ieo oiia. O
noaoe?iia?iiio aeiaaeeo ae??oaeueiei iiiaioii i?e aea/aii? oaeiai ?iaeo
caaea/ ? aeyaeaiiy aa??aoe?eii? i?e?iaee ?ica’yce?a. Oea i?ecaaei aei
aea/aiiy ?ioaa?aeueieo ooieoe?iiae?a ?c ci?iiith iaeanoth ?ioaa?oaaiiy.
Aeine?aeaeaiith oaeeo ooieoe?iiae?a i?enay/ai? ?iaioe
E.O?eae?eona,.I.Aa?aaaaeyia, O. Eaa?, I. Oeooa?a, ?.?.Aeaieethea, A.A
Aacae?y, A.TH.Oae?iiaa oa ?i. A oeeo ?iaioao aaeaeiny anoaiiaeoe
?nioaaiiy ?ica’yceo c eeane/ieie aeeoa?aioe?aeueieie aeanoeainoyie, aea
oiiae ia a?euei?e a?aieoe? aeeiiothoueny iaeaea anthaee. Iio?i a
na?aaeei? ainueieaeanyoeo ?ie?a a ?iaioao O.Aeueoa, E.Eaoa?aee? oa
A.O?eaeiaia aoei aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo o ieineiio oa
inaneiao?e/iiio aeiaaeeao.

Ianoaoe?iia?i? caaea/?, ye? iienothoue i?ioean iioe?aiiy aeeooc?eiiai
iieoi’y a iaoaiaoe/i?e oai??? ai??iiy, aeine?aeaeoaaeeny o ?iaioao
O.Aaiooeaeue, A.I.Iae??iaiiaa, E.A.Eaoa?aee?, Ae.E.Aaneana,
A.A.Aaeaeo?iiiaa, Aeae.Oaenoioa oa ?i. Iniiaieie ?acoeueoaoaie oeeo
?ia?o yaeythoueny oae?:

1) ?nioaaiiy eeane/ieo ?ica’yce?a o iaeiiaei??ieo aeiaaeeao;

2) ?nioaaiiy neaaeeo ?ica’yce?a o aaaaoiaei??ieo aeiaaeeao;

3) noaa?e?caoe?y ?ica’yce?a;

4) ?nioaaiiy eeane/iiai ?ica’yceo o aaaaoiaei??iiio aeiaaeeo o iaeiio
ca /anii.

Ia?aoo?, a aeena?oaoe?? aea/athoueny ieoaiiy ?nioaaiiy ?ica’yceo a
eaac?noa-oe?iia?i?e caaea/? Noaoaia. Oey caaea/a ia? oaieio?ce/ia
iioiaeaeaiiy. Aiia iiaeaeth? i?ioean iioe?aiiy oaiea o na?aaeiaeu?, yea
ciaoiaeeoueny o aeai-oaciaiio noai?, yeui i?eionoeoe, ui o?iio
e?enoae?caoe?? ?ooa?oueny ??a-iii??ii ?c noaeith oaeaee?noth oa o
a?aeiia?aei?e ?ooii?e nenoai? eii?aeeiao oaiia?aoo?a ia caeaaeeoue a?ae
/ano. A ?acoeueoao? iaea?aeo?ii aeaioacio eaac?-noaoe?iia?io caaea/o
Noaoaia.

Aia?oa ?nioaaiiy eeane/iiai ?ica’yceo a iaeiioaci?e eaac?noaoe?iia?i?e
caaea/? Noaoaia aoei aeiaaaeaii a ?iaioao aaoi?a. Aeey oeueiai aoea
aeei?enoaia ?aeay Aaeiee?. Caaea/o aaeaeiny ?aaeoeoaaoe aei ae?ioe/ii?
aa??aoe?eii? ia??aiino?, a iio?i, aeei?enoiaoth/e iaoiae eieaeueieo
aa??aoe?e oa iaoiae neiao?oaaiiy, aeiaanoe aeaaee?noue a?eueii?
a?aieoe?. Oaeei aea iaoiaeii aoea aeiaaaeaia eeane/ia ?ica’yci?noue a
inaneiao?e/iiio aeiaaeeo. Aea/aiith ??cieo aniaeo?a eaac?noaoe?iia?ii?
caaea/? i?enay/ai? ?iaioe ?.?.Aeaieethea, A.A.Aacae?y,
A.TH.Oae?iiaa,N.I.Ae?aoy?ueiaa oa ?i.

Ioaea, caaea/? ?c a?eueieie a?aieoeyie aoee a oeaio?? oaaae aaaaoueio
aeaeaoieo iaoaiaoee?a. Ca inoaii? o?e aeanyoee?ooy iioae?eiaaii a?eueoa
oeny/? ?ia?o, i?enay/aieo oe?e oaiaoeoe?. A ?acoeueoao? i?iaaaeaieo
aeine?aeaeaiue caciaee iiaeaeueoiai ?icaeoeo iaoiae ?ioaa?aeueieo
ooieoe?iiae?a ?c ci?iiith iaeanoth ?ioaa?oaaiiy, iaoiae aa??aoe?eieo
ia??aiinoae, iaoiae eieaeueieo aa??aoe?e, iaoiae neiao?oaaiiy oa ?i. Aea
a oeeo ?iaioao ia aaeaeiny noai?eoe iaoiaea, yeee ae aeicaieea
aeine?aeeoe aeinoaoiuei oe?ieee eean noaoe?iia?ieo oa ianoaoe?iia?ieo
caaea/ ?c a?eueieie a?aieoeyie. E??i oiai, ieoaiiy ?nioaaiiy aeiaaeueieo
eeane/ieo ?ica’yce?a o aaaaoiaei??ieo caaea/ao aoee aea/ai?
iaaeinoaoiuei.

Iaoa ? caaea/a aeine?aeaeaiiy. Aeine?aeaeaiiy ?nioaaiiy aeiaaeueieo
eeane/ieo ?ica’yce?a o aeaioaci?e aaaaoiaei??i?e caaea/? Noaoaia aeey
e?i?eiiai oa eaac?e?i?eiiai ??aiyiue oaieii?ia?aeiinoe a caaea/ao, ye?
iienothoue i?ioeane ai??iiy, ?nioaaiiy eeane/iiai ?ica’yceo a
noaoe?iia?ieo caaea/ao, ye? aeieeathoue i?e aea/aii? no?oieiieo oa
eaa?oaoe?eieo oa/?e ??aeeie, eaac?noaoe?iia?ii? caaea/? Noaoaia.

Noai?aiiy iaoiaea, yeee aeicaiey? aeine?aeeoe aeaaee?noue a?eueii?
a?aieoe? aeey oe?eiai eeano noaoe?iia?ieo oa ianoaoe?iia?ieo caaea/.

