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622.24.05 aea ..

BPOHAPEHHOCT HHOO OOPOBAH P PEH CBAH ucu ocyapcme emo mexuecu yuepcumem poecc ype cope c paoo poa ocoeoc:

eoopooc cpyyp caax cay aco, yeee yeee oax yc ye ooa c aoe, eocaoa cooc oopye oop ooa, pxa ypx py .. yoe oopyoae p o oepaec ec pax o aye acoe oea. p ooe eex c copoe, acx o copoc e ypo oo, ooa, ypoypa, oy paac ae aooea. Bpa, aoecye py c pyo poecce ype, cococy eoy papye opx opo, o e pe apy epeay ep a ao, cya po ooe epeex ape eeax ypo oo x ycaocoy opee. Bocpe yp cpyeo cooo oec, aeo pa, oo oxapaepoa a poapyeoc.

Aa poapyeoc eoxo p oee pao execoo coco yoo oopyoa poecce ype.

ee yoo oopyoa oy e eey paoy xapaepcy [1]. Ocoeoc x eex exaecx cce coco o, o x oea e oaeo o pxo e. O oy oa ycoo oepac cao oeaeo ccee.

ece pa eex cceax po coeopa, aco eoa ea, oope, c oo copo, oy cooa exooeco poecce ype ca, c pyo copo, oy c po eeaex ae aacpoecx cya. o ece pa oeaeo ccee oy cey pee oc oe e cce, cec xapaep ooe paoec (.e. x ycooc eycooc), ec aco ax coox oea ooe ycooo paoec, oy e paoo c p. [2].

p oao paoe poapyeoo yoo oopyoa popeo aae ec oee exaeco copoc ype poxo a ooo p peoxpae ypoo cpyea o pex oec pa, o aco peaec c oo ee poax ycpoc ypo oo. B acoc, p ype epx opx opoax pocxo oco ooa, ae yap peo eo pao, a ycaoa aopaopa oea ypo oo ooe ycpa ye oco ooa coee eo xoocy [3, 4].

pe aece ooo apaepo poapyeoc yoo oopyoa ey ocoa ooa c eo oo oe e cyax oec a paoy opoopapyaeo cpyea.

Paccop ypo oo c aopaopo e ypoeo oe (pcyo 1). Opee aoe pe ocoa ooa p oec cyax oea.

Ha ypo oo ecy ocoa ca a cyaa ca (t), oopa ec cya poecco c ye aeaec oae [5]. paee e ooa:

x = a + (t), (1) e x ca copoe.

Pcyo Tpeyec opee epooc oo, o cyaa y x(t) (ceee ooa) eee opeeeoo pee T ye aee, ec p t = 0 x0 = 0.

coye ypaee ooopoa:

f a f 2 f + - = 0. (2) t x 2 x Bocoyec eoo ype pee ypae (2) f = X (x)T (t). (3) B peyae oyae a ypae 2 dT d X 2a dX 2 + 2T = 0 ; - + X = 0, (4) 2 dt dx dx pee oopx e :

a x T = c1e- t ; X = e (c1 cos1x + c2 sin1x), (5) 2 2 2 ae 1 = -.

2 ooc epooc f (x,t) oa yoeop pae yco:

f (x1,t) = f (x2,t) = 0, ooy pxo y ypae:

c1 cos1x0 + c2 sin 1x0 = 0, c1 cos1x0 - c2 sin1x0 = 0. (6) Ccea (6) ee peee c2 = 0, cos1x0 = 0, oya ceye (2k + 1) (2k + 1) 1x0 =, 1k =. (7) 2 2 x (5) axo (2k + 1)2 2 a2 2 = +. (8) k 2 2 2 x Ooaeo oyae ceyee paee pee ypae (2):

a x 2 (2k + 1) x -2t k f (x,t) = e cos e. (9) ck 2 x k = B aa oe pee (t = t0 = 0 ) oyeoe paee (9) oo yoeop aaoy yco a x 2 (2k +1) x -2t k e cos e = (x), (10) ck 2 x k =oopoe ooe oy coooee x0 x0 - a x (2k + 1)x 2 (2k + 1)x ck cos2 dx = e cos (x)dx. (11) 2x0 2x- x0 - x (11): ck =.

x Bpaee (9) pae a x 1 2 -2t k f (x,t) = e (12) cos (2k2+1) x e.

x0 k =0 x coa epooc pee aeeo ocoa ooa T paa:

x0 a x 1 P(T ) = e dx cos (2k2+1) x e-kT. (13) x0 - x0 k =0 x oce eppoa oy oo oea pee t a 2ch x0 -2T k bk + P(t) = e (-1)2+k, e bk = (2k2x01). (14) x0 k =0 2a2 + bk oyeoe aee epooc P(T ) e ac o aa ocoo c a. Ec aa oe pee x0 ec cyaa ea c eco ooc epooc x f (x0) = cos, (15) 4x0 2xo peee (9) ec, p t = 0 oo oo yoeop yco a x 2 (2k + 1)x x e cos = cos, (16) ck 2x0 4x0 2xk =e x0 - a x 2 kx (k + 1)x ck = e cos x0 + cos x0 dx.

