1

IAOe?IIAEUeIA AEAAeAI?ss IAOE OE?A?IE

?inoeooo i?iaeai iiaeaethaaiiy a aia?aaoeoe? IAI Oe?a?ie

IEEEIAIEI Iaoae?y Ieeiea?aia

OAeE 681.3

IAOAIAOE*IA IIAeAETHAAIIss

AeEIAI?*IEO NENOAI

A IIA?AOe?EI?E IAEANO?

IA INIIA? ?AEO?AIOIEO NI?AA?AeIIOAIUe

(C I?IA?AIAIE A NENOAI? “MATHEMATICAO”)

Niaoe?aeuei?noue 01.05.02 – Iaoaiaoe/ia iiaeaethaaiiy oa ia/enethaaeuei?
iaoiaee

A a o i ? a o a ? a o

aeena?oaoe?? ia caeiaoooy iaoeiaiai nooiaiy

eaiaeeaeaoa oaoi?/ieo iaoe

Ee?a – 1999

Aeena?oaoe??th ? ?oeiien.

?iaioa aeeiiaia a A?aeae?eaii? a?a?eaeieo iiaeaethth/eo oa ea?oth/eo
nenoai a aia?aaoeoe? ?inoeoooo i?iaeai iiaeaethaaiiy a aia?aaoeoe? IAI
Oe?a?ie.

Iaoeiaee ea??aiee: /eai-ei?aniiiaeaio IAI Oe?a?ie,

aeieoi? oaoi?/ieo iaoe, i?ioani?

AANEEUe?A Anaaieiae A?eoi?iae/,

ea??aiee A?aeae?eaiiy a?a?eaeieo iiaeaethth/eo oa ea?oth/eo nenoai a
aia?aaoeoe? ?inoeoooo i?iaeai iiaeaethaaiiy a aia?aaoeoe? IAI Oe?a?ie

Io?oe?ei? iiiiaioe: aeieoi? oaoi?/ieo iaoe, noa?oee iaoeiaee
ni?a?ia?oiee

IIOI? Aieiaeeie? Aieiaeeie?iae/,

canooiiee aee?aeoi?a ?inoeoooo i?iaeai iiaeaethaaiiy a aia?aaoeoe? IAI
Oe?a?ie

eaiaeeaeao oaoi?/ieo iaoe, aeioeaio

AEOIAAAOeUeEEE Aieiaeeie? ?aaiiae/,

canooiiee caa?aeoth/iai eaoaae?ith aeaeo?ii?ee oa ia/enethaaeueii?
oaoi?ee Oe?a?inueeiai O?ainii?oiiai Oi?aa?neoaoo

I?ia?aeia onoaiiaa: ?inoeooo i?iaeai iaoaiaoe/ieo iaoei oa nenoai IAI
Oe?a?ie, i. Ee?a

Caoeno a?aeaoaeaoueny 28 a?oaeiy 1999 ?. i_14_ aiaeei?

ia can?aeaii? niaoe?ae?ciaaii? a/aii? ?aaee Ae26.185.01 ?inoeoooo
i?iaeai iiaeaethaaiiy a aia?aaoeoe? IAI Oe?a?ie ca aae?anith i. Ee?a,
aoe. Aaia?aea Iaoiiaa, 15

C aeena?oaoe??th iiaeia iciaeiieoeny o a?ae?ioaoe? ?inoeoooo i?iaeai
iiaeaethaaiiy a aia?aaoeoe? IAI Oe?a?ie ca aae?anith i. Ee?a, aoe.
Aaia?aea Iaoiiaa, 15

Aaoi?aoa?ao ?ic?neaiee “_26_”_eenoiiaaea_ 1999 ?.

A/aiee nae?aoa?

niaoe?ae?ciaaii? a/aii? ?aaee,

eaiaeeaeao oaoi?/ieo iaoe Naiaa?ia A.I.

CAAAEUeIA OA?AEOA?ENOEEA ?IAIOE

Aeooaeuei?noue oaie. Ao?oeeaee ?icaeoie iaoee iai?ee?ioe? aeaaaeoeyoiai
noi??//y, noai?aiiy iiaeo aenieeo oaoiieia?e, aaoiiaoecaoe?y on?o noa?
ethaenueei? ae?yeueiino? i?ecaiaeyoue aei ana a?eueo ?ioaineaiiai
aeei?enoaiiy ??ciiiai?oieo iaoiae?a iaoaiaoe/iiai iiaeaethaaiiy.
Aeine?aeaeoaai? nenoaie, ye i?aaeei, iienothoueny
?ioaa?i-aeeoa?aioe?aeueieie ??aiyiiyie, iaoiaee aea/aiiy yeeo
iio?aaothoue iiooaeiiai iaoaiaoe/iiai aia?aoo. ?aae?caoe?y oaeeo
iaoiae?a aeiaaa? cia/ieo ia/enethaaeueieo aeo?ao. Ia?iaea ?acoeueoao?a
ianooaaieo aenia?eiaio?a oa aaoiiaoecaoe?y iaoeiaeo aeine?aeaeaiue
ni?e/eiythoue iaiao?aei?noue noai?aiiy iiaeo aoaeoeaieo cania?a
oeeo?iai? ia?iaee neaiae?a.

A aeine?aeaeaii? aeeiai?/ieo nenoai oe?iea ?iciianthaeaeaiiy caeiaoee
iia?aoi?i? iaoiaee, oae? ye ia?aoai?aiiy Oo?’?, Eaieana, Oieoa, Oaa?a oa
?i. Oae, Iooiaei A.?. oa Aa?aaiaaiei A.ss. ?ic?iaeyeeny iia?aoi?i?
iaoiaee ia iniia? no?ae/anoeo cia?aaeaiue. Na?aae aeeoa?aioe?aeueieo
iia?aoi?ieo iaoiae?a oe?ieiai ?iciianthaeaeaiiy caeiaoee
AeO-ia?aoai?aiiy, cai?iiiiiaai? oa ?icaeioo? Iooiaei A.?. A inoaii? ?iee
aeoeaii ?icaeaathoueny aaeaeao-nenoaie. A ?iaioao N?iae E.I. ?ic?iaeai?
aeaye? iaoiaee i?aaenoaaeaiiy neaiae?a ocaaaeueiaieie iie?iiiaie, ui
i?eaiaeyoue aei ai?ieneioth/eo iie?iii?aeueieo niaeo??a (AIN), ia iniia?
yeeo iiaeooue aooe iiaoaeiaai? ??ci? iia?aoe?ei? /eneaiiy. Na?aae AIN
iniaeeaa i?noea caeiathoue ai?ieneioth/? ?iioeueni? niaeo?e (A?N), a
iniia? yeeo eaaeaoue i?oiaiiaeuei? aaceni? nenoaie
eieaeueii-?iioeueniiai oeio. Aeei?enoaiiy niaoeeo?/ieo aeanoeainoae
iao?eoeue iia?aoi?ieo ??aiyiue aeicaiey? ?ic?iaeoe iaoiaee, ye? cia/ii
ni?iuothoue aeine?aeaeaiiy neaiae?a, caa??aathoue ia/enethaaeuei?
?ano?ne, ciaioothoue /an ?ica’ycaiiy ??aiyiue.

Aeena?oaoe?eia ?iaioa i?enay/aia aeine?aeaeaiith oa ?ic?iaoe? iaoiae?a
iaoaiaoe/iiai iiaeaethaaiiy aeeiai?/ieo nenoai ia iniia? AIN,
i?ecia/aieo aeey aeei?enoaiiy i?e ?ica’ycaii? oe?ieiai eeano ?iaeaia?ieo
oa oaoi?/ieo caaea/. Aeooaeuei?noue ?iaioe iieyaa? a i???ioaoe?? ia
aeei?enoaiiy no/anieo IAII oa iaoaiaoe/iiai iiaeaethaaiiy.

Iiooaeiei caniaii iaoaiaoe/iiai iiaeaethaaiiy ? i?ia?aiia ?aae?caoe?y
iaoiae?a. Nenoaia MathematicaO (Wolfram Research Inc.) ? iaei??th c
iaea?eueo oi?aa?naeueieo na?aae ?nioth/eo iaoaiaoe/ieo i?ia?ai.
Ae?oaei?e ?ioa?oaen, /oaeiaa a?ao?ea, ai?iaoe?ei? iiaeeeaino?, iiooaeia
iiaa i?ia?aioaaiiy oa oe?ieee niaeo? i?aaenoaaeaieo iaoaiaoe/ieo
?icae?e?a ?iaeyoue ?? i?ae?aniith iniiaith aeey iaoaiaoe/iiai
iiaeaethaaiiy. Oe? iniaeeaino? coiiaeee aeei?enoaiiy nenoaie
MathematicaO ye iniiae aeey iiaeaethaaiiy, aeine?aeaeaiiy oa i?iaaaeaiiy
aenia?eiaio?a a aeena?oaoe?ei?e ?iaio?.

