Бутковский А.Г. Методы управления системами с распределенными параметрами

518

·"'r,..;,

C

-

\.'

.:

..

'··

,,'

......

UHc%>POBOE

MO.llEnHPOBAHHE

•

519

c'

MO.llEnHPOBAflHE

TEn.tlOBoro·

npOc%>HnS!

BAnKOD

t), onpe-

-

(1)

§

2):'

pyeM

(2)

(3)

.'

t;ll.e

PI

~

t,,+ tt,

Pa

= t,,+ t

1

+t

2

,

Ps

=

tit+tl

+t

2

+t

a

,

k=O,

1,

...

,

k·.

nOinopeHHH

yKaaaHHOH:

nocne.noBaTeJIbHOCTH

tIpH

YCJlOBHSlX

«cKneHKiu:

.

,A-TeqJIOnpoBo.nHOCTb

BaJIKa;

n(r.

z.

t)/I=O

=T

o

•

2)

Ha

rpaHH

gd:

Tlm)(r.z.

t)/

m

=T~m+l)(r.z.t).m=I.2.3jk=I.2

•.•.•

k••

1=1,,+

1:

1

m

.

-A

~~

1,=,

=aw[T(r

w

•

z,

t)-T:f],

zE[Lc,

L5

];

(4) .

n=1

w

'"

c,\

TW

(r.

z.

t)

11=1"

=

T~ll

(r.

z.

t).

k

==

1.2

•...•

k*.

3)

Ha

rpaHHX OR,

be,

eLl'

OL

1'

·eg,

de:

rJ{e

tIl.

k=0.1,2

•

...•

k·,-TeKyuiee

BpeMSI

npol.{ecca,

k-HoMep

~~

=

O.

r.ne

f1,-HOpMaJIb K

nOBepxHocTH.

t >

O.

(Q)

o6opoTa BanKa. t

np

--:-3aj{aHHOe

BpeMSI

Mo.neJlHpOBaHHH,

tm-,lVIHTeJib-'

HOCTb

Tennoo6MeHa

B

m-A

30He,

To-Ha'IanbHOe

pacnpe.neneHHe

TeMnepaTypbI B

BanKe,

T~m)

(r,

z,

t)-Ha'IanbHOe

pacnpe.neJIeHHe

np"

npHMeHHM

JIOKaJIbHo-o.nHoMepHbIH

MeTO.n,

.nnH

qer6

.n

HcK

peTH3H-

peweHliH

m-A

3a.na'IH.

3a.naHHyIO

HenpepblBHYIO

o6nacTb'

H3MeHeHHH

apryMeHTOIJ

I1taK,

'rpe6yeTcSI

HaH:TH

4>yHKI.{HIO

T

(M,

t)

= T (r,

z,

.neJIeHHyIO

npH'

t >

0.

BHyTpH

3a.naHHOH:

06JIaCTH

G C

rpaHHl.{eH:

f,

y.noBneTBOpSlIOIIJ.YIO

npH t >°

BHyTpH

G

ypaBHeHHIO

TennonpoBok

"

.CeKl(lIl/

HOCTH

B lI.

HnHH

.npHqeCKHX

Koop.nHHaTax

npH

OCeBOH:

cHMMeTpHH:

OX.I1U

JKiJeHl/R

-.,--r---,

oT (

o2T

1 oT

o2T

)

, .

at = a

az

2

+r

ar

+

ora

,

If

#$

Z

r.ne

a-TeMnepaTyponpoBo.nHOCTb BanKa, C

Ha'IaJIbHbiM

yCJIOBHeM

T (M,

0)

=

To

=const,

M

(r,

z) E G

H

CJIe.nYIOIIJ.HMH

rpaHH'IHbIMH

ycnoBHHMH

Ha

f:

1)

Ha

rpaHH r =

R:

-A7ff

aT

I

'=R =a(m)

(t.

t)(T

(R,

z.

t)-e(m)

(z,

t»;

m =

1,

2. 3, 4,

I.{HKJIH'IeCKHj

a,. z E

[0,

L»J,

m =

1:

aU>

(z,

t)

=

{

E [L L J

at,

z

:I'

a,

,

{'

Tn' i E

[0,

L~J,

eU>

(z,

t)

= T tr. [L

L]

t E

[t

k

,

ttl,

,

co

z

<;;;

:I'

a,

m=2:

a(2)(z,t)=a

t

,

zErO,

LaJ,

tE(.

