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MATEMATECOE MOEPOBAHE P.H. y, H.. Caeoa,.. poo (Moca) POCTPAHCTBEHHO HEOHOPOHA MOE PABT COAHO HAPEHHOCT OTEHO BTOO PEOHA B cae peaaec coco

ee epepx coax epe ex, oox cpo pocpaceo eoopoe oe a ee coao apeoc. poc cpaee peyao cox paceo pocpaceo oopoo pocpac eo eoopoo oee, ao eocppyc peyeca ocee.

ee coa: pocpaceo oopoa oe, pocpaceo eoopoa oe, acep, pyaoe pocpaco, epepa epeea, paoca cxea, eo p o poo.

He peaaec oe, ooa paa ac oc coao apeoc (CH) eoopo oeo o peoe o pa ecyx o peoe coao-o Pyap Hoae y aae PAEH, oop o-aeaecx ay, poeccop ecoo ayea M, peop cya cepe PAH.

Haea epoa Caeoa aa o-aeaecx ay, cap ay copy ayea BM M.

ae opeo poo cye V ypca ayea BM M.

Cooo: 4M. 2003. 16.

pocmpacmeo eoopoa oe...

oecx aopo. Aop e peey a ooy oe , a o, o ye peaaeo apae oa oe a py coooa ope aapa o pacey ypo CH. Ha po, eoxoo oepy, o pe e o eoo po ae, cxee, oopa aee oa aoc opeee coooec coepae. Te e eee, peaaea oe oe eca oea cooo , ocoy, a a , oa aae eoopoe apaee, paax oopoo ooa coeca paoa coooo a eao, e oopo oyee ocpyoo exa a epe co cooo e, a CH (oop oxo o oa epe CH c. [1]).

B ocoe peaaeo oe axoc coeo poa a co, ocae aoeoe oe oae ao pa poecca oppoa CH:

(1) peo ec opoao cceo: ae ao p e e oecy a ypoe CH;

o peooee, oeo, peo peayec paec, o eo ecoee po peoy ycoe oe;

(2) a aopo, oppy ypoe CH eo eee o pee, ec oece a ee peoa cooyoc pocxox e co;

ooa o o ece epe S;

ao oe yoo e aapyc coco e ex co, oope eceo oe cy a CH, e e pe o cocoax oe aoo oe c;

o oax ec poeax oox oxoax x pee c., apep, [2];

opoo ye oca c ooa oe apa cooecyx paceo;

(3) peo oo pae a ac (oac, opeo), ao oopx eec co cpe ypoe CH, oop, coceo, o ep aa ac ec a ee epe;

e ye ca ac peoa oac oea eoey eppopaoy ee oce P.H. yu, H.. Caeoa,.. pouo eo;

o eceoc o oy e ca c eopa e: cae, ac peoa oe coco ye a eo ep pop cyeeco ooe cecx ecoepo ..;

(4) ca ec paccapaex co a ay CH, co oepe, opeeec oece pa pyx a opo, exc o oac oac. He aece a oo aopa coyec ypoe oeceeoc ao o ac cpeca c co coe peoa (peoaaec, o eo epe cpeca c o ee ao oac oxo opa o oppyx ypoe CH cox). o ypoe ooa yo q;

(5) ao ee CH o oac oac oe yooe aoy pacea oc o eoopoo oepx oc o oece eoopo yae c (po ). p o eoopooc aaec ye q, ya a ca aaoa ec S. eee c oec co, ax CH, o oac oac (ao, o yaaoe eee ocec ece y q) oe eppepoao a y S. ecye ec o oo coe peooee, caoe c e, o acpa pyec o oo, o co oace peoe oeo, pac capae q a epepy epeey.

B paoax [3;

4] a peoea a aaea oopo a oe ee CH (oe ocoaa a eco ypa e HaeCoca pacea oc):

x = S - x, T e x CH, S yaa ca (po), T pe peaca (.e. pe, eee oopoo yaa ca oec ye a aceee oac;

ece o c, a pao, opaeo o pee). ec e pac o ae cop ypoae e ooe (3) (4). Caoc, o x S pocmpacmeo eoopoa oe...

ac oo o pee:

x = x(t), S = S(t), aee S ec cpe o cey peoy. Hae pao pox oace peoe e peoaaoc. pee aax paoax ce pace o o oe cp pye pae apeoc x yae c S o pe e p yco oopooc paccapaeoo peoa.