Aea?aiee iaoeiaee iai?yiie ? aaaeo/ei a iaoeiai-aeine?aeieo ?iaioao
A?aeae?eaiiy iiaeaethth/eo oa ea?oth/eo nenoai a aia?aaoeoe? ?IIA IAI
Oe?a?ie, aea aeeiioaaeany aeena?oaoe?eia ?iaioa.

Iaoa ? caaea/? aeine?aeaeaiiy. Iniiaiith iaoith aeena?oaoe?eii? ?iaioe
? ?ic?iaea iaoiae?a aeine?aeaeaiiy aeeiai?/ieo nenoai a iia?aoe?ei?e
iaeano? ia iniia? AIN, i???ioiaaieo ia ?aae?caoe?th caniaaie IAII oa
aeaeo?iiiiai iiaeaethaaiiy.

Aeacaia iaoa aecia/eea iaiao?aei?noue ?ica’ycaiiy oaeeo caaea/:

· aea/aiiy AIN oa aeyaeaiiy ?o iniaeeainoae oeyoii i?iaaaeaiiy
ii??aiyeueiiai aiae?co ai?ieneiaoe?eieo iaoiae?a;

· ?ic?iaea iaoiae?a ?aeo?neaiiai aiae?co iia?aoe?eieo iiaeaeae
aeeiai?/ieo nenoai ia iniia? ai?ieneioth/eo ?iioeuenieo niaeo??a;

· ?ic?iaea iaoiaeo ioe?ithaaiiy ao?aoe ?ioi?iaoe?? i?e ai?ieneiaoe??;

· ?ic?iaea iaoiaeo aeai?a i?aenenoaie c caaeaiiai aaceno aeey
iioeiaeueii? ai?ieneiaoe?? aeia?eueii? ooieoe??;

· aeai?eoi?caoe?y ?ic?iaeaieo iaoiae?a;

· ?ic?iaea aoaeoeaieo o aeei?enoaii? i?ia?ai, ui ?aae?cothoue aeai?eoie
aeine?aeaeaiiy aeeiai?/ieo nenoai ca aeiiiiiaith AIN a nenoai?
MathematicaO.

Iaoiaee aeine?aeaeaiiy iineee oai?aoe/iee oa aenia?eiaioaeueiee
oa?aeoa?. I?e i?iaaaeaii? aeine?aeaeaiue oa ?ic?iaie ii aeena?oaoe?ei?e
?iaio? aeei?enoiaoaaeenue iaoiaee oai??? iaoaiaoe/iiai iiaeaethaaiiy,
oai??? oeeo?iai? ia?iaee neaiae?a oa ai?ieneiaoe??, i?eeeaaeii? oa
ia/enethaaeueii? iaoaiaoeee, oai??? aeeoa?aioe?aeueieo ??aiyiue,
canoiniaoaaany aia?ao e?i?eii? aeaaa?e, cie?aia, oai??? iao?eoeue.
Io?eiai? ?acoeueoaoe ia?aa??yeeny oeyoii i?iaaaeaiiy ia/enethaaeueieo
aenia?eiaio?a ia IAII.

Iaoeiaa iiaecia aeena?oaoe?? iieyaa? a ianooiiiio:

1. ?ic?iaeaii i?ia?aie a nenoai? MathematicaO, ui ?aae?cothoue
ai?ieneiaoe?ei? iaoiaee oa aeicaieythoue i?iaiaeeoe ia/enethaaeuei?
aenia?eiaioe;

2. ?ic?iaeaii oa aeine?aeaeaii iaoiaee ?aeo?neaiiai aiae?co iia?aoe?eieo
iiaeaeae aeeiai?/ieo nenoai ia iniia? ai?ieneioth/eo ?iioeuenieo
niaeo??a;

3. cai?iiiiiaaii, oai?aoe/ii iaa?oioiaaii oa aeine?aeaeaii iaoiae
ioe?ithaaiiy ao?aoe ?ioi?iaoe?? i?e ai?ieneiaoe?? ooieoe?? c iaeiiai
ne?i/aiiiaei??iiai i?aei?inoi?o L2 ([a,b]) ooieoe?yie c ?ioiai
ne?i/aiiiaei??iiai i?aei?inoi?o;

4. cai?iiiiiaaii, oai?aoe/ii iaa?oioiaaii oa aeai?eoi?ciaaii iaoiae
aeai?o i?aenenoaie ooieoe?e c caaeaiiai aaceno aeey iioeiaeueii?
ai?ieneiaoe?? aeia?eueii? ooieoe??. ?ic?iaeaii aeai?eoi, ui aeicaiey?
cae?enieoe oeae aea?? ca ne?i/aiiee /an;

5. oi/i?noue ?iaioe noai?aieo i?ia?ai i?aeoaa?aeaeaia /eneaiieie
i?eeeaaeaie ?ic?aooieo caaea/ oaieiianiiai?io, oai?aoe/ii? iaoai?ee oa
oai??? aeaeo?e/ieo eaioetha?a.

I?aeoe/ia oe?ii?noue. ?ic?iaeai? a ?iaio? aeai?eoie aoee ?aae?ciaai? a
nenoai? neiaie?/ii? iaoaiaoeee MathematicaO a ianooiieo i?ia?aiao:

· i?ia?aie iiooeo iaeii- oa aeaiaei??ieo AIN;

· i?ia?aie ai?ieneiaoe?? ooieoe?e iaei??? oa aeaio ci?iieo ia iniia? A?N
oa AeO-ia?aoai?aiue;

· i?ia?aie ?icaycaiiy aeeoa?aioe?aeueieo ??aiyiue c ii/aoeiaeie oiiaaie
oa ?o nenoai iia?aoe?eieie iaoiaeaie c aeei?enoaiiyi ?aeo?aioieo
aeai?eoi?a ?ica’ycaiiy iia?aoi?ieo ??aiyiue;

· i?ia?aia ioe?ithaaiiy ao?aoe ?ioi?iaoe?? i?e ai?ieneiaoe??;

· i?ia?aia aeai?o i?aenenoaie c caaeaiiai aaceno aeey iioeiaeueii?
ai?ieneiaoe?? aeia?eueii? ooieoe??.

I?aeoe/ia oe?ii?noue io?eiaieo ?acoeueoao?a i?iaeaiiino?iaaia ia
i?eeeaae? ?icaycaiiy caaea/ oaieio?ceee, iaoai?ee oa oai??? aeaeo?e/ieo
eaioetha?a.

Cai?iiiiiaai? a ?iaio? iaoiaee aeei?enoiaoaaeeny a i?ioean? aeeiiaiiy
iaoeiai-aeine?aeieo ?ia?o AAIENA ?IIA IAI Oe?a?ie ca oaiith
“Iiaeaethaaiiy aia?aaoe/ieo ia’?eo?a oa ioe?iea ?o oaoi?/iiai noaio a
noaoe?iia?ieo oiiaao aenieoaoaoe??” (Oeo? oaie “Ciiae”. Aeeiioaaeany a
1995-1997 ??. Ca?aeii iinoaiiae A?aeae?eaiiy o?ceei-oaoi?/ieo i?iaeai
aia?aaoeee IAI Oe?a?ie), oa a i?iaeo? ? 602 Oe?a?inueeiai
Iaoeiai-Oaoiieia?/iiai Oeaio?o “Aeine?aeaeaiiy oa ?ic?iaea eiiieaeno
aia?aoii-i?ia?aiieo cania?a iii?oi?eiao i?ioeano aeoaiaiai caa?thaaiiy
aeey iiia?aaeiuei? ae?aaiinoeee i?ioeano, i?aeaioiaee oa aeiioneiiai
eiio?ieth caa?thaa/?a a aoiii?e aia?aaoeoe?”.

Ai?iaaoe?y ?acoeueoao?a aeena?oaoe??. Iniiai? iieiaeaiiy oa ?acoeueoaoe
?iaioe aeiiia?aeaeeny, iaaiai?thaaeeny ? io?eiaee iiceoeaio ioe?ieo ia
3-?e oe?a?inuee?e eiioa?aioe?? c aaoiiaoe/iiai ea?oaaiiy
(“Aaoiiaoeea-96”), i. Naaanoiiieue, 9-15 aa?aniy 1996 ?. oa iaoeiaeo
nai?ia?ao o AAIENA ?IIA IAI Oe?a?ie.