].

e(2)

(z,

t)

= T

e

,

zE[0,

La],

Pu

P:I

,

a~,

z E

[0,

l

J,

m = 3:

ala)

(z,

t)

=

{

CX~I,

Z E[ItI 1

t

2

];

a~",

z E [l"

La],

T~,

z E [0, It],

e1a)(z.

t)=

T}J.,

zE[/

t

,

1

2

],

tE[Pa,

Pal,

{

]:.1

1

,

zE(l:l,

LaJ,

.

'/h=4:

a(C)(z,

t)=a

1

,

elCl(z.t)=T

e

•

zE[0,L

8

J.

tE[P.,

tHIJ.

_~L!L....l

JIl-l

l,

L2

l?

L

3

.

L

lf

L

5

N

5

0

#3

#If

i

(li';

ZS)

r

w

[+1

c g d

R

5

5+1

e

11

b

N

f

r

PHe.

10.2,1.

(pHe. 10.2.1).

BBe.neM

ceTKy

(B

1/2

06JIaCTH

B

CHJlY

cHMMeTpHH)

\',

ooh={r;E[O.

R],

i=O,I

•.

·

..•

N

I

;

ro=O, rN,=R},

ooj={zsE[O, L

1

],

8=0,1,

...•

N.;

zo=O;

zN.=L

t

},

a

TaKJKe

BpeMeHHyIO

ceTKy

ro~={tkE[O,

t

k

.),

k=O,

1•

...

,k*;

to=O,

tk.=t

np

}·

B

npocTpaHcTBeHHo-BpeMeHHOH:

06JIaCTH

G

{O

~

r

~

R, 0

~

z

~

L,

o

~

t

~

tnp}

nOJIyqHM

ceTKy

OOhf't=rohxro/xro~={(r{,

zs,

t,,).

i=O. 1",

...•

N

I

;

5=0.1,

..

1

••

Nfl.;

.

k=O.I,

•••

,k·}.

__

020

·''

...

,,01

MO,l(EJU1POBA'HHE

tEnJl013of.~/POcI>HJUI

BAJlI(OB

"':'::.i

{fJl.)o,

UHcI>POBOE

MO,l(EJIHPOBAHHE

C

521

I-

3Ha'leHHRM

HCKOMOH

q,yHKUHH

T

(r.

z,

t)

conOCTaBHM

SHa''leHHH

CeTO'lHOH

q,yHKUHH

Y,

onpe.l{eJIeHHoH

Ha

(f)hf't.

060SHa'lHM

'lepes

y~s

SHa'leHHe

B

Y3JIe

(rio

zs, t

k

).

BY.l{eM

HCKaTb

npH6J1H.H<eHHoe

pellIeHHe

ypaBHeHlfR

(1)

Vk+l

npH

t . t

k

+

1

•

nOCJIe.l{OBaTeJIhHO

pellIc'm

O.l{HO-

MepHMe

ypaBHeHHH

TennonpOBO.l{aOCTH

a~6

=

AoV

o

•

'tk~t~tk+h

6=r.

z,

r.he

(6)

I a

aa

A =

__

+~

(J.

r ar

2

'r

ar •

Az=a;r,

~

.

ycJIoBJhlMH

V

k=Vk+l

H

Vi_Vi

z r r

H eCTeCTBeHHhlMH

rpaHH'lHhIMH

YCJlOBHHMH

(3).:;..,!(5).