Paeee ceo peoa a aeceo paopoe oa c, eee ocaeo y paoaeceoc aopa q eae oe eoopoo. peoaaec, o pocxoe peoe co eoaoo ocpac e pa x oace peoa. CH, opeeee ee oece S a c e oo o pee, o o yoyoo aopa:

x = x(t, q), S = S(t, q).

oee aea peaoc pecaec coae pocpa ceo eoopox aecx oee a:

2 x x = D + S - x.

q2 T 2 x B ypae oc o e D, oo yec q eoopooc peoa o epepo epeeo q. D oe y (D > 0).

peoo, o epeeo, aae eoopooc oe, ec pae coco cpec c ey coe peoa oe eo oac. ep a y epeey oo o-paoy. peaae a coco epe coco ceye: o oc pe xa paep, aop e aac cepe papaoa a.

P.H. yu, H.. Caeoa,.. pouo pee ceo e oee cpeca c (opo, eee opo, ae opa, apoopa ..) acoc o coco aoo x o o oac pe o oac co (a) cooec c a. 1 ( e yppy 9 cpec c, aoe oopx oe axoc 3 cocox).

Taua CPECTBA CB A, PCBAEME X COCTOHM Cpec o c 01 2 opo poce o O M c p c c p opo .

e e He Ooo e A oec opo op He Mec p Apoop He Mec , p oe cc pc Te eee Me e 50% Mec oe H o oo C ooe e P o He Mec oe H o oo e pecc He Mec H o oo e epe He Ec Te eo Me e 50% Mec opoco Me po a, ao oac ye ocaea cooece oc eoaeoc k (o cy paccapaex cpec c ;

ao cyae k = 9) ce. yc i- oac oeae i (qij,..., qk ), oceoaeoc a epy peoa oceoa (q0,..., qk ).

eoc B cooec co caa e, i- j oac o pca co, paaee paccoe o ee o epa peoa, .e. paccoe ey yaa e oceoaeoc. Eo, a eco, oo c pa eoa [3]. ye cooa ca pacpocpa pocmpacmeo eoopoa oe...

e eoo paccoe. Toa i- oac pec co qi, paoe:

k qi = (q - qij )2, j j = e k oeco paccapaex cpec c. peee ooo eoa pacco oe opaao o cyae, oa ce oopa yaex oace oopo o eco y ccy oaoo a pee ocaeo aa.

ye oaa, o o ee eco.

Ta opao, oy eoopoe ooepoe py aoe pocpaco, aaoe epepo coao epee o q, opaae oc ao oac epy peoa o cpoa coa xapaepca. Taoe pya oe pocpaco oe ooep. Tpyoc ec oy oy cy cooc pee cooecyx coooecx aa: ee aopo, eeppyx coay apeoc ooe yay cy (co oyoc co, pocxox peoe), a ye oe a, ec epoco aae.

oaae, o ocpoey ay oo ca epa o, .e. ao, oopo ec occe c ooe oeca opa a ey ca-peyaa epe, a ey cooecy co epaa. B ao cyae caoc oo poo op aaa ocea ax ae. ooy e opae ooc ye ca, o ep (coa) peoa a xoc a ye paec a oc q ce oe, aae a o aeo pyaoo pocpaca. Hy cooecye coa, pacco qs oe (oac peoa), oco o epa a paccoe qs.

a, ye ca, o x = x(q, t), ae pocpace o oopoy y S(t) a pocpaceo eoopoy P.H. yu, H.. Caeoa,.. pouo S = S(q, t) ocpeco epo y S = S(q, t) a ce ee [0, qn] c oo ooa apaa:

(q - q2 ) (q - qn ) (q - q1 ) (q - qn-1 ) SL (q,t) = S0 (t) + + Sn (t).

(q0 - q2 ) (q0 - qn ) (qn - q1 ) (qn - qn-1) B aa oe pee pae apeoc, c xo peyao pocpaceo oopoo oe:

x(q, 0) = const. Ece opao ypae coe o ac ec coa, oopa axoc yp aao o peoa (ccea aya). ooy oo apeoc epe pay peoa pae y. pocpaceo eoopo a oe ocaec ceye cceo ypae:

2 x x = D q2 + S - T x x(q, 0) = const (1) x x (0, t) = (N, t) = q q ae paocy cxey (1). ye cooa ey cxey, a a oa acoo ycoa (o cocoax pee cce (1), o ce o eo paoco cxee po ee acoo ycooc c. [5]).