Iaoeiaa iiaecia io?eiaieo ?acoeueoao?a. Iaoeiaa iiaecia ?acoeueoao?a
aeena?oaoe?? iieyaa? o ianooiiiio:

1) cai?iiiiiaaii iiaee iaoiae aeine?aeaeaiiy oe?eiai eeano iae?i?eieo
caaea/ ?c a?eueieie a?aieoeyie aeey ae?ioe/ieo oa ia?aaie?/ieo
aeeoa?aioe?aeueieo ??aiyiue ae?oaiai ii?yaeeo;

2) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo o oe?eiio ca /anii o
aeaioaci?e aaaaoiaei??i?e caaea/? Noaoaia aeey e?i?eiiai oa
eaac?e?i?eiiai ??aiyiue oaieii?ia?aeiino?, a oaeiae o eiioaeoi?e caaea/?
Noaoaia;

3) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo a oe?eiio ca /anii o
caaea/?, yea iiaeaeth? i?ioean iioe?aiiy aeeooc?eiiai iieoi’y a oai???
ai??iiy, a oaeiae eeane/ia ?ica’yci?noue noaoe?iia?ii? caaea/?, yea
aeieea? i?e iieno no?oieiieo oa eaa?oaoe?eieo oa/?e ??aeeie o
a?ae?iaeeiai?oe?;

4) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo a iaeiioaci?e ieine?e oa
inaneiao?e/i?e eaac?noaoe?iia?i?e caaea/? Noaoaia aeey e?i?eiiai
??aiyiiy, neaaeiai ?ica’yceo aeey eaac?e?i?eiiai ??aiyiiy, a oaeiae
?nioaaiiy neaaeiai ?ica’yceo o aeaioaci?e ieine?e eaac?noaoe?iia?i?e
caaea/? Noaoaia.

Iniaenoee aianie caeiaoaa/a. O noi?nieo ?iaioao [4], [6] aaoi?o
iaeaaeeoue iinoaiiaea caaea/? oa iaoiae aeine?aeaeaiiy.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. ?acoeueoaoe aeena?oaoe??
aeiiia?aeaeeny ia Ananithci?e eiioa?aioe?? “Iaoaiaoe/ia iiaeaethaaiiy
i?ioean?a ioaa?ae?iiy.iaoae?a oa nieaa?a.” o i. Iiainea??nuee (1983 ?.),
ia ?anioae?eainuee?e eiioa?aioe?? “Eiiieaeni? iaoiaee o iaoaiaoe/i?e
o?ceoe?” a i. Aeiiaoeuee (1984 ?.), ia ?aaeyinueei-/aoineiaaoeuee?e
ia?aae? “Canoinoaaiiy ooieoe?iiaeueieo iaoiae?a oai??? ooieoe?e aei
caaea/ iaoaiaoe/ii? o?ceee” a i.Aeiiaoeuee (1986 ?.), ia
?anioae?eainuee?e eiioa?aioe?? “Iae?i?ei? caaea/? iaoaiaoe/ii? o?ceee” a
i.Aeiiaoeuee (1991 ?.), ia I?aeia?iaeieo eiioa?aioe?yo ii iae?i?eieo
caaea/ao iaoaiaoe/ii? o?ceee a i.Ee?a (1995,1997 ??.), ia I?aeia?iaei?e
eiioa?aioe?? “Aeeoa?aioe?aeuei? ??aiyiiy oa noi?aei? ieoaiiy” a i.
Iineaa (1996 ?.), ia nai?ia?ao ?i. ?. A. Iao?ianueeiai a i. Iineaa
(1976, 1980, 1983 ??.), ia I?aeia?iaei?e eiioa?aioe?? “Iaoai?ea
nooe?eueiiai na?aaeiaeua ?c a?eueieie a?aieoeyie” a i. Iiainea??nuee
(1991 ?.), ia I?aeia?iaeieo eiioa?aioe?yo “Iaoiaee iaoaiaoe/ii? o?ceee”
a i.?ao?a (1995 ?.), a i.Ee?a (1997 ?.), ia nai?ia?ao o
I.A.Eaaeeaeainueei? a i. Eai?ia?aae, ia nai?ia?ao ?.?.Aeaieethea,
?.A.Ne?eiieea, A.A.Aacae?y a ?III IAI Oe?a?ie oa ?i.

Ioae?eaoe??. ?acoeueoaoe aeena?oaoe?? iioae?eiaaii a ?iaioao [1] — [28].

No?oeoo?a aeena?oaoe??. Aeena?oaoe?y neeaaea?oueny ?c anooio, o?ueio
aeaa oa nieneo e?oa?aoo?e (166 iaeiaioaaiue). Caaaeueiee ia’?i
aeena?oaoe?? neeaaea? 282 noi??iee.

CI?NO ?IAIOE

Ia?oa aeaaa aeena?oaoe?? i?enay/aia aeine?aeaeaiith aeiaaeueii?
eeane/ii? ?ica’yciino? aeaioacii? aaaaoiaei??ii? i?iaeaie Noaoaia aeey
e?i?eiiai oa eaac?e?i?eiiai ??aiyiue oaieii?ia?aeiino?, a oaeiae
eiioaeoii? caaea/? Noaoaia. Aeey aeine?aeaeaiiy oeeo caaea/
cai?iiiiiaaii iiaee iaoiae, nooue yeiai a oiio, ui: iiaoaeiaaii aeayeo
iine?aeiai?noue ae?ioe/ieo aeeoa?aioe?aeueii-??cieoeaaeo ai?ieneioth/eo
caaea/, onoaiiaeaii ?o ?ica’yci?noue, aeiaaaeaii ??aiii??i? ioe?iee, a
iio?i cae?eniaii a?aie/iee ia?ao?ae.

Iaoae

D=x3:01,

ia a?aeii?e a?aieoe?

u(x,t)=(x,t) ia B1(0,T)BT(0,T); (2)

ia iaa?aeii?e a?aieoe? T=TDT=GTDT

[u]=0, wmetafile8? ?????????????
???????????yyy????.????1?????? ???????
 a???&??yyyy?????AyyyFyyy ??C?? ???&?
?MathType??a????u?y??????ii/aoeia? oiiae

u(x,0)=(x) a D, (x)=(x,0) ia B1B2, 0<(x)<1 ia B1, (x)>1 ia B2, (4)

Ooo b(u) eoneiai-noaea ooieoe?y, ??aia b1 a Toa b2 a GT,(x,t), (x,t)
caaeai? ooieoe??, n ii?iaeue aei iiaa?oi? T — iai?yieaia a noi?iio
c?inoaiiy u(x,t), [u(x,t)], [ux(x,t)]

n ??cieoe? i?ae a?aie/ieie cia/aiiyie ia T ye? acyoi c iaeanoae Toa
GT a?aeiia?aeii,

n , b1,b2,R1,R2, caaeai? aeiaeaoi? eiinoaioe.