Aaoi? aeena?oaoe?eii? ?iaioe aoa iaeiei c aaoi??a i?iaeoo ? 602 OIOOe
“Aeine?aeaeaiiy oa ?ic?iaea eiiieaeno aia?aoii-i?ia?aiieo cania?a
iii?oi?eiao i?ioeano aeoaiaiai caa?thaaiiy aeey iiia?aaeiuei?
ae?aaiinoeee i?ioeano, i?aeaioiaee oa aeiioneiiai eiio?ieth caa?thaa/?a
a aoiii?e aia?aaoeoe?”. Cai?iiiiiaai? a aeena?oaoe?ei?e ?iaio? iaoiaee
aeei?enoiaoaaeeny i?e ?ic?iaoe? i?ia?aiiiai eiiieaeno iii?oi?eiao.

Ioae?eaoe??. Ii iaoa??aeai aeena?oaoe?eii? ?iaioe iioae?eiaaii 8
ae?oeiaaieo ?ia?o, c yeeo 2 aeeiiai? naiino?eii, 6 — o ni?aaaoi?noa?.

No?oeoo?a ? ia’?i ?iaioe. Aeena?oaoe?eia ?iaioa ianyaii 183 iaoeiiienieo
noi??iie neeaaea?oueny c anooio, o?ueio ?icae?e?a, aeniiae?a, nieneo
aeei?enoaieo aeaea?ae, ui i?noeoue 131 iaeiaioaaiiy, oa i’yoe
aeiaeaoe?a. Iniiaiee oaeno i?noeoue 116 noi??iie iaoeiiieniiai oaenoo,
?ethno?iaaiee 44 ?enoieaie oa 3 oaaeeoeyie.

Aaoi? aeneiaeth? ue?o iiaeyeo ?inoeoooo iaoaiaoe/iiai iiaeaethaaiiy
“O?aen?i” ca iaaeaiiy aia?aoieo oa i?ia?aiieo cania?a aeey ?iaioe iaae
aeena?oaoe??th.

CI?NO ?IAIOE

A ia?oiio ?icae?e? aeena?oaoe?eii? ?iaioe i?iaaaeaii ii??aiyeueiee
aiae?c oa ?ic?iaeai? i?ia?aii? ?aae?caoe?? aeayeeo ?nioth/eo iaoiae?a
ai?ieneiaoe?? iaia?a?aieo iaeii- oa aeaiaei??ieo neaiae?a oa neaiae?a,
caaeaieo oaaee/ii.

Iaoae caaeaiee aeayeee neaiae x(t), tI[a,b]. Eiai ai?ieneiaoe?y ii
nenoai? aacenieo ooieoe?e
picscalex1000100090000039f0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e002e0061200000026060f001a00ffffffff00001000
0000c0ffffffbfffffffa00600009f0200000b00000026060f000c004d61746854797065
0000700010000000fb02bbfd0201000000009001000000020002001053796d626f6c0002
040000002d01000008000000320aff01650201000000280010000000fb02bbfd02010000
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08000000320aff018c0301000000290010000000fb02bbfd020100000000900100000002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,
tI[a,b] ia? aeaeyae:
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,
(1)

aea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 — ai?ieneioth/ee
iie?iii?aeueiee niaeo? (AIN) neaiaeo x(t).

Aaeoi?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 ciaoiaeeoueny oeyoii i?i?i?caoe?? ii?ie
ooieoe?? iiieeee ? iiaea aooe aecia/aiee ca oi?ioeith:
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 (2)

Aeaiaioe iia?aoe?eii? iao?eoe? niaeo?o W ? iia?aoe?eiiai aaeoi?a niaeo?o
Q ciaoiaeyoue ca oi?ioeaie:

(3)

Iaeiei c ??ciiaeae?a ai?ieneioth/eo iie?iii?aeueieo niaeo??a ?
ai?ieneioth/? ?iioeueni? niaeo?e (A?N), ui aoaeothoueny ia iniia? nenoai
aei/ii-?iioeuenieo ooieoe?e, cai?iiiiiaaieo N?iae E.I.

Nenoaia A?O ioeueiaiai oa ia?oiai ii?yaeeo aaiaeeoueny ianooiiei /eiii.
Iaoae a?ae??cie [0,T] ?icaeoi ia m ??aieo /anoei. Caaeaii nenoaio
ooieoe?e:

, i=0, 1, …, m-1, (4)

aea s(t) — ooieoe?y iaeeie/iiai no?eaea:
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Aeey oaei? aacenii? nenoaie oi?ioee ciaoiaeaeaiiy AIN ni?iuothoueny:

, i=0,…,m-1. (5)

Noeoii?noue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?oueny A?N.

A ?iaioao N?iae E.I. A?N ooieoe?? aeaio iacaeaaeieo ci?iieo aaiaeeoueny
ianooiiei /eiii. Aaaaeaii nenoaio aeaiaei??ieo A?O ia iniia? (4) :

(6)

Neaiae f(x,y) i?aaenoaaey?oueny iie?iiiii ii nenoai? oa??ieo ooieoe?e
(6):

, (7)

aea

i=1,…,m, j=1,…,n. (8)

*enea F(i,j), Fo (i,j), Fo (i,j) iaceaathoueny A?N aeaiaei??iiai neaiaeo
f(x,y).

I?iaaaeaii ii??aiyeueiee aiae?c ai?ieneiaoe?eieo iaoiae?a aeey ooieoe??
iaei??? ci?iii? oa iaoiaeo AeO-ia?aoai?aiue, cai?iiiiiaaiiai aeaaeai?eii
Iooiaei A.?., ca oi/i?noth ai?ieneiaoe??, no?ee?noth uiaei ia?aoeiae,
caeaaei?noth a?ae aeaeyaeo oa iiooaeiino? nenoaie aacenieo ooieoe?e oa
nenoai i?ae?ioa?aae?a ?icaeooy ?ioa?aaeo [a,b] ia m /anoei. Aecia/aii,
ui a a?eueoino? aeiaaee?a aeei?enoaiiy A?N aea? iaee?aua iaaeeaeaiiy i?e
o?eniaai?e e?eueeino? ooieoe?e o aacen?.

Ia iniia? aenia?eiaioaeueieo aeine?aeaeaiue iieacaii, ui AIN oa,
cie?aia, A?N iiaeooue aeei?enoiaoaaoeny aeey o?eueo?aoe?? neaiaeo, ui
i?noeoue ooiia? neeaaeia?. Iaoiae AeO-ia?aoai?aiue ia iiaeia
canoiniaoaaoe aeey ai?ieneiaoe?? cao?aiiai neaiaeo.

?icaeyiooi ai?ieneiaoe?th neaiae?a, caaeaieo oaaee/ii. I?iaaaeaii
?ethno?aoeai? aenia?eiaioe. Iieacaii, ui AAeN iiaeia aoaeoeaii
canoiniaoaaoe aeey ai?ieneiaoe?? neaiae?a, caaeaieo ianeaii.

Ina?oeaii ieoaiiy ai?ieneiaoe?? aeaiaei??ieo ooieoe?e ia iniia?
aeaiaei??ieo A?O.

O ae?oaiio ?icae?e? ?ic?iaey?oueny oa aeine?aeaeo?oueny ?aeo?neaiee
iaoiae ?ica’ycaiiy iia?aoi?ieo ??aiyiue, ye? aeieeathoue i?e
canoinoaaii? iaoiaeo ai?ieneioth/eo ?iioeuenieo niaeo??a aei aeeiai?/ieo
nenoai, ui iienothoueny aeeoa?aioe?aeueieie ??aiyiiyie c? noaeeie
eiao?oe??ioaie oa ?o nenoaiaie.

Canoinoaaiiy A?N aei iiaeaethaaiiy aeeiai?/ieo nenoai aeicaiey?
aeaaa?a?coaaoe ?ioaa?i-aeeoa?aioe?aeuei? ??aiyiiy nenoaie. Aeey
?ica’ycaiiy io?eiaieo aeaaa?a?/ieo caaea/ iaiao?aeia iaa?iaiiy
iao?eoeue, ui aecia/athoue A?N nenoaie. Ii?yaeie iao?eoeue aecia/a?oueny
aeiaooeii ii?yaeeo aeeoa?aioe?aeueiiai ??aiyiiy, /enea ci?iieo nenoaie
oa e?eueeino? /anoei ?icaeooy ?ioa?aaeo ?ica’yceo a noaiai? ?ici??iino?
i?inoi?o. Oea coiiaeth? iaiao?aei?noue iaa?iaiiy iao?eoeue aenieiai
ii?yaeeo oa ?noioii iaiaaeo? iiaeeeaino? canoinoaaiiy iaoiaeo.