- xkj-1 xkj+1 - 2xkj + xkj-1 xkj = D + Skj - xkj ;

j = 0,..., n;

k = 0,..., t q2 Tk x0 = x1 ;

xk = xk ;

x0j = const n-1 n (2) k k j = 0,..., n;

k = 0,..., t pocmpacmeo eoopoa oe...

a ccea peaec p eoo poo [5]:

D D > 0 ai = bi = > q 2D 1 1 2D T > 0, > 0 ci = + + = ai + bi q2 T q a1 =1, b1 = eo poep, o yco ycooc poo (2) ce a oe.

peoo, o ee p oac A, B C co ae y q, pa, cooeceo, q1 = 4,236, q2 = 6,472, q3 = 7,292. ye ca, o oac A pooo oe co , aeo x a CH, yaa ca o oac paec S, e S cpe apeoc paccap aeo peoe (apeoc, oyea p cooa oopoo oe);

oac B pooo oe co, e x cpeee e a CH, yaa ca o ac epe paec S;

acepe C pooo oe co, oa ax eooe e a CH, yaa ca o acepe paec S. Ta opao, cpe yaa ca o peoy, cocoey pex oace A, B, C, paec S.

a, o eec p acepa A, B, C, oope oy axoc a pacco q1, q2, q3 o co, oyae ec apao paoo pacooe acepo ooce o co, aoy oopx cooecye co pa apeoc. p o pc. 1.1 2.1 cooecy apay, oa A axoc a pacco q1 o epa, B a pacco q2, C a pacco q3;

pc. 1.2 2.2 cooecy apay, P.H. yu, H.. Caeoa,.. pouo oa A axoc a pacco q1 o epa, B a pacco q3, C a pacco q2;

pc. 1.3 2.3 cooecy apay, oa A axoc a pacco q2 o epa, B a pacco q1, C a pacco q3;

pc. 1.4 2.4 cooecy apay, oa A axoc a pacco q3 o epa, B a pacco q1, C a pacco q2;

pc. 1.5 2.5 cooecy apay, oa A axoc a pacco q2 o epa, B a pacco q3, C a pacco q1;

pc. 1.6 2.6 cooecy apay, oa A axoc a pacco q3 o epa, B a pacco q2, C a pacco q1.

Hao, o aa ccea ee a pee: x = x(q, t) S = S(q, t). oo, o e oooc cooca pee pocpaceo oopoo eoopoo oe e, acpye oe pee. Hapep, ye ca, o t = 5, cpa, ae cee o peoe peoca a yo ye oe eo oo oea pee.

Peee cce pae :

x = x(q).

Hecoo co eoxoo caa o o, a opao paccaac y S = S(q).

a oea e, a y opaae oece a ee peoa cooyoc pocxox peoe co . Moe apa oe o y paccac ce y opao.

Ha ocoe aaa CM cocae cco co, o ope e eco peoe a oce o , cooec c oea cceoaee, oope o oeao o ecoa a CH. C oo cepoo opoca (coo ac eo apx cpae) co pca eca. aee p pacee ae S aoo-o opeoo opeoa coc, ae eo co yoyoo cca poo o opeoe a oce o (aoe pocmpacmeo eoopoa oe...

coe yaoc coo pa, coo oo eo eco) cypoac eca x co. oyea cya cy a aee yae c, oeae paccapaeo y opeoy.

Ha pc. 1.11.6 opaea acoc x = x(q), a pc. 2.1 2.6 acoc S = S(q). py coa, a cex pcyax opoaa oc cooecye eeoy e pacco oace o epa (epeeo q);

epaa e a pc. 1.1 1.6 CH (epeeo x), a a pc. 2.1 yae ce (e peeo S).

Puc. 1.1 Puc. 2. Puc. 1.2 Puc. 2. P.H. yu, H.. Caeoa,.. pouo Puc. 1.3 Puc. 2. Puc. 1.4 Puc. 2. Puc. 1.5 Puc. 2. pocmpacmeo eoopoa oe...