Iiaoaeo?ii nenoaio ai?ieneioth/eo caaea/. O ca’yceo c oeei ?ic?a’?ii
oeee?iae? DTieiueiaie t=kh, hN =T, ooo N — aeayea aeiaeaoi? /enei,
k=1,2, …..N. Aeey aeia?eueiiai >0 aecia/eii ooieoe?th (x)C2(1) oae:

(x)=1 x1, (x)=0 x1+, ’(x)0.

Iicia/eii b(x)=b2+(b1-b2)(x), k(x)=(x,kh). Iaaeeaeaoeii ooieoe?th
u(x,t) ooieoe?yie uk(x,h,), ye? aecia/eii oae:

uk — 1hwmetafile8? ??,???????????
???????????yyy????.????1?????? ??????? 
 ???&??yyyy?????Ayyy?yyy`??O?? ???&?
?MathType??????u?y??????uk=(x,kh)=k(x) ia B1B2, u0=(x) a
D, (6)

Fk — 1hwmetafile8? ?????????????
???????????yyy????.????1?????? ??????? 
@???&??yyyy?????Ayyy?yyy???O?? ???&?
?MathType??????u?y??????Fk=0 ia B1B2, F0 = 0 a D.
(8)

I?e o?eniaaieo >0, h>0wmetafile8? ??T?????????????
???????????yyy????.????1?????? ???????  
???&??yyyy?????Ayyy¬yyya???I?? ???&?
?MathType??P????&??yyyy??????????ia? i?noea

Oai?aia 1.12. Iaoae aeeiiothoueny oiiae:

wmetafile8? ??T?????????????
???????????yyy????.????1?????? ???????  
???&??yyyy?????Ayyy¬yyya???I?? ???&?
?MathType??P????&??yyyy??????????(x)C2+ (D) k(x) C2+(D),
(0,1),

aeey ooieoe?e (x) e k(x) i?e xD oa k=0 aeeiiothoueny
a?aeiia?aei? oiiae ocaiaeaeaiiy. Oiae? caaea/a (5)-(8) ?ica’ycia oa
uk(x,h,) C2+(D), Fk(x,h,) C2+(D).

Iicia/eii /a?ac

wk(x,h,)=uk(x,h,) — Fk (x,h,) (9)

oa a?aei?iaii (7) c (5). Oiae? ooieoe?? wk(x,h,) aoaeooue ?ica’yceaie
ianooiii? caaea/?

wk — 1h wmetafile8? ??4???????????
???????????yyy????.????1?????? ???????
` ???&??yyyy?????Ayyy?yyy`???? ???&? ?MathType??
???u?y??????C ??aiino? (9) aeieeaa?, ui ooieoe?? uk(x,h,)
i?aaenoaaeai? noiith aeaio aeiaeaie?a, iaeei c yeeo wk(x,h,)
iaia?aae ? a?eueo aeaaeeith ooieoe??th , i?ae ?ica’ycie caaea/? Noaoaia,
a ae?oaee —

Fk (x,h,) i?noeoue ?ioi?iaoe?th i?i iiaaae?ieo ?icayceo iiaeeco
a?eueii? a?aieoe?, aea ? ?ica’yceii a?eueo i?inoi? caaea/?.

Oai?aia 1.15. Iaoae aeeiiothoueny oiiae oai?aie 1.12 oa

-(x) c1>0, 00 , yea ia caeaaeeoue a?ae oa h, ui
ia? i?noea ioe?iea

wmetafile8? ??Y???????????
???????????yyy????.????1?????? ???????
?a???&??yyyy?????Ayyy¬yyy ??¬?? ???&?
?MathType??A????u?th?????? C oeeo oaa?aeaeaiue aeieeaa? ??aiii??ia
iaiaaeai?noue ii?i wkC1+(D), (0,1)

oa yeui >0 a D, oi wmetafile8? ??Y???????????
???????????yyy????.????1?????? ???????
?????&??yyyy?????Ayyy¬yyyA??¬?? ???&?
?MathType??A????u?th?????? Aeae? anoaiiaeaii ??aiii??i? ioe?iee aeey
ooieoe?e uk(x,h,) Ioe?iee aeey ia?oeo iio?aeieo iaea?aeai? c ??aiyiiy
(5) ca aeiiiiiaith i?eioeeia iaeneioio.

Oai?aia 1.19. Iaoae a?eiiothoueny oiiae oai?aie 1.12, 4 h12.Oiae?

xD0() uk(x,h,)xic,

aea eiinoaioa wmetafile8? ?? Oai?aia 1.20. Iaoae

(x) C2+(D), k(x) C2+(D), 0, wk-1(x,h,) — wk(x,h,) c1h.

Oiae?

0uk-1(x,h,) — uk(x,h,) c2h,

aea ci ia caeaaeaoue a?ae h, , k.

Aeae? anoaiiaeaii, ui ciai? aeayei? iiiaeeie, i??a yei? i?yio? aei
ioey i?e h0, 0

i?inoi?ia? iio?aei? ooieoe?e Fk (x,h,) oa ??cieoeaaa iio?aeia
??aiii??ii i?yiothoue aei ioey i?e h0, 0.

Iaoae ooieoe?y (x,t)C2,1(DT) ??aia ioeth ia a?aieoe? iaeano? D oa i?e
t=T, k(x)=(x,kh). I?aaenoaaeii ??aiyiiy (5) o aeaeyae?

uk — 1hwmetafile8? ??,???????????
???????????yyy????.????1?????? ??????? 
 ???&??yyyy?????Ayyy?yyy`??O?? ???&?
?MathType??????u?y??????Iiiiiaeeii oea ??aiyiiy ia wmetafile8?
????????????? ???????????yyy????.????1??????
??????? ` ???&??yyyy?????Ayyy3/4yyy`??

?? ???&? ?MathType??`????u@th??????wmetafile8? ??=???????????
???????????yyy????.????1?????? ???????
Aa???&??yyyy?????Ayyy·yyy ??w?? ???&?
?MathType??????u?th??????wmetafile8? ??±????????????
???????????yyy????.????1?????? ??????? ?
???&??yyyy?????Ayyy?yyya????? ???&?
?MathType??????u?y??????Iicia/eii /a?ac u(x,t,h,) eoneiai-e?i?ei?
?ioa?iieyoe?? ii ci?ii?e t ooieoe?e uk(x,h,).

Iaoae 0C2+,wmetafile8? ??1???????????
???????????yyy????.????1?????? ??????? a
???&??yyyy?????Ayyy¬yyya????? ???&?
?MathType??°????u?th??????C iaea?aeaieo ??aiii??ieo ioe?iie aeieeaaa?,
ui anthaee aDT, aea

u(x,t,h,)1++h, u(x,t,h,)1-h,

/anoeii? iio?aei? ux(x,t,h,), ut(x,t,h,) ??aiii??ii iaiaaeai? oa ia?
i?noea ioe?iea -utc>0.

Oiio iiaa?oi? ??aiy T-(h,), T+(h,) iiaeia caaeaoe, a?aeiia?aeii,
yaieie ??aiyiiyie

t=+(x,h,), t=-(x,h,), i?e/iio ooieoe?? +(x,h,), -(x,h,), ??aiii??ii
iaiaaeai?. Ooieoe??