Eia?/iee aeo?ae c oe??? neooaoe?? i?aeeaco? no?oeoo?a iao?eoeue
iia?aoe?eieo iiaeaeae. Oae? iao?eoe? ?ic??aeaeai?, neeaaeathoueny c
o?eeooieo oueiie?oeaaeo ee?oei. Aeae? aoaeaii iaceaaoe iao?eoe? oaeiai
oeio eaac?-oueiie?oeaaeie. Ii?yaeie iiaoaeiae iia?aoe?eii? iiaeae?
aeeiai?/ii? nenoaie ?icaeyiaii ia i?eeeaae? nenoaie aeeoa?aioe?aeueieo
??aiyiue ia?oiai ii?yaeeo, ?icaeyiooeo o ia?aa?ao? 2.1.

Iaoae iaoaiaoe/io iiaeaeue aeeiai?/ii? nenoaie iiaeia i?aaenoaaeoe o
aeaeyae? nenoaie aeeoa?aioe?aeueieo ??aiyiue ia?oiai ii?yaeeo c
ii/aoeiaeie oiiaaie:

tI[0, T] . (9)

aea y(t) –iaa?aeiiee neaiae; fi (t), i=1,…,n – cao?aiiy, ui ae?thoue
ia nenoaio; ai j, i,j=1,…,n — aeaye? ae?eni? ia?aiao?e, y0 i ,
i=1,…, n – ii/aoeia? oiiae.

Aeey a?eueo eiiiaeoiiai caieno ia?aieoaii nenoaio o aaeoi?i?e oi?i?:

(10)

Ia?aeaeaii a iia?aoe?eio iaeanoue, oiaoi cai?ieii ooieoe??
a?aeiia?aeieie niaeo?aie, a iaoaiaoe/i? iia?aoe?? — iia?aoe?yie aeaaa?e
niaeo??a:

, (11)

aea —
picscalex100010009000003890000000200120000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02e001c0011200000026060f001a00ffffffff00001000
0000c0ffffffd0ffffff80010000b00100000b00000026060f000c004d61746854797065
0000300010000000fb0240fe0000000000009001000000020002001053796d626f6c0002
040000002d01000009000000320a80012d0001000000c40058010a00000026060f000a00
ffffffff01000000000010000000fb021000070000000000bc02000000cc010202225379
7374656d00cc040000002d01010004000000f00100000300000000004b00000100000000
000000ffffff000000000000000000000000000000730000000000010000000000000001
010000000000000aec000000010000007e0000ffffff00ffffff00ffffff00ffffff00ff
ffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ff
ffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ffffff00ff
ffff00000000000000000000000000ffffff0000000000ffffff000a500200ffffff0000
31000000000000010000002c0100000032000000320a00ffffff00010100000000000001
010000000800 — ciae oaici?iiai iiiaeaiiy, aai

, (12)

Ia?aieoaii (12), aeacaaoe ?ici??i?noue:

. (13)

sseui ia?aeoe aei oi?ie caieno aaeoi??a

oi ??aiyiiy (13) iaaoaea aeaeyaeo:

, (14)

. Iicia/eii

.

Iao?eoey L – eaac?-oueiie?oeaaa, ?? aeieaie ? 4m2 aa?oiuei-o?eeooieo
i?aeiao?eoeue ?ici??iino? n?n. Ioaea, noaiaoe/ii ??aiyiiy (14) iiaeia
caienaoe o aeaeyae?:

I?ney aeine?aeaeaiiy ??aiyiiy (14) io?eiaii oi?ioeo ?aeo?aioiiai
iaoiaea ?ica’ycaiiy iia?aoi?iiai ??aiyiiy nenoaie cae/aeieo e?i?eieo
aeeoa?aioe?aeueieo ??aiyiue ia?oiai ii?yaeeo:

(15)

Oaeei /eiii, yeui i?yiee iaoiae ?ica’ycaiiy iia?aoi?iiai ??aiyiiy aeey
nenoaie n e?i?eieo aeeoa?aioe?aeueieo ??aiyiue ia?oiai ii?yaeeo
iia?aoi?iei iaoiaeii i?e m aacenieo ooieoe?yo iio?aao? ii?yaeeo 4m2n2
ae?e, oi aeey ?aeo?aioiiai iaoiaeo aeineoue eeoa 4n2m ae?e.

A ia?aa?ao? 2.2 ?icaeyaea?oueny aeeiai?/ia nenoaia, ui iieno?oueny
aeeoa?aioe?aeueiei ??aiyiiyi n-ai ii?yaeeo c ii/aoeiaeie oiiaaie,
iaeeaaeaieie ia ooieoe?th oa iio?aei? aei (n -1) ii?yaeeo aeeth/ii:

(16)

Ooo y(t) – iaa?aeiiee neaiae, f(t) – cao?aiiy, ui ae?? ia nenoaio, bi ,
i=1,…,n-1 — aeaye? ae?eni? ia?aiao?e,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, i=1,…,n-1 – ii/aoeia? oiiae.

Cai?iith ci?iieo caaea/o (16) iiaeia caanoe aei caaea/? (9), ui
?icaeyiooa a iiia?aaeiueiio ia?aa?ao?. I?ney i?iaaaeaiiy cai?ie ??aiyiiy
(16) ia?aoai?th?oueny a nenoaio:

(17)

Ioaea, ye ? aeey nenoaie n aeeoa?aioe?aeueieo ??aiyiue ia?oiai ii?yaeeo,
i?yiee iaoiae ?ica’ycaiiy e?i?eiiai aeeoa?aioe?aeueiiai ??aiyiiy n-ai
ii?yaeeo iia?aoi?iei iaoiaeii i?e m aacenieo ooieoe?yo aeiaaa? ii?yaeeo
4m2n2 ae?e, a aeey ?aeo?aioiiai iaoiaeo — eeoa 4n2m ae?e.

A ia?aa?ao? 2.3 aeine?aeaeo?oueny aeeiai?/ia nenoaia, yea iieno?oueny
e?i?eiei aeeoa?aioe?aeueiei ??aiyiiyi a /anoeiieo iio?aeieo ia?oiai
ii?yaeeo c ii/aoeiaeie oiiaaie:

(18)

aea 0F o F tx, 0F y F ty, u(x,y) – iaa?aeiiee neaiae aeaio iacaeaaeieo
ci?iieo x, y, f(x,y) — cao?aiiy, ui ae?? ia nenoaio, a,b,c – aeaye?
ae?eni? ia?aiao?e, a*b>0.

Iia?aoi?ia ??aiyiiy ia? aeaeyae:
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. (19)

Iicia/eii

.

A nenoai? (19) iao?eoey L ?ici??o 3mn?3mn ? aei/ii-ae?aaiiaeueiith. ??
aeieaie ? 9 aa?oiuei-o?eeooieo i?aeiao?eoeue ?ici??iino? mn?mn, aea m, n
– e?euee?noue /anoei ?icaeooy ?ioa?aae?a ?ica’yceo ii o oa o
a?aeiia?aeii. Cia?aceii ??aiyiiy (19) noaiaoe/ii:

Io?eiaii oi?ioeo ?aeo?aioiiai iaoiaea ?ica’ycaiiy aeeoa?aioe?aeueiiai
??aiyiiy a /anoeiieo iio?aeieo ia?oiai ii?yaeeo (20).

A ?ic?iaeaiiio iaoiae? mn ?ac iaa?oa?oueny iao?eoey 3?3. Ioaea, a oie
/an, eiee i?yiee iaoiae ?ica’ycaiiy iia?aoi?iiai ??aiyiiy aeiaaa?
ii?yaeeo 9m2n2 ae?e, aeey ?aeo?aioiiai iaoiaeo aeinoaoiuei ii?yaeeo 9mn
ae?e.

, k=1,…,mn (20)

A ia?aa?ao? 2.4 aeine?aeaeo?oueny aeeiai?/ia nenoaia, yea iieno?oueny
aeeoa?aioe?aeueiei ??aiyiiyi a /anoeiieo iio?aeieo ae?iaiaiai ii?yaeeo c
ii/aoeiaeie oiiaaie:

(21)

aea 0F o F tx, 0F y F ty; u(x,y) – iaa?aeee neaiae aeaio iacaeaaeieo
ci?iieo x,y; f(x,y) – cao?aiiy, ui ae?? ia nenoaio, g(y), q(x) –
ii/aoeia? cia/aiiy; a,b,c – aeaye? ae?eni? ia?aiao?e. Neiaieii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 /anoeiia
iio?aeia ii o ii?yaeeo a, neiaieii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 – /anoeiia iio?aeia
ii o ii?yaeeo b, a,bIC.

Iia?aoi?ia ??aiyiiy, a?aeiia?aeia (21), ia? aeaeyae:

. (22)

Iicia/eii

,

aea m, n – /enei /anoei ?icaeooy ?ioa?aae?a ?ica’yceo ii o oa o
a?aeiia?aeii.