Puc. 1.6 Puc. 2. Aa cex paceo pocpaceo eoopoo oe oaae, o oa ooe oee o ypa aec poecco.

opoaa epa a paax (pc. 1.11.6) cooecye cpeey o peoy ae apeoc, paoy 0,985, o opoe o aeo oea pee t = 5 c oo po cpaceo oopoo oe. Paccop pc. 1.1. Pace o pocpaceo eoopoo oe oaae, o yaa ypoe apeoc (0,985 e) a cao ee xapaepe oo ex oace, oopx q o pae opey [3,3;

q2], o pao 1,45 (paa, aee apeoc p q = 1,45 y ee cpeeo ypo, X = 0,978). A ocax oa cx ypoe apeoc X opao ee eeoo. o oy ec ca, o o ypoe ec pec pe ye ceaoo ypae cyae, oo cea o o o, o ocoe pecypc ypae ceye pe eo e oea, apeoc oopx axoc yp ceea [3,3;

q2] paa 1,45. co, o c oo pacea a ocoe pocpaceo oopoo oe e oy a oo e peoea.

Ha pc. 1.2 aee apeoc a ceee [0,7;

1,6], oop c oo oopoo oe oe o e P.H. yu, H.. Caeoa,.. pouo ooo. pe oea, axoc o ceee, cpeecacecoe ypaee eeo. cya o o cea ceye o: acaoe oeco cpec ypae eecoopao cocpeoo a oeax peoa c apeoc, axoec a ceee [0;

1,6].

Paccopea cya ae epeca e, o a coo ecye pae yae c S (pc. 2.2) ax o a eeo ceee e o. aec a aee S, oa p o ae yae c (cy a aec cao) a cao ee apeoc pao e oa peye ooex pecypco ypae.

Cya a pc. 1.3 oxoa a cya, opaey a pc.

1.1. Too ec cyae, opaeo a pc. 1.1, p q = 1,45 ap eoc X paec pac pecoy ae, o a pc. 1.3 p q = 1,4 apeoc opaea (X < 0 a cee e [0;

2,3]) ec ye e ec apeoc, a copee xapaepye cee caoc. Taoe aee X oyaec a ce opaeoo ae yae c S a o e opee. o o pc. 2.3. Opaey yay cy S copee ceye ca co caa. ocao cya oo cea ceye o: oac, oopx a ee q e opee [0;

2,3], oo paec c paccope, cocpeoo ocoe cpeca ypae a ex oacx, oopx CH oe y, pacpee cpeca ypae o oe oac, poopoao oea oac paepa apeoc.

Pc. 1.4 paec coaae c pc. 1.3 opa pao ea, o a pc. 1.4 pa oe cee pao. Taa e apa aaec ooceo pao yae c: pc. 2.3 pc. 2.4 paec e.

Cya, opaea a pc. 1.5, paec ea cya, opaeo a pc. 1.2 (paa, pa a pc. 1.5 e coo cee ). Bo aao.

pocmpacmeo eoopoa oe...

Caa epeca, a a , cya coac a pc. 1.6, oa paec ec pa axoc e cpeeo (oyc oo) ae CH, oo oe q2 = 6,472 pa epecea e cooecyy opoay py.

p o aa ee yae c, opae a a pc. 2.6, oaae ooo poc co ca ayxae ocepee, o paec ooc coao c ao apeoc, ec y ocee e o peoo caa a ce ee [0;

1,6]. cya oo cea ceye o: oc ooe oeco cpec ypae eecoopao cocpeoo a oacx peoa, axoxc o epa a pacco, pe ae ey q2. p o ocax oeo peoa paec e yo pe cpeca ypae.

Ta opao, aa cex paceo eocppye oe peyeco pocpaceo eoopoo oe. Ee cooae ooe opoa pacpeeee pecyp co ypae.

TEPATPA 1. Bopoua H.. pec aa ecx cocoo epe co ao apeoc // Ayae poe copeeo cooo. M.:

MAC-pecc, 2001.

2. Tocmoa .H. oeyaoe oepoae peeo oac c ceoa p ye coao apeoc // Tpa copee oc cooo. M.: MAC-pecc, 2001.

3. yu P.H., Caeoa H.., epeo M.B. Maeaecoe oe poae opaox ooo coye // Maeaecoe oepoae coax poecco. M.: -o M, 2001. B. 3.

4. Caeoa H.., yu P.H. Maeaecoe oepoae paax copeco cooo // Tpa copeeoc cooo. M.: MAC pecc, 2001.

5. ypo A.M., Mxumap B.C., Tpou .. Mooepe cacece eo. M.: ac caca, 2000.




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