Fk(x,h,)- Fk-1(x,h,)h

??aiii??ii iaiaaeai? a D, a ia iiiaeei? uk-1<1- 2h, uk> 1++2h
caaeiaieueiythoue ??aiyiiy

(Fk-Fk-1)- 1hwmetafile8? ??4???????????
???????????yyy????.????1?????? ???????
` ???&??yyyy?????Ayyy?yyy`???? ???&? ?MathType??
???u?y??????Oiio ia oe?e iiiaeei? ia? i?noea ioe?iea wmetafile8?
??T????????????? ???????????yyy????.????1??????
???????   ???&??yyyy?????Ayyy¬yyya???I?? ???&?
?MathType??P????&??yyyy?????????? Fk(x,h,)- Fk-1(x,h,)hch, >0.

E??i oiai, i??a o??? /anoeie aeacaii? iiiaeeie, ia ye?e

Fk(x,h,)- Fk-1(x,h,)0

i?yio? aei ioey i?e h0, 0.Iicia/eii /a?ac

u(x,t)= wmetafile8? ??M???????????
???????????yyy????.????1?????? ???????
a`???&??yyyy?????Ayyy¬yyy ????? ???&?
?MathType??°????u?th??????Iaea?aeai? ?acoeueoaoe aeicaieythoue
cae?enieoe a?aie/iee ia?ao?ae a ?ioaa?aeuei?e oioiaeiino?.

Ioaea, ia? i?noea

wmetafile8? ??U????????????
???????????yyy????.????1?????? ???????
@`???&??yyyy?????Ayyy?yyy ??e?? ???&? ?MathType??
???u?y??????+wmetafile8? ??U????????????
???????????yyy????.????1?????? ???????
?@???&??yyyy?????Ayyy?yyy?????? ???&?
?MathType??????u?y??????aea DT1=DT(u=1)Ft0).

C iaea?aeaii? ?ioaa?aeueii? oioiaeiino? aeieeaa?, ui ooieoe?y u(x,t)
caaeiaieueiy? ??aiyiiy (1), a ia a?euei?e a?aieoe? iaeaea ne??cue ia?
i?noea (3).

Oai?aia 1.23. Iaoae aeeiiothoueny oiiae

wmetafile8? ??T?????????????
???????????yyy????.????1?????? ???????  
???&??yyyy?????Ayyy¬yyya???I?? ???&?
?MathType??P????&??yyyy?????????? (x)C2+(D), <0, a D, (x,t)H2+,1+(D), wmetafile8? ??TH??????????? ???????????yyy????.????1?????? ???????  ????&??yyyy?????Ayyy±yyyA ??Q?? ???&? ?MathType?? ????u y??????t<0 ia D[0,T], (x)=0 ia B1, x)>1 ia B2.

oa a?aeiia?aei? oiiae ni?yaeaiiy. Oiae? T>0 ?nio? ?aeeiee ?ica’ycie
caaea/? (1)-(4) oa

u(x,t)C(DT)H2+,1+(T) H2+,1+(GT),

a?eueia iiaa?oiy caaea?oueny ??aiyiiyi t=(x)C1+ oa aeey eiaeii?
oi/ee T yea eaaeeoue o DT, ?nio? ie?e, a yeiio ??aiyiiy T iiaeia
i?aaenoaaeoe o aeaeyae?

xi=f(x1, …,xi-1, xi,…,x3,t) H2+,1+.

A ioieo? 1.3 aea/a?oueny eiioaeoia aeaioacia caaea/a Noaoaia. Aeey
aeine?aeaeaiiy caaea/? aeei?enoiao?oueny oaeee naii iaoiae, ui e aeua.
Iicia/eii /a?ac

sd=xD: x12+x12=b2, x3=d, s0= xD: x12+x12=b2, x3=0.

Iniiaiee ?acoeueoao iieyaa? a ianooiiiio

Oai?aia 1.34. Iaoae aeeiiothoueny ianooii? oiiae

(x)C2+(D), 0, a D, x3>0 a D, (x,t)H2+,1+(D), (x,t)>1 ia 3,

aeeiiothoueny a?aeiia?aei? oiiae ocaiaeaeaiiy. Oiae? ?nio? ?aeeiee
?ica’ycie caaea/?, i?e/iio

u(x,t)C(DT)H2+,1+(Ts0) H2+,1+(GTsd),

a?eueio a?aieoeth iiaeia caaeaoe ??aiyiiyi x3=(x1,x2,t) H2+,1+(T ),

aea T=(x1, x2, ): x12+x12 b2(x), T=(0,T).

O ae?oa?e aeaa? aea/athoueny caaea/?, iniiaia neeaaei?noue yeeo ia a
?ioaa?oaaii? iniiaieo ??aiyiue, a a oiio, uia caaeiaieueieoe a?aie/i?
oiiae, ye? c iaeiiai aieo, iae?i?eii i?noyoue ooeai? aaee/eie, a c
ae?oaiai aieo, iiaeii? aeeiioaaoeny ia iaa?aeii?e a?aieoe?.

O?aaa ciaeoe ooieoe?th u(x,t) oa iaeano? T, GT ca oaeeie oiiaaie

u-aut = 0 a TGT,
(13)

Ia a?aeii?e /anoei? a?aieoe?

u(x,t)=i(x,t) ia Bi(0,T)
(14)

Ia iaa?aeii?e (a?euei?e) /anoei? a?aieoe? T=TDT=GTDT

u+=u-=1, u-2-u+2=-(ut++ut-)+Q(x)
(15)

Ii/aoeia? oiiae

u(x,0)=(x) ia D, (x)<1 ia B1, (x) >1 ia B2,
(16)

Ooo Ri, T, a, caaeai? aeiaeaoi? eiinoaioe, (x), Q(x), i(x,t) -caaeai?
ooieoe??, u+, u-,- icia/a? a?aie/i? cia/aiiy ooieoe?? u(x,t) ia T
ye? acyoi, a?aeiia?aeii, c? noi?iie iaeanoae GT oa T.

Ioaea, caaea/a (13)-(16) i?e Q(x)0 ni?aiaaea? c aeaioaciith caaea/ath
Noaoaia. sseui iieeaaeaii a=0, =0, oi iaea?aeeii a?aeiio noaoe?iia?io
caaea/o, yea aeieea? i?e aea/aii? no?oieiieo oa eaa?oaoe?eieo oa/?e ?c
aeaiia ??aeeiaie a a?ae?iiaoai?oe?. sseui ae o?eueee =0, oi oaea
caaea/a iieno? i?ioean iioe?aiiy iieoi’y a oai??? ai??iiy.