A nenoai? (22) iao?eoey L ? aei/ii-ae?aaiiaeueiith, ?? aeieaie ? 9m2
aa?oiuei-o?eeooieo i?aeiao?eoeue ?ici??iino? n?n. Nenoaia (22)
noaiaoe/ii cia?aaea?oueny ianooiiei /eiii:

Noaia ?icae?ey?oueny ia 9 aaeeeeo aeie?a, eiaeai c yeeo neeaaea?oueny c
m2 ee?oei ?ici??iino? n?n. Oaea no?oeoo?a nenoaie (22) iaaaaeo?
no?oeoo?o nenoaie (19).

Ioaea, aeey ?ica’ycaiiy iia?aoi?iiai ??aiyiiy (22) iiaeia
aeei?enoiaoaaoe ?aeo?aioiee aeai?eoi, aiaeia?/iee ?ic?iaeaiiio aeey
iia?aoi?iiai ??aiyiiy (19). Eiai iniiaiith oi?ioeith oaeiae aoaea (20):

,k=1,…,mn

sse ? aeey aeeoa?aioe?aeueiiai ??aiyiiy a /anoeiieo iio?aeieo ia?oiai
ii?yaeeo, aeey i?yiiai iaoiaeo ?ica’ycaiiy iaiao?aeii ii?yaeeo 9m2n2
ae?e, a aeey ?aeo?aioiiai iaoiaeo — eeoa ii?yaeeo 9mn ae?e.

?ic?iaeaia i?ia?aiia ?aae?caoe?y oeeo iaoiae?a a nenoai? MathematicaO.
?icaycaii ?yae caaea/ iaoaiaoe/ii? o?ceee, a oiio /ene? oaieiianiiai?io
oa oai??? aeaeo?e/ieo eaioetha?a.

O?ao?e ?icae?e i?enay/aiee aeine?aeaeaiith aeayeeo ieoaiue oi/iino?
ai?ieneiaoe?? neaiaeo.

A ia?aa?ao? 3.1 cai?iiiiiaaii ? ?ic?iaeaii aeai?eoi ioe?ithaaiiy ao?aoe
?ioi?iaoe?? i?e ai?ieneiaoe?? ooieoe?? c iaeiiai ne?i/aiiiaei??iiai
i?aei?inoi?o L2 ([a,b]) ooieoe?yie c ?ioiai ne?i/aiiiaei??iiai
i?aei?inoi?o.

C iaoaiaoe/ii? oi/ee ci?o ciaoiaeaeaiiy AIN ooieoe?? ? ia ui ?ioa ye
ciaoiaeaeaiiy i?iaeoe?? oe??? ooieoe?? ia ne?i/aiiiaei??iee i?aei?ino??
L2 ([a,b]). Ae?th ai?ieneiaoe?? iiaeia ?icaeyaeaoe ye iia?aoi?
i?iaeooaaiiy c iaeiiai i?aei?inoi?o a ?ioee. Oiae? ao?aoo ?ioi?iaoe??
iiaeia ioe?ieoe ye ii?io ??cieoe? i?ae ooieoe??th oa ?? iaaeeaeaiiyi a
i?inoi?? L2 ([a,b]).

Iinoaiiaea caaea/?.

Iaoae I1 =L(e1 (t), e2 (t), … , en (t)),
I2 = L(h1 (t), h2 (t), … , hn (t)), aea ei ,hj  — aea? e?i?eii
iacaeaaei? nenoaie ooieoe?e c L2 ([a,b]), L — e?i?eia iaieiiea.

Iaoae fIH1, P1f — i?oiaiiaeueia i?iaeoe?y f ia H2, oiaoi P1f — oaea
ooieoe?y c H2 , ui ||P1f-f ||=min|| g-f ||, gIH2 , aai P1f — iaee?aua
ai?ieneiaoe?y f ooieoe?yie c H2. Ooo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 — ii?ia a
i?inoi?? L2 ([a,b]), oiaoi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.

Iio??aii ioe?ieoe i?i?ioi ? iaeneioi ii an?i fIH1 , || f ||=1 aaee/eie
||P1f – f ||. Oaeei /eiii, ie aoaeaii ciaoe i?i?iaeueio oa iaeneiaeueio
a?aeiinio ao?aoo ?ioi?iaoe?? i?e i?iaeooaaii?.

?ica’yceii oe??? caaea/? ? ianooiiee aeai?eoi.

Aeai?eoi.

1. Ciaeoe iao?eoe?. neaey?ieo aeiaooe?a S, G, R:

2. S=(si j ), i,j=1, …,n, si j=(ek , ej ),

3. R=(ri j ), i,j=1, …,n, ri j =(hi, hj ),

4. G=(gi j ), i,j=1, …,n, gi j =(hi , ej ).

5. Ciaeoe iao?eoeth A= S-1 GTR-1 G.

6. Ciaeoe aeani? /enea li, ?=1, …, n iao?eoe? A.

7. Ia/eneeoe
a=picscalex134010009000003c301000004001500000000000500000009020000000004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,
b=picscalex1310100090000039e01000004001500000000000500000009020000000004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.

8. Oiae? aeey an?o ooieoe?e f c H1 , oaeeo ui || f ||=1, aeeiio?oueny

||P1 f — f||I[a,b]I[0,1].

9. *ei iaioa b , oei iaioa ?ioi?iaoe?? ao?a/a?oueny i?e ai?ieneiaoe??.

Ia iniia? i?ia?aiii? ?aae?caoe?? oeueiai aeai?eoio iiaoaeiaaio oaaeeoeth
?ioa?aae?a oi/iino? aca?iiiai iaaeeaeaiiy ?yaeo nenoai ooieoe?e.

O ia?aa?ao? 3.2 cai?iiiiiaaii, oai?aoe/ii iaa?oioiaaii ? aeine?aeaeaii
iaoiae aeai?o i?aenenoaie c caaeaiiai aaceno aeey iioeiaeueii?
ai?ieneiaoe?? ooieoe??, yeui e?euee?noue ooieoe?e a i?aenenoai?
o?eniaaia ? caaeaia ca?ai? ei?enooaa/ai.

Iinoaiiaea caaea/?. Iaoae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 —
i?oiii?iiaaiee aacen i?inoi?o L2 ([a,b]), x(t)IL2 ([a,b]) — aeayea
ooieoe?y, mIN — o?eniaaia /enei.

Iaiao?aeii aecia/eoe ooieoe??
picscalex129010009000003820100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c02a002200b1200000026060f001a00ffffffff00001000
0000c0ffffffb0ffffffe00a0000500200000b00000026060f000c004d61746854797065
0000800010000000fb0240fe0000000000009001010000020002001053796d626f6c0002
040000002d01000009000000320aa0011a00010000006a000e0109000000320aa0010603
010000006a000e0109000000320aa0015f08010000006a000e0115000000fb02e0fe0000
000000009001010000000002001054696d6573204e657720526f6d616e00007304000000
2d01010004000000f001000009000000320a10023d01010000006e00900009000000320a
10022904010000006e00900009000000320a10028209010000006e00900015000000fb02
20ff0000000000009001010000000002001054696d6573204e657720526f6d616e000073
040000002d01000004000000f001010009000000320a58021a0a010000006d00a1001500
0000fb0220ff0000000000009001000000000002001054696d6573204e657720526f6d61
6e000073040000002d01010004000000f001000009000000320a5802c701010000003100
700009000000320a5802c104010000003200700015000000fb0240fe0000000000009001
000000000002001054696d6573204e657720526f6d616e000073040000002d0100000400
0000f001010009000000320aa0018402010000002c00700009000000320aa00193050100
00002c00700009000000320aa001dd07010000002c00700011000000fb0240fe00000000
0000900100000002000200104d542045787472610002040000002d01010004000000f001
000009000000320aa0010a06010000004b00c0010a00000026060f000a00ffffffff0100
0000000010000000fb021000070000000000bc02000000cc0102022253797374656d00cc
040000002d01000004000000f00101000300000000000000000000001500000000000000
000000000000000000000000000000000000006573200065732000000000000000000000
000000ffffff0000000000ffffff00000000000a9b0500ffffff00000000000210000000
0000000af2020001800000003d00000009000000320a00018503000000000000cd00 c
aaceno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 oa /enea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, oae? ui

, (23)

aea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 — ii?ia a
i?inoi?? L2 ([a,b]). Oaeo ai?ieneiaoe?th ooieoe?? x(t) ia ?ioa?aae?
[a,b] i?e caaeai?e e?eueeino? aacenieo ooieoe?e m aoaeaii aaaaeaoe
iaee?auith.

?iceeaaeaii ooieoe?th x(t) ii aaceno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:

.