- x Opee oe ck, axo f (x,t), a ae epooc P(T ) o opye (13). p f (x,t) = (x - x0) pe ocoa ooa ye:

a 2ch x(-1)2+k bk T =. (17) x0 k =0 2a2 2 4 + bk k Bocoyec opyo (17), o ye c yeo c ep, aaa pao yoo oopyoa. peca paaec ypo oo (cocpeooey accy ooa aopaopa) e aa c acae a eo co (pcyo 2). Ha ay aecy ccey ecye oco oe a cya oe (t). peepee oeo copoe, oa ypaee pae ca acco m ye:

J = a + (t). (18) (t)+a J, m Pcyo B aa oe pee yoa copoc aa paa y. Bpe T, p oopo yoa copoc oce ae 0, opee o pae (17), oopo ceye x0 ae a 0, a a J :

aJ 2ch (-1)2+k bk T =. (19) 0 k =0 J 2a2 2 4 + bk k C oo opy (19) opee acoc pee ocoa ooa o yoo copoc pae e ac poapyeo ypo oo p ee aooo acc. paceo pe accy ooa 50 , accy ooa ece c aopaopo 300 , copoc pae ypoypa n = 200 800 o/ (ce o opye ocpoee pao oeo c oo popa MathCAD 2000).

B poecce ype a ypo oo oecye epeea aeca apya, o aopo opeee oopo ec epoa oepxoc ao. Xapaepco popo ao (pe) ye ca cpeeapaoe ooee, oopoe eec peeax o = 7c o = 25c. Hapep, paoe [6] pecae xapaepc popoe pyo pao cee epoc: pcyo 3, a x pyo, pcyo 3, epx opo. Macaa coa epooc h opeee oe pee t ec cpeeapaoe ooee xapaepca cyaoo oec ao a ooo.

Pcyo p cpoao cpeeapao ooe p ype ca coaco opye (19) aaec ceya acoc pee ocoa ooa T o aco pae aa aooo ae n, pecaea a pcyax 4 5.

Pcyo 4 acoc pee ocoa ooa o copoc pae aooo ae p = 7c Pcyo 5 acoc pee ocoa ooa o copoc pae aooo ae p = 25c o paa o, o epx opo (pcyo 4) pe ocoa T = 0,0057 c oe, e x opo (pcyo 5) T = 0,0014 c.

acoc pee ocoa ooa T o xapaepc aoo pe (pcyo 6) oaae, o cooae aopaopa ye oee e epx opoax.

Pcyo 6 acoc pee ocoa ooa o cee yxacoc ao cceoae ocoo ooa aece xapaepc poapyeoc yoo oopyoa oaae, o eoo ype pe ocoa oo ee [3]. Aa poapyeoc pax oooo (c aopaopo e) oaa, o yeee pee ocoa ycpaee coe xoocy ooa caoc oo p cooa aopaopa.

ypee c aopaopo paoax pao-oo, e pape ca coe, ocoo, ep opoa, oaao, o exaeca copoc poxo yeac a 3,4 5,1%, poxoa a ooo yeac a 8 9%.

p ca oooo c aopaopo a ecopoex aao Cp, e peoaa e opo, exaeca copoc ocac a 2,5 3%, poxoa a ooo yeac a 7 7,5%.

TEPATPA 1. ey E.. Heee oea eeo ypox a:

eoe ocoe. a: . c. e. -a, 1988. 98 c.

2. exa .. Bpaoa exaa. M.: a, 1994. 400 c.

3. oo B.., oo B.E. ypee c aopaopo Teco oac//PHTC BHOH. Cep. ypee. 1969. 3. C. 3-6.

4. oo H.M., Mao M.P. oea e ac ypoo cpyea p paoe ooa//. yo. He a, 1964. 10. C. 19-5. Ce B.A. Cyae oea exaecx cce. M.:

Maocpoee, 1976. 216 c.

6. Cae A.A. Cepaa eop opeccopa pacopx a. M.: Maocpoee, 1972. 192 c.




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