Aeey aeine?aeaeaiiy oeeo caaea/ aaoi? ?icaeaa? iaoiae, cai?iiiiiaaiee o
ia?o?e aeaa?. Ia?aeaeaii aei iiaoaeiae ai?ieneioth/eo caaea/. ?ic?a’?ii
oeee?iae? DT ieiueiaie t=kh, aea hN=T, N — caaeaia oe?ea /enei,
k=1,2,…,N. Iicia/eii /a?ac uk(x,h,), Fk(x,h,) ooieoe??, ye? ?
?ica’yceaie ianooiii? caaea/?:

uk-a(uk-uk-1)h = — [(uk)- (u0)]h — 12 Q2(x) ’(ul)+ ah Fk-1 xD,
(17)

uk(x,h,)=i(x,kh) xBi, u0(x)=(x),
(18)

Fk-aFkh = -[(uk)- (u0)]h — 12 Q2(x) ’(ul)+ ah Fk-1 xD,
(19)

Fk(x,h,)=0 xB1, F0 = 0 xD.
(20)

Oai?aia 2.1. Iaoae DC2+, i(x,kh) C2+(D), Q(x)C(D), (x)C2+(D),
(0,1).

Oiae? h>0, >0 caaea/a (17)-(20) iaeiicia/ii ?ica’ycia, a ooieoe??

uk(x,h,) C2+(D), Fk(x,h,) C2+(D).

Iicia/eii /a?ac

wk(x,h,)=uk(x,h,) — Fk (x,h,)
(21)

A?aei?iaii oaia? ?c (17) ??ai?noue (19) oa a?aoo?ii (21). A
?acoeueoao? iaea?aeeii, ui ooieoe?? wk(x,h,) ? ?ica’yceaie oaeeo
caaea/:

wk -a(wk-wk-1)h=0 xD,

wk (x,h,)=ki(x,h) xB1, w0(x)=(x) xD,

A?aei?oeii ua aeaye? aeanoeaino? ooieoe?e wk (x,h,) Iaoae
aeeiiothoueny oiiae oai?aie 2.1 oa

<0 a D, 01 ia B2,

(x,t) H2+,1+(DT), Q(x)0, Q(x)C1+(D),

aeeiiothoueny a?aeiia?aei? oiiae ocaiaeaeaiiy ooieoe?e (x,t) oa (x)
i?e t=0 ia D. Oiae? T>0 ?nio? eeane/iee ?ica’ycie caaea/? (13)-(16)
i?e =0 i?e/iio

u(x,t)C(DT)H2+,1+(T0) H2+,1+(GT0),

a?eueia a?aieoey T. H2+,1+.

Iiaeia oaeiae iaea?aeaoe eeane/io ?ica’yci?noue a?aeiii? noaoe?iia?ii?
caaea/?, yea aeieea? o a?ae?iaeeiai?oe? i?e iiaeaethaaii? no?oeieo oa
eaa?oaoe?eieo oa/?e c aeaiia ??aeeiaie.

Iaoae

D=x3:01.

Ia a?aeii?e /anoei? a?aieoe?

u=0 ia B1, u=1+q ia B2.

Ia iaa?aeii?e /anoei? a?aieoe? =D=GD

u+=u-=1, u+2+u-2=Q2(x).

Iaoae Q(x)C1+(D), Q(x)0, q>0. Oiae? aeua noi?ioeueiaaia caaea/a ia?
eeane/iee ?ica’ycie, i?e/iio

u(x)C(D)C2+()C2+(G).

Oea oaa?aeaeaiiy aeieeaa? c iiia?aaeiuei? oai?aie, yeui iieeanoe
1(x,t)=0, 2(x,t)=1+q=const.

oa c?iaeoe a?aie/iee ia?ao?ae i?e a0.

A ioieo? 2.2 iaaoa iiaeaeueoiai ?icaeoeo iaoiae, cai?iiiiiaaiee
aaoi?ii. Caaaeyee oeueiio aaeaeiny iineaaeoe i?eiouaiiy oai?aie 2.13.

Iicia/eii /a?ac u(x,t,,h,) eoa?/io ?ioa?iieyoe?th ooieoe?e
uk(x,t,h,) ii ci?ii?e a

u(x,t,)wmetafile8? ??n???????????
???????????yyy????.????1?????? ???????
?????&??yyyy?????Ayyy?yyyA??9?? ???&?
?MathType???????u?th??????C?iaeii a?aie/iee ia?ao?ae o a?aeiia?aei?e
?ioaa?aeuei?e oioiaeiino? , a iio?i ni?yio?ii a, 0 . A ?acoeueoao?
iaea?aeeii

Oai?aia 2.20. Iaoae aeeiiothoueny oae? i?eiouaiiy

(x,t)H2+,1+2(DT), (x)C2+(D), Q(x)C1+(D), Q(x)0,

0C2+, (x,t)<0 ia D1, (x,t)>0 ia D2,

aeeiiothoueny a?aeiia?aei? oiiae ocaiaeaeaiiy. Oiae? T>0 caaea/a
?ica’ycia, i?e/iio

u(x,t)C(DT)H2+,1+(TTD1) H2+,1+(GTTD2),

T- iiaa?oiyue eeano H2+,1+.

O?aoy aeaaa i?enay/aia aeine?aeaeaiith ?ica’yciino? eaac?noaoe?iia?ii?
caaea/? Noaoaia. Oey caaea/a aeieea? c ianoaoe?iia?ii? caaea/? Noaoaia o
i?eiouaii?, ui o?iio e?enoae?caoe??, yeee ?icae?ey? ??aeeo oa oaa?aeo
oace, ?ooa?oueny ??aiii??ii ?c noaeith oaeaee?noth oa o a?aeiia?aei?e
nenoai? eii?aeeiao oaiia?aoo?a ia caeaaeeoue a?ae /ano. Iaoae

D=(x,y)2: 00, (1)=b, =(x,y): 0< x <1, 0b .

Oea a?aee?eoa iiiaeeia, i?e/iio w(x,y) caaeiaieueiy? a ??aiyiiy

w+awy=a+.

E??i oiai,

wmetafile8? ??T?????????????
???????????yyy????.????1?????? ???????  
???&??yyyy?????Ayyy¬yyya???I?? ???&?
?MathType??P????&??yyyy??????????1 oa 2 , wx +w = 0 ia 1 ,
w y =0 ia 2

oa anthaee a D iathoue i?noea ia??aiino?

wx(x,y)0, wy(x,y)0.

C oeeo oaa?aeaeaiue aeieeaa?, ui ooieoe?y u(x,y)=wy(x,y) ? neaaeei
?ica’yceii caaea/?, a?eueia a?aieoey — iiiioiiia e?eaa, yea ia ia?
aa?oeeaeueieo oa ai?eciioaeueieo ae?eyiie, a oiiae ia i?e aeeiiothoueny
iaeaea ne??cue. Aeae? iaoiaeaie eiiieaeniiai aiae?ca aeiaaaeaii, ui
a?eueia a?aieoey ? aiae?oe/iith aeoaith.Oaeei /eiii, anoaiiaeaii
?nioaaiiy eeane/iiai ?ica’yceo.