Aeaa?aii ?iaeaene
picscalex1200100090000034d0100000200150000000000050000000902000000000400
000002010100050000000102ffffff00040000002e011800050000003102010000000500
00000b0200000000050000000c022003c0041200000026060f001a00ffffffff00001000
0000c0ffffffb6ffffff80040000d60200000b00000026060f000c004d61746854797065
0000800010000000fb029ffd0c01000000009001000000020002001053796d626f6c0002
040000002d01000008000000320a4a022700010000007b0010000000fb029ffd0c010000
00009001000000020002001053796d626f6c0002040000002d01010004000000f0010000
08000000320a4a023e02010000007d0015000000fb0240fe000000000000900101000000
0002001054696d6573204e657720526f6d616e000073040000002d01000004000000f001
010009000000320a2002ed00010000006e00e00015000000fb02e0fe0000000000009001
010000000002001054696d6573204e657720526f6d616e000073040000002d0101000400
0000f001000009000000320a9002bf01010000006900500009000000320ad10215030100
00006900500009000000320a22011e03010000006d00cf0010000000fb02e0fe00000000
00009001000000020002001053796d626f6c0002040000002d01000004000000f0010100
09000000320ad1027a03010000003d009e0015000000fb02e0fe00000000000090010000
00000002001054696d6573204e657720526f6d616e000073040000002d01010004000000
f001000009000000320ad102100401000000310090000a00000026060f000a00ffffffff
01000000000010000000fb021000070000000000bc02000000cc0102022253797374656d
00cc040000002d01000004000000f001010003000000000040000002c603000090000000
000000000000000000000000000000000000000000000002100000021000000000000000
00000000000000ad0ce80000000000000000000000000000000000000000000000000000
00650000000000ffffff000000000000000000002d01000004000000f001000009000000
4000 oaeei /eiii, ui

. (24)

Iieeaaeaii

, i=1, … , m. (25)

Iieacaii, ui aeey aea?aieo oaeei /eiii ooieoe?e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?
a?aeiia?aeieo eiao?oe??io?a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 ai?ieneiaoe?y ooieoe?? x(t)
iaee?aua o nain? (23) aeey o?eniaaiiai m.

?ic?iaeaii aeai?eoi, yeee aeicaiey? ca ne?i/aiiee /an aea?aoe c
iane?i/aiiiai aaceno i?aenenoaio iaea?eueo cia/eieo aacenieo ooieoe?e
o?eniaaii?, caaeaii? ei?enooaa/ai iiooaeiino?.

Ia iniia? aeai?eoio aeine?aeaeaia ai?ieneiaoe?y ?yaeo ooieoe?e.
Iieacaii, ui ai?ieneiaoe?y ia iniia? i?aenenoaie ooieoe?e, aea?aii? ca
aeai?eoiii, ? a?eueo oi/iith, i?ae ai?ieneiaoe?y ia iniia? aeia?eueii?
?ioi? i?aenenoaie.

?ic?iaeaiee aeai?eoi ? nain aeei?enoiaoaaoe aeey aeine?aeaeaiiy ooieoe?e
i?e iaiaaeaieo ia/enethaaeueieo ?ano?nao.

A aeiaeaoeao aei ?iaioe iaaaaeai?: aeaiaioe aeaaa?e A?N; aiae?oe/i?
icia/aiiy oa a?ao?/i? cia?aaeaiiy aacenieo nenoai ooieoe?e ??cieo oei?a;
aeaye? icia/aiiy oa oi?ioee ae?iaiaiai ?ioaa?i-aeeoa?aioe?aeueiiai
/eneaiiy; e?noeiae i?ia?ai ia iia? nenoaie Mathematica ®; ?acoeueoaoe
ia/enethaaeueieo aenia?eiaio?a ii ai?ieneiaoe?? neaiae?a, caaeaieo
ianeaii, c aeei?enoaiiyi AAeN.

INIIAI? ?ACOEUeOAOE OA AENIIAEE

Iniiaiei ?acoeueoaoii aeena?oaoe?eii? ?iaioe ? ?ic?iaea aoaeoeaieo
iaoiae?a iiaeaethaaiiy aeeiai?/ieo nenoai, a iniia? yeeo eaaeeoue
i?aaenoaaeaiiy neaiae?a ai?ieneioth/eie iie?iii?aeueieie niaeo?aie.
Cie?aia, io?eiai? ianooii? ?acoeueoaoe:

1. Ia iniia? ?ic?iaeaieo i?ia?ai a nenoai? MathematicaO i?iaaaeaiee
ii??aiyeueiee aiae?c oi/iino? ai?ieneiaoe?? aeey AIN, A?N oa
AeO-ia?aoai?aiue. Aeine?aeaeaii ai?ieneiaoe?? ooieoe?e ii aaceniei
nenoaiai ??ciiai oeio. Iiaoaeiaaii oaaeeoeth iioeaie ai?ieneiaoe?? aeey
?yaeo ooieoe?e. Aeine?aeaeaii ai?ieneiaoe?ei? iaoiaee ia no?ee?noue
uiaei ooio, yeee noi?iaiaeaeo? neaiae. Iieacaii, ui AIN oa, cie?aia, A?N
iiaeooue aeei?enoiaoaaoeny aeey o?eueo?aoe?? neaiaeo, yeee
noi?iaiaeaeo?oueny ooiii, a iaoiae AeO-ia?aoai?aiue a oaeeo aeiaaeeao ia
ni?aoeueiao?. Aeine?aeaeaii ?iaioo iaoiaeo ??aieo ieiu aeey ??cieo
nenoai i?ae?ioa?aae?a.

2. ?ic?iaeaia iaoiaeeea ai?ieneiaoe?? ooieoe?e, caaeaieo ianeaii, ia
iniia? AIN. Ia iniia? ?ic?iaeaiiai aeai?eoia io?eiaia ai?ieneiaoe?y
ciai?oi?o aieueo-aiia?ieo oa?aeoa?enoee aeayeeo aeaea?ae aeeaeaiiy aeey
?o/iiai caa?thaaiiy.

3. ?ic?iaeai?, ?aae?ciaai? oa aeine?aeaeai? ?aeo?aioi? aeai?eoie
?ica’ycaiiy iia?aoi?ieo ??aiyiue aeey aeeiai?/ieo nenoai, ye?
iienothoueny nenoaiaie e?i?eieo aeeoa?aioe?aeueieo ??aiyiue ia?oiai oa
aeueo ii?yaee?a oa e?i?eieo aeeoa?aioe?aeueieo ??aiyiue a /anoeiieo
iio?aeieo ia?oiai ? aeueo ii?yaee?a.

4. Ia iniia? ?ic?iaeaieo aeai?eoi?a aeine?aeaeaii ?yae caaean
oaieiianiiai?io, oai?aoe/ii? iaoai?ee, oai??? aeaeo?e/ieo eaioetha?a.

5. Cai?iiiiiaaii ? ?ic?iaeaii iaoiae ioe?iee ao?aoe ?ioi?iaoe?? i?e
ai?ieneiaoe?? ooieoe?? c iaeiiai ne?i/aiiiaei??iiai i?aei?inoi?o
L2 ([a,b]) ooieoe?yie c ?ioiai ne?i/aiiiaei??iiai i?aei?inoi?o.

6. Cai?iiiiiaaii oa ?icaeiooi iaoiae aeai?o i?aenenoaie c caaeaiiai
aaceno aeey iioeiaeueii? ai?ieneiaoe?? aeia?eueii? ooieoe??. ?ic?iaeaii
aeai?eoi, yeee aeicaiey? ca ne?i/aiiee /an aea?aoe c caaeaiiai
iane?i/aiiiai aaceno i?aenenoaio, ui caaacia/o? iaea?eueo oi/ia
iaaeeaeaiiy caaeaii? ooieoe??, yeui e?euee?noue ooieoe?e a i?aenenoai?
o?eniaaia ? caaeaia ca?ai? ei?enooaa/ai.

Cai?iiiiiaai? oa ?icaeioo? a ?iaio? iaoiaee iiaeaethaaiiy ia iniia? AIN
iiaeia canoiniaoaaoe aeey aeine?aeaeaiiy aeeiai?/ieo nenoai, ia?aaea/?,
noeniaiiy oa caa?aaeaiiy ?ioi?iaoe?? a oaeeo aaeocyo iaoee ? oaoi?ee ye
?iaioioaoi?ea, ia/enethaaeueia oaoi?ea, ?aae?iaeaeo?ii?ea,
i?eeaaeiaoaeoaaiiy, oiui.

IOAE?EAOe?? II OAI? AeENA?OAOe??