Aiaeia?/ia oaa?aeaeaiiy aeiaaaeaii oaeiae a inaneiao?e/iiio aeiaaeeo.

Oaeei naii iaoiaeii aeiaaaeaii ?nioaaiiy neaaiai ?ica’yceo a
eaac?noaoe?iia?i?e caaea/? Noaoaia aeey eaac?e?i?eiiai ??aiyiiy
wmetafile8? ??T?????????????
???????????yyy????.????1?????? ???????
`@???&??yyyy?????Ayyy3/4yyy???

?? ???&? ?MathType??`????&??yyyy??????????

[k(u)u] + a(u)uy=0

Oai?aia 3.11. Iaoae aeeiiothoueny i?eiouaiiy

wmetafile8? ??a????????????
???????????yyy????.????1?????? ??????? 
 ???&??yyyy?????Ayyy¬yyy`??I?? ???&?
?MathType???????u?th??????b(x)=A[-1(x)], (x) wmetafile8?
??????????????? ???????????yyy????.????1??????
??????? ?????&??yyyy?????Ayyy?yyyA????? ???&?
?MathType??????u?y??????aea g(y)=wmetafile8? ???????????????
???????????yyy????.????1?????? ??????? 
????&??yyyy?????Ayyy?yyyA??O?? ???&?
?MathType?????u?y??????un=cos(n,y).

Aea/aeany oaeiae aeaioacia eaac?noaoe?iia?ia caaea/a Noaoaia. Aoea
canoiniaaia aeeoa?aioe?aeueii-??cieoeaaa ai?ieneiaoe?y caaea/?.
Aeiaaaeaii ?nioaaiiy neaaeiai ?ica’yceo.

AENIIAEE

Aeenna?oaoe?y i?enay/aia aea/aiith iae?i?eieo caaea/ iaoaiaoe/ii?
o?ceee c a?eueieie a?aieoeyie. Aaaeeea?noue oeeo caaea/ aecia/a?oueny
oei, ui aiie yaeythoueny iaoaiaoe/ieie iiaeaeyie i?ioean?a, oa?aeoa?iith
iniaeea?noth yeeo ? iayai?noue ??cieo ca nai?ie oa?aeoa?enoeeaie oac,
a?aeie?aieaieo a?eueiith a?aieoeath. Oae? i?ioeane a?aeaoaathoueny a
aeayeeo no/anieo iaoaeo?a?eieo oaoiieia?yo, i?e ooai?aii? e?eae, i?e
ae?iuoaaii? iiiie?enoae?a, a oai??? i?oaeiino?, a?ae?iaeeiai?oe?, oai???
ai??iiy oa a ?ioeo iaeanoyo iaoee oa oaoi?ee. C ?ioiai aieo, oe?
i?iaeaie i?aaenoaaeythoue ci?noiaiee iaoaiaoe/iee ia’?eo, aeine?aeaeaiiy
yeiai i?eaaei aei noai?aiiy iiaeo oa aei a?eueo aeeaieiai aea/aiiy
a?aeiieo iaoiae?a.

Cia/aiiy aeena?oaoe?? iieyaa? o ianooiiiio

1) cai?iiiiiaaii iaoiae aeine?aeaeaiiy oe?eiai eeano iae?i?eieo caaea/
?c a?eueieie a?aieoeyie aeey ae?ioe/ieo oa ia?aaie?/ieo
aeeoa?aioe?aeueieo ??aiyiue ae?oaiai ii?yaeeo;

2) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo a oe?eiio ca /anii o
aeaioaci?e aaaaoiaei??i?e caaea/? Noaoaia aeey e?i?eiiai oa
eaac?e?i?eiiai ??aiyiue oaieii?ia?aeiino?, a oaeiae o eiioaeoi?e caaea/?
Noaoaia;

3) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo a oe?eiio ca /anii a
caaea/?, yea iiaeaeth? i?ioean iioe?aiiy iieoi’y a oai??? ai??iiy, a
oaeiae aeiaaaeaia eeane/ia ?ica’yci?noue noaoe?iia?ii? caaea/?, ui
aeieea? i?e iiaeaethaaii? no?oieiieo oa eaa?oaoe?eieo oa/?e a
a?ae?iaeeiai?oe?;

4) aeiaaaeaii ?nioaaiiy eeane/iiai ?ica’yceo a iaeiioaci?e ieine?e oa
inaneiao?e/i?e eaac?noaoe?iia?i?e caaea/? Noaoaia aeey e?i?eiiai
??aiyiiy, neaaeiai ?ica’yceo aeey eaac?e?i?eiiai ??aiyiiy, a oaeiae
?nioaaiiy neaaeiai ?ica’yceo a aeaioaci?e ieine?e eaac?noaoe?iia?i?e
caaea/? Noaoaia;

5) iaoiae, cai?iiiiiaaiee a aeena?oaoe??, iiaea noaoe iniiaith aeey
?ic?iaee aoaeoeaiiai /enaeueiiai iaoiaea;

6) iaoiae, cai?iiiiiaaiee a aeena?oaoe??, iiaeia aeei?enoaoe i?e
aeine?aeaeaii? ?ioeo caaea/ ?c a?eueieie a?aieoeyie.

NIENIE IIOAE?EIAAIEO AAOI?II I?AOeUe CA OAIITH AeENA?OAOe??

1. Ai?iaeei I. A. I iaeioi?uo iaeeiaeiuo caaea/ao oaieii?iaiaeiinoe. //
Iao. oeceea. — E.: Iaoeiaa Aeoiea, aui. 14, 1973.- n. 8 — 14.

2. Ai?iaeei I. A. Oai?aia nouanoaiaaiey ?aoaiey iaeiioaciie
eaacenoaoeeiia?iie caaea/e Noaoaia. // Aeiee. AI ONN?.- Na?. A.-1976.-
N7.- n. 582 — 585.

3 . Ai?iaeei I. A. Iaeiioaciay eaacenoaoeeiia?iay caaea/a Noaoaia. //
Aeiee. AI ONN?.- Na?.A.-1977.- N9.- n. 775 — 777.

4. Ai?iaeei I. A., Oaeueaaioaoy? O. Iaeiioaciay eaaceeeiaeiay caaea/a
Noaoaia. // Aeiee. AI ONN?.- Na?.A.-1978.-N2.- n. 99 — 102.

5. Ai?iaeei I. A. Iaeiioaciay eaacenoaoeeiia?iay caaea/a Noaoaia. //
E?aaaua caaea/e aeey o?aaiaiee a /anoiuo i?iecaiaeiuo. — Eeaa: Iaoeiaa
Aeoiea, 1978.- n. 13 — 21.

6. Ai?iaeei I. A., Oaeueaaioaoy? O. Inaneiiao?e/aneay iaeiioaciay
caaea/a Noaoaia. // Iao. oeceea. — E.: Iaoeiaa Aeoiea, aui.24, 1978.- n.
74 — 76.

7. Ai?iaeei I. A. I?ino?ainoaaiiay iaeiioaciay eaacenoaoeeiia?iay
caaea/a Noaoaia. // OII.-1980.-O. 35.- aui. N4.-n.177.

8 . Ai?iaeei I. A. I ?ac?aoeiinoe aeaoooaciie eaacenoaoeeiia?iie caaea/e
Noaoaia. // Aeiee. AI ONN?.- Na?. A.-1982.-N2.- n. 3-5.

9. Ai?iaeei I. A. I ?ac?aoeiinoe aeaoooaciie ianoaoeeiia?iie caaea/e
Noaoaia. //AeAI NNN?.- 1982.- o.263,- N5.- n. 1040-1042.

10. Ai?iaeei I. A. I eeanne/aneie ?ac?aoeiinoe aeaoooaciie
ianoaoeeiia?iie caaea/e Noaoaia. // OII.-1983.-o. 38.- aui.5.-n.152.

11. Ai?iaeei I.A. Aeaoooaciay eaacenoaoeeiia?iay caaea/a Noaoaia.//
O?aaiaiey a /anoiuo i?iecaiaeiuo e caaea/e ni naiaiaeiuie a?aieoeaie. —
Eeaa: Iaoeiaa Aeoiea, 1983.- n. 28 — 30.

12. Ai?iaeei I. A. Nouanoaiaaiea eeanne/aneiai ?aoaiey a iiiaiia?iie
caaea/a Noaoaia ia eiia/iii i?iiaaeooea a?aiaie. //Oe?. Iao. AE.- 1992.-
o.4 — N12. — n.1652 — 1657.

13. Borodin M. A. Existense of the classic solution of a two-phase
multidimensional Stefan problem on any finite time interval. // Intern.
Ser. Numer. Math.- 1992.- v.106. — p. 98 — 103.

14. Ai?iaeei I. A. Aeaoooaciay eiioaeoiay caaea/a Noaoaia. // Oe?. Iao.
AE.- 1995. — o. 47. — N2. — n. 158 — 167.

15. Borodin M. A. The two-phase Stefan problem. // Nonlinear boundary
value problems. — 1997. — v.7. — p. 37 — 46.

16. Ai?iaeei I. A. Iiaue iaoiae enneaaeiaaiey iaeioi?uo caaea/ ni
naiaiaeiuie

a?aieoeaie aeey ia?aaiee/aneeo o?aaiaiee. // A?ni. Aeii.Oi. Na?. A. —
1997.- N1. — n.21 — 26.

17. Borodin M. A. A new method of studying for free boundary problems.
// Nonlinear boundary value problems. — 1998. — v.8. — p. 64-69.

18. Ai?iaeei I.A.Nouanoaiaaiea aeiaaeueiiai eeanne/aneiai ?aoaiey a
caaea/a, aicieeathuae a oai?ee ai?aiey. // A?ni.Aeii.Oi. Na?. A. —
1998.- N2. — n. 14-22.

19. Ai?iaeei I. A. Nouanoaiaaiea aeiaaeueiiai eeanne/aneiai ?aoaiey a
iaeioi?ie iaeeiaeiie ia?aaiee/aneie caaea/a ni naiaiaeiie a?aieoeae.//
Aeiee. IAI. Oe?. Na?. A. — 1999. — N6 -n.7-12.

20. Ai?iaeei I. A. ?aoaiea iaeiioaciie eaacenoaoeeiia?iie caaea/e
Noaoaia.// O?. Ananithciie eiio. ii o?aaiaieyi n /anoiuie
i?iecaiaeiuie.- I.: Ecae-ai IAO, 1978.- 275-276.

21. Ai?iaeei I. A. I iaeioi?ie aeaoooaciie ianoaoeeiia?iie caaea/a
Noaoaia. // Oacenu Ananithci. eiio. “ Iaoaiaoe/aneia iiaeaee?iaaiea
i?ioeannia caoaa?aeaaaiey iaoaeeia e nieaaia — Iiaineae?ne. — 1983. — n.
142 — 143.

22. Ai?iaeei I. A. I aeaaeeinoe naiaiaeiie a?aieoeu a aeaoooaciie
caaea/a Noaoaia. // Oacenu Ananithci. eiio. “ Eiiieaeniua iaoiaeu
iaoaiaoe/aneie oeceee. “ — Aeiiaoee. — 1984. — n. 124.

23. Ai?iaeei I. A. I iaeioi?ie aeaoooaciie ianoaoeeiia?iie caaea/a
Noaoaia. // Oacenu Niaaonei-*aoineiaaoeiai Niaau. “ I?eiaiaiea
ooieoeeiiaeueiuo iaoiaeia e iaoiaeia oai?ee ooieoeee e caaea/ai
iaoaiaoe/aneie oeceee. “ — Aeiiaoee. — 1986. — n. 18.

24. Ai?iaeei I. A. I eeanne/aneie ?ac?aoeiinoe aeaoooaciie caaea/e
Noaoaia. // Oacenu ?anioae. eiio. “ Iaeeiaeiua caaea/e iaoaiaoe/aneie
oeceee.” — Aeiiaoee. — 1987. — n. 16.

25. Ai?iaeei I. A. Nouanoaiaaiea eeanne/aneiai ?aoaiey a aeaoooaciie
iiiaiia?iie caaea/a Noaoaia a oeaeii ii a?aiaie. // Oacenu ?anioae.
eiio. “ Iaeeiaeiua caaea/e iaoaiaoe/aneie oeceee e caaea/e ni
naiaiaeiuie a?aieoeaie.” — Aeiiaoee. — 1991. — n. 20.

26. Borodin M. A. Existence of global classical solution of a two-phase
multidimensional Stefan problem in any finite time interval. // Abstr.
International Conf. “ Free boundary problems in continium mechanics.” —
Novosibirsk. — 1991.- p. 24 — 25.

27. Borodin M. A. The two-phase Stefan problem. // Abstr. International
Conf. “ Nonlinear partial differential equations” — Kiev. — 1995. — p.
26.

28. Borodin M. A. A new mathod of studying for free boundary problems.
// Abstr. International Conf. “ Nonlinear partial differential
equations” — Kiev. — 1997. — p.33.

AIIOAOe??

Ai?iae?i I. I. Caaea/? c a?eueieie a?aieoeyie aeey ae?ioe/ieo oa
ia?aaie?/ieo ??aiyiue. — ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy aeieoi?a o?ceei-iaoaiaoe/ieo
iaoe ca niaoe?aeuei?noth 01.01.02 — aeeoa?aioe?aeuei? ??aiyiiy. —
?inoeooo i?eeeaaeii? iaoaiaoeee ? iaoai?ee IAI Oe?a?ie, Aeiiaoeuee,
1999.