1. Aaneeueaa A.A., Neiae E.A., Ieeeiaiei I.I. ?aaeecaoeey
aii?ieneiaoeeiiiuo iaoiaeia iaoaiaoe/aneiai iiaeaee?iaaiey neaiaeia a
i?ia?aiiiie n?aaea nenoaiu Mathematica O// Yeaeo?iiiia iiaeaee?iaaiea. —
1997. — o.19. — ?3. — n.3-13.

2. Neiae E.A., Ieeeiaiei I.I., Einiaa A.I. Iaeioi?ua aii?inu ?aaeecaoeee
aii?ieneiaoeeiiiuo iaoiaeia ia?aaioee neaiaeia aeeiaie/aneeo nenoai a
i?ia?aiiiie n?aaea iaeaoa MathematicaO // I?iaeaiu oi?aaeaiey e
eioi?iaoeee. — 1997. — ?4. — n. 123-132.

3. Ieeeiaiei I.I. Auai? iiaenenoaiu ec caaeaiiiai aacena aeey
iioeiecaoeee aii?ieneiaoeee ooieoeee iaeiie ia?aiaiiie // Yeaeo?iiiia
iiaeaee?iaaiea. — 1998. — o.19. — ?6. — n.3-13.

4. Aaneeueaa A.A., Neiae E.A., Ieeeiaiei I.I. ?aeo?neaiue aiaeec
iia?aoeeiiiuo iiaeaeae aeeiaie/aneeo nenoai ia iniiaa aii?ieneie?othueo
eiioeueniuo niaeo?ia. — Eeaa, 1999. – 76 c. — (Iaoeeiiaeueiay Aeaaeaiey
iaoe Oe?aeiu. Ioaeaeaiea aea?eaeiuo iiaeaee?othueo e oi?aaeythueo nenoai
a yia?aaoeea EIIY). – ISBN 966-02-0956-8.

5. Vasilyev V.V., Simak L.A., Pilipenko N.N. Realization of
Approximation Methods for Signal Simulation Using the “Mathematica”
Software // Engineering Simulation. — 1998. — o.15. — pp.255-268.

6. Ieeeiaiei I.I. I?ia?aiiiay ?aaeecaoeey aii?ieneiaoeeiiiuo iaoiaeia
iaoaiaoe/aneiai iiaeaee?iaaiey iai?a?uaiuo neaiaeia a n?aaea iaeaoa
MathematicaO. — Eeaa, 1997. — 35 n. (I?ai?eio/ IAI Oe?aeiu Ioaeaeaiea
aea?eaeiuo iiaeaee?othueo e oi?aaeythueo nenoai a yia?aaoeea EIIY;
?01/97).

7. Aaneeueaa A.A, Neiae E.A., *a/ue A.A., Ai?iiiaa I.N., Einiaa A.I.,
Ieeeiaiei I.I. Iaoiaeu aii?ieneiaoeee e oeeo?iaie ia?aaioee neaiaeia
iiieoi?eiaa — Eeaa, 1998. – 72 c. — (I?ai?eio / IAI Oe?aeiu. Ioaeaeaiea
aea?eaeiuo iiaeaee?othueo e oi?aaeythueo nenoai a yia?aaoeea EIIY;
? 02/98).

8. Neiae E.A., Einiaa A.I., Oea/oe (Ieeeiaiei) I.I. Aii?ieneie?othuea
iaoiaeu ia?aaioee neaiaeia aeeiaie/aneeo nenoai e eo ?aaeecaoeey a
i?ia?aiiiie n?aaea nenoaiu Mathematica // 3-a oe?a?inueea eiioa?aioe?y c
aaoiiaoe/iiai ea?oaaiiy (“Aaoiiaoeea-96”): Naaanoiiieue, 9-15 aa?aniy
1996. — Naaanoiiieue: NaaAOO, 1996. — o.2. — n.207-208.

Iniaenoee aianie aaoi?a. O ?iaioao, iaienaieo o ni?aaaoi?noa?, aaoi?o
iaeaaeaoue: [1,5] — aeai?eoi?caoe?y iiooeo ai?ieneioth/eo
iie?iii?aeueieo niaeo??a oa ?ic?iaea a?aeiia?aeieo i?ia?ai, iinoaiiaea
ia/enethaaeueieo aenia?eiaio?a oa i?iaaaeaiiy ii??aiyeueiiai aiae?co
ai?ieneiaoe?eieo iaoiae?a; [2] — iinoaaeai? ia/enethaaeuei? aenia?eiaioe
oa ia ?o iniia? i?iaaaeaiee ii??aiyeueiee aiae?c ai?ieneiaoe?eieo
iaoiae?a; [4] — cai?iiiiiaai? oa ?ic?iaeai? ?aeo?aioi? iaoiaee
aeine?aeaeaiiy aeeiai?/ieo nenoai, noai?ai? a?aeiia?aei? i?ia?aie,
i?iaaaeaii na??th aenia?eiaio?a, ui ?ethno?othoue ?ic?iaeai? iaoiaee;
[7] — ?ic?iaea i?ia?ai ? iinoaiiaea ?ethno?aoeaieo aenia?eiaio?a a
nenoai? MathematicaO; [8] — i?ia?aii? aaia?aoe?? ai?ieneioth/eo
iie?iii?aeueieo niaeo??a.

Ieeeiaiei I.I. Iaoaiaoe/ia iiaeaethaaiiy aeeiai?/ieo nenoai a
iia?aoe?ei?e iaeano? ia iniia? ?aeo?aioieo ni?aa?aeiioaiue (c i?ia?aiaie
a nenoai? “Mathematica ®”). – ?oeiien.

Aeena?oaoe?y ia caeiaoooy iaoeiaiai nooiaiy eaiaeeaeaoa oaoi?/ieo iaoe
ca niaoe?aeuei?noth 01.05.02 – iaoaiaoe/ia iiaeaethaaiiy oa
ia/enethaaeuei? iaoiaee. – ?inoeooo i?iaeai iiaeaethaaiiy a aia?aaoeoe?
IAI Oe?a?ie, Ee?a, 1999.

Aeena?oaoe?th i?enay/aii ieoaiiyi aiae?co aeeiai?/ieo nenoai ia iniia?
iia?aoe?eieo iaoiae?a. ?icaeioo? ai?ieneiaoe?ei? iaoiaee aeine?aeaeaiiy
aeeiai?/ieo nenoai ia iniia? ?aeo?neaieo ni?aa?aeiioaiue, ye?
?icoe?ythoue iiaeeeaino? canoinoaaiiy ai?ieneiaoe?? ca ?aooiie ciyooy
iaiao?aeiino? iaa?iaiiy iao?eoeue aenieiai ii?yaeeo. ?aeo?aioi?
aeai?eoie ?ic?iaeai? aeey nenoai, ui iienothoueny e?i?eieie
aeeoa?aioe?aeueieie ??aiyiiyie oa ?o nenoaiaie, a oaeiae ??aiyiiyie a
/anoeiieo iio?aeieo ia?oiai ? ae?iaiaiai ii?yaeeo. Cai?iiiiiaaii oa
?icaeiooi iaoiae ioe?ithaaiiy ao?aoe ?ioi?iaoe?? i?e ai?ieneiaoe??
ooieoe?? c iaeiiai ne?i/aiiiaei??iiai i?aei?inoi?o L2 ([a,b]) ooieoe?yie
c ?ioiai ne?i/aiiiaei??iiai i?aei?inoi?o. Cai?iiiiiaaii oa iaa?oioiaaii
iaoiae aeai?o ne?i/aiii?, caaeaii? ei?enooaa/ai e?eueeino? ooieoe?e c
iane?i/aiiiai aaceno aeey iioei?caoe?? ai?ieneiaoe?? ooieoe?? iaei???
ci?iii?. A nenoai? Mathematica® noai?ai? i?ia?aii? ?aae?caoe?? an?o
iaoiae?a. Iniiai? ?acoeueoaoe ?iaioe aeei?enoiaoaaeeny a i?ioean?
aeeiiaiiy iaoeiai-aeine?aeieo ?ia?o AAIENA ?IIA IAI Oe?a?ie.

Eeth/ia? neiaa: iaoaiaoe/ia iiaeaethaaiiy, iia?aoe?ei? iaoiaee,
ai?ieneiaoe?y, /eneia? iaoiaee.

Pylypenko N.M. Mathematical modeling of dynamic systems in operational
field based on recurrent ratios (with programs in “Mathematica ®”
language). – Manuscript.

Thesis for a candidate’s degree by speciality 01.05.02 – mathematical
modeling and calculation methods. – The Institute of Simulation Problems
in Power Engineering of National Academy of Science of Ukraine, Kyiv,
1999.