Aeena?oaoe?th i?enay/aii ieoaiiyi eeane/ii? ?ica’yciino? a oe?eiio ca
/anii aaaaoiaei??ieo noaoe?iia?ieo oa ianoaoe?iia?ieo caaea/ ?c
a?eueieie a?aieoeyie. Oae? caaea/? yaeythoue niaith iaoaiaoe/i? iiaeae?
i?ioean?a, oa?aeoa?iith iniaeea?noth yeeo ? iayai?noue ??cieo ca nai?ie
oa?aeoa?enoeeaie oac, a?aeie?aieaieo a?eueiith ( iaa?aeiiith)
a?aieoeath. A aeena?oaoe?? cai?iiiiiaaiee iiaee iaoiae, ca aeiiiiiaith
yeiai aaeaeiny aeiaanoe ?nioaaiiy aeiaaeueieo eeane/ieo ?ica’yce?a aeey
oe?ei? iecee a?aeiieo caaea/ ?c a?eueieie a?aieoeyie. Oeei iaoiaeii
iiaeia nei?enooaaoeny i?e aeine?aeaeaii? ?ioeo caaea/ ?c a?eueieie
a?aieoeyie; a?i oaeiae iiaea noaoe iniiaith aeey noai?aiiy iiaeo
aoaeoeaieo /enaeueieo iaoiae?a i?e aea/aii? aaaeeeaeo i?eeeaaeieo
caaea/.

Eeth/ia? neiaa: Aeaioacia i?iaeaia Noaoaia, caaea/? ?c a?eueieie
a?aieoeyie, aeiaaeuei? eeane/i? ?ica’ycee, aeaaee?noue a?eueii?
a?aieoe?.

Ai?iaeei I. A. Caaea/e ni naiaiaeiuie a?aieoeaie aeey yeeeioe/aneeo e
ia?aaiee/aneeo o?aaiaiee. — ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie aeieoi?a
oeceei-iaoaiaoe/aneeo iaoe ii niaoeeaeueiinoe 01.01.02 —
aeeooa?aioeeaeueiua o?aaiaiey. — Einoeooo i?eeeaaeiie iaoaiaoeee e
iaoaieee IAI Oe?aeiu, Aeiiaoee, 1999.

Aeenna?oaoeee iinayuaia aii?inai eeanne/aneie ?ac?aoeiinoe a oeaeii ii
a?aiaie iiiaiia?iuo noaoeeiia?iuo e ianoaoeeiia?iuo caaea/ ni
naiaiaeiuie a?aieoeaie. Oaeea caaea/e i?aaenoaaeytho niaie
iaoaiaoe/aneea iiaeaee i?ioeannia, oa?aeoa?iie iniaaiiinoueth eioi?uo
yaeyaony iaee/ea ?aciuo ii naiei oa?aeoa?enoeeai oac, ?acaeaeaiiuo
naiaiaeiie (iaecaanoiie) iiaa?oiinoueth.A aeenna?oaoeee i?aaeeiaeai
iiaue iaoiae, i?e iiiiue eioi?iai oaeaeinue aeieacaoue nouanoaiaaiea
aeiaaeueiuo eeanne/aneeo ?aoaiee aeey oeaeiai ?yaea ecaanoiuo caaea/ ni
naiaiaeiuie a?aieoeaie. Yoio iaoiae iiaeao auoue eniieueciaai i?e
enneaaeiaaiee ae?oaeo caaea/ ni naiaiaeiuie a?aieoeaie; a oaeaea noaoue
iniiaie aeey nicaeaiey iiauo yooaeoeaiuo /eneaiiuo iaoiaeia i?e eco/aiee
aaaeiuo i?eeeaaeiuo caaea/.

Eeth/aaua neiaa: Aeaoooaciay i?iaeaia Noaoaia, caaea/e ni naiaiaeiuie
a?aieoeaie, aeiaaeueiua eeanne/aneea ?aoaiey, aeaaeeinoue naiaiaeiie
a?aieoeu.

Borodin M. A. Free boundary problems for parabolic and elliptic
equations. — Manuscript.

Thesis for a doctor’s degree by speciality 01.01.02 — differential
equations. — The Institute of Applied Mathematics and Mechanics of
National Academy of Science of Ukraine, Donetsk, 1999.

In the dissertation the problem of classical solvability on the whole in
time of multidimensional stationary and nonstationary free boundary
problems is studied.Such problems represent mathematical models of
processes whose characteristic property is the presence of two different
phases separated by a free (unknown) surface. The presence of the free
boundary makes these models substantially nonlinear and particularly
difficult for study.

Stefan problem occupies the central place among the free boundary
problems. It was formulated over a hundred years ago when Austrian
physicist J. Stefan tried to construct a model of formation and
evolution of the ice in the world ocean. Since then the problem was
under attention of many prominent mathematicians. Classical solvability
on the whole in time of the one phase problem and classical solvability
for small times of the two phase problem have been proved. However, the
problem of existence of a global classical solution remained open.

In the first chapter of the dissertation we prove the existence of a
global classical solution for the two phase multidimensional Stefan
problem for quasilinear heat equation. This result has been proved by
applying a new method invented by the author.The method consists in the
following: first a special difference—differential elliptic system of
approximating problems is constructed, then certain uniform estimates
are proved, and the passage to the limit is performed.

Another class of problems studied in the dissertation arises in
mathematical models of combustion processes and in the studies of wave
and jet flows in hydrodynamics. These problems are different from the
Stefan problem: they are nonlinear not only because of the free boundary
but also because of the nonlinearity of boundary conditions. In the
stationary case the key point in the study of these problems was the
discovery of the variational nature of solutions. This led to the
investigation of integral functionals with variable domain of
integration and allowed to prove the existence of a classical solution
in the flat case. In the nonstationary case the problem of existence of
a classical global solution remained unsolved.

In the second chapter of the dissertation we prove the existence of a
global classical solution in multidimensional two phase nonstationary
problem modeling the combustion process, and the existence of the
classical solution in the hydrodynamic problem mentioned above. This is
done by the method used in the first chapter.

Finally, we also study problems of existence of solutions in the
quasistationary Stefan problem. This problem originates from heat
physics. It models the spreading of heat in a media which has two
phases, if we assume that the crystallization front moves uniformly with
a constant speed, and that in the corresponding moving system of
coordinates the temperature does not depend on time. In the third
chapter of the dissertation we prove the existence of a classical
solution for one phase quasistationary Stefan problem in the flat case.
By a change of the unknown function the problem is reduced to an
elliptic variational inequality. Then we use method of local variations
and symmetrization method. We also prove the existence of a weak
solution in two phase quasistationary Stefan problem.

Thus, we have suggested a new method which allowed to prove the
existence of a global classical solution in a series of known free
boundary problems. This method can also be applied to other free
boundary problems. It can also become a base for development of new
effective numerical methods in studying of important applied problems.

Key words: two phase Stefan problem, free boundary problems, global
classical solution, smoothness of the free boundary.

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