The dissertation is devoted to the questions of dynamic systems analysis
based on operational methods. Approximation methods for dynamic systems
analysis based on recurrent ratios row are developed. These methods
extend application of approximation at the expense of removing necessity
of converting high order matrixes. Recurrent algorithms are developed
and realized for the systems described by linear differential equations
and their systems and partial differential equations of first and
fractional order. The method for estimation of information loss after
approximation of function from one finite-measured subspace of
L2 ([a,b]) by functions of another finit-measured subspace is proposed.
The method of finding the finite subsystem from infinite basis for the
optimisation of approximation of one-variable function is proposed and
proved. Program realizations of all methods in Mathematica® system are
prepared. The basic results of the dissertation were used in research
work of Department of Hybrid Modeling and Controlling Systems in Power
Engineering of ISPP of NAS of Ukraine.

Key words: mathematical modeling, operational methods, approximation,
numerical methods.

Ieeeiaiei I.I. Iaoaiaoe/aneia iiaeaee?iaaiea aeeiaie/aneeo nenoai a
iia?aoeeiiiie iaeanoe ia iniiaa ?aeo??aioiuo niioiioaiee (n i?ia?aiiaie
a nenoaia “Mathematica ®”). – ?oeiienue.

Aeenna?oaoeey ia nieneaiea o/aiie noaiaie eaiaeeaeaoa oaoie/aneeo iaoe
ii niaoeeaeueiinoe 01.05.02 – iaoaiaoe/aneia iiaeaee?iaaiea e
au/eneeoaeueiua iaoiaeu. – Einoeooo i?iaeai iiaeaee?iaaiey a yia?aaoeea
IAI Oe?aeiu, Eeaa, 1999.

Aeenna?oaoeey iinayuaia aii?inai aiaeeca aeeiaie/aneeo nenoai n
eniieueciaaieai iia?aoeeiiiuo iaoiaeia.

I?iaaaeai n?aaieoaeueiue aiaeec aii?ieneiaoeeiiiuo iaoiaeia (ia iniiaa
aii?ieneiaoeeiiiuo iieeiiieaeueiuo niaeo?ia e aii?ieneiaoeeiiiuo
eiioeueniuo niaeo?ia) e iaoiaea AeO-i?aia?aciaaiee ii ?yaeo i?eciaeia:
caaeneiinoue oi/iinoe aii?ieneiaoeee io aaceniie nenoaiu ooieoeee,
iiiaoionoie/eainoue e o. ae. Enneaaeiaaia aii?ieneiaoeey aeaoia?iuo
ooieoeee. I?eaaaeaiu eeethno?aoeaiua i?eia?u. ?ac?aaioaia iaoiaeeea
aii?ieneiaoeee ooieoeee, caaeaiiuo ianneaii. Ia iniiaa ?ac?aaioaiiiai
aeai?eoia iieo/aia aii?ieneiaoeey aiaoieo aieueo-aiia?iuo oa?aeoa?enoee
iaeioi?uo enoi/ieeia ieoaiey aeey ?o/iie naa?ee.

?acaeou aii?ieneiaoeeiiiua iaoiaeu enneaaeiaaiey aeeiaie/aneeo nenoai ia
iniiaa aii?ieneiaoeeiiiuo eiioeueniuo niaeo?ia (AEN). I?e i?eiaiaiee
aii?ieneiaoeeiiiuo iaoiaeia aicieeatho iia?aoi?iua o?aaiaiey aunieiai
ii?yaeea. I?yiua iaoiaeu ?aoaiey iiaeiaiuo o?aaiaiee i?eaiaeyo e
a?iiicaeeei au/eneaieyi, /oi a ?yaea neo/aaa ia iicaieyao aeinoeaiooue
aeaeaaiie oi/iinoe i?e ia?aaioea neaiaeia. Iia?aoeeiiiua iao?eoeu,
iino?iaiiua ia iniiaa AEN, eiatho eaace-oaieeoeaaue aeae (o.a. ninoiyo
ec o?aoaieueiuo oaieeoeaauo eeaoie). Yoi naienoai iicaieyao ?ac?aaioaoue
?aeo?neaiua oi?ioeu ia?auaiey iao?eoe eioaa?e?iaaiey. Iineaaeiea
?anoe?ytho aiciiaeiinoe i?eiaiaiey aii?ieneiaoeee ca n/ao niyoey
iaiaoiaeeiinoe ia?auaiey iao?eoe aunieiai ii?yaeea. ?aeo??aioiua
aeai?eoiu ?ac?aaioaiu e ?aaeeciaaiu aeey nenoai, iienuaathueony
eeiaeiuie aeeooa?aioeeaeueiuie o?aaiaieyie e eo nenoaiaie, a oaeaea
o?aaiaieyie a /anoiuo i?iecaiaeiuo ia?aiai e ae?iaiiai ii?yaeea. A
nenoaia neiaieueiie iaoaiaoeee Mathematica® nicaeaiu i?ia?aiiiua
?aaeecaoeee anao ?ac?aaioaiiuo iaoiaeia. Ia iniiaa i?ia?aii enneaaeiaai
?yae caaea/ iaoaiaoe/aneie oeceee (oaieiianniiaiaia, oeeueo?aoeee,
oai?ee yeaeo?e/aneeo oeaiae).

A aeenna?oaoeee enneaaeiaaiu iaeioi?ua aii?inu oi/iinoe aii?ieneiaoeee
neaiaea. I?aaeeiaeai e ?ac?aaioai iaoiae ioeaieaaiey iioa?e eioi?iaoeee
i?e aii?ieneiaoeee ooieoeee ec iaeiiai eiia/iiia?iiai iiaei?ino?ainoaa
L2 ([a,b]) ooieoeeyie ec ae?oaiai eiia/iiia?iiai iiaei?ino?ainoaa. Ia
iniiaa i?ia?aiiiie ?aaeecaoeee iaoiaea a nenoaia Mathematica® i?iaaaeai
?yae yenia?eiaioia, ?acoeueoaoii eioi?uo noaei iino?iaiea oaaeeoeu
iioa?e eioi?iaoeee aeey ?yaea nenoai ooieoeee (noaiaiiuo,
iieacaoaeueiuo, o?eaiiiiao?e/aneeo, aei/ii-eiioeueniuo, *aauoaaa e
ae?.).

I?aaeeiaeai e iainiiaai iaoiae auai?a eiia/iiai, caaeaiiiai
iieueciaaoaeai eiee/anoaa iaeaieaa cia/eiuo ooieoeee ec aaneiia/iiai
aacena aeey iioeiecaoeee aii?ieneiaoeee ooieoeee iaeiie ia?aiaiiie.
?ac?aaioai aeai?eoi, eioi?ue iicaieyao aua?aoue oaeoth iiaenenoaio ca
eiia/iia a?aiy. Aeai?eoi ?aaeeciaai a aeaea i?ia?aiiu a nenoaia
Mathematica®. I?eaaaeaiu eeethno?aoeaiua i?eia?u. Iiaeoaa?aeaeaii, /oi
aii?ieneiaoeey ia iniiaa aua?aiiie nenoaiu ooieoeee yaeyaony aieaa
oi/iie ii n?aaiaieth n aii?ieneiaoeeae ia iniiaa i?iecaieueiie ae?oaie
iiaenenoaiu.

Iniiaiua ?acoeueoaou ?aaiou eniieueciaaeenue a i?ioeanna auiieiaiey
iao/ii-enneaaeiaaoaeueneeo ?aaio Ioaeaeaiey aea?eaeiuo iiaeaee?othueo e
oi?aaeythueo nenoai a yia?aaoeea EIIY IAI Oe?aeiu ii oaia
“Iiaeaee?iaaiea yia?aaoe/aneeo iauaeoia e ioeaiea eo oaoie/aneiai
ninoiyiey a noaoeeiia?iuo oneiaeyo yenieoaoaoeee” (Oeo? oaiu “Ciiae”.
Auiieiyeanue a 1995-1997 aa. ii iinoaiiaeaieth Ioaeaeaiey
oeceei-oaoie/aneeo i?iaeai yia?aaoeee IAI Oe?aeiu), a oaeaea a i?iaeoa
? 602 Oe?aeineiai Iao/ii-Oaoiieiae/aneiai Oeaio?a ii oaia “Enneaaeiaaiea
e ?ac?aaioea eiiieaena aiia?aoii-i?ia?aiiiuo n?aaenoa iiieoi?eiaa
i?ioeanna aeoaiaie naa?ee aeey i?aaeaa?eoaeueiie aeeaaiinoeee i?ioeanna,
iiaeaioiaee e aeiioneiiai eiio?iey naa?ueeia a aoiiiie yia?aaoeea”.

Eeth/aaua neiaa: iaoaiaoe/aneia iiaeaee?iaaiea, iia?aoeeiiiua iaoiaeu,
aii?ieneiaoeey, /eneaiiua iaoiaeu.

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