PellIeHHeM

:nOH

Sa.l{aIJH,

KOTOpylO

yCJIOBHO

sanHllIeM

B BH.l{e A

_

A

,

HBJIHeTCH

Vk+l

= V:+l.

r

z

I\a)K.l{oe

HS

yparmeHHHHoMepa

6

saMeHHM

.i{BYXCJIOHHOH lJeThl-

peXTOIJe'lHOH

',;HeHBHOH

cxeMOH.

.

1I.JIH 6 =

z;

3aMeHHS!-

npoHSBO.l{HMe,

BXO.l{Hm:He B

ypaBHeHHe

(6),

pasHoCTHhIMH

OTHOllIeHHH~fH

av

i+l

k J

"V

J k+x

k+1

--L"",

H,

-y,

_~""'_

V'+l-Y'

.

at

Tk+I'

r ar r h'+1 '

Jlav J

(!J.

k

+.

1 .

"k+

1

.

.k+1

"k+

1

)

v

2

r

1+1

-:Ii

V,

-'#-1

nOJIylJHM

ar

"'"

7If

h'+I

-

h,

'

A

k+x . C

k+

1

+

B

k+l

_

Dk

,Yr,

i-I

-

,Yr,

{ .

,Yr,

+1

- -

.r,

i'

r.l{e

i =

1,

...

, N

I

-I,

(7)

A,,='~~+I

,

h{h,

B

,.

= aTk+l

.+.

aTk+I

-

.

,

C,==A,+B,+r

h'

,h,+Ii'+I,

r,h'+I

hih'+1

,, {

:2

AH8JIOr»lJHO

.l{JIH 6

'z

npH

ali .

ki-I .

ii'

2

..

, , " - k+J

---!.

<"'-'

at

!Is

-!Is

.

~

"'"

-

2-

(!lS+I

-!Is

k+..

k+

l )

!Is

-!ls-

1

Tk+1

az

2

f f +

s

S

1

f

,

. ,

(8)

nOJIyIJHM.

,Lt~!I:tLI-CsY:,+l

+Bsy::J:i+1

=

-D:,

s

==

-

y~,

so

s=

I,

...

,

f.l{e

Ne-l.

'"

A

·=~k_:tt

..

B'=,QTk+1

,

.

:':

,~.,

f.,li

',

..

: s 7l +

'

s

,

I

Cs'=A~+B~+l,

+

1

f

~

"+'

,

.

I.

2 •

!I.acKpeTHSaI:l,HH

rpaHH'lHMX

yCJIOBHH

III

po.l{a

k+1

k+

I

!!-\

"'"

UN.

-!IN.-I

ar ,

=,R

hN.

npHBO)J.HT K Bhlpa.H<eHHIO BH)J.a:

1)

Ha

rpaHH

r=R:

k+\

m m 1 4

Y',N.~XsY"N,-I+Vs,

m=

•

....

;

5=1,2,

....

Ns-I;'

k

=0,

I,

...•

k*,

(9)

r)J.e

amhN.

X~

=

"+a?h

"

N

, •

v~

=

"+a:ahN.

2)

Ha

rpaHH

gd:

k+1

_

k+1

-N

N 1

N'

-0

...

Yr,N.-')(sY"N.-I+V

s

,5-

.'

4+

,...,

.

&.k-

.

,1,

...

,k

,(10)

r)J.e

, awhN.

')(.

s

"

A+awh

N

.'

vS="+awhN.;

,

I

3)

Ha

rpaHHX'

eg, de.

bL

lO

OR,

be, eLl COOTBeTCTBeHHO

)J.JIH

JII06oro

k:

k k

{5=N

s

'-

•••

,

N.,

Y,:N =

Y,.+R-l

N = N2

)J.JIH

cg;

5=N5'

....

, N

e

, {

5=0.1,

...

,N

e

,

{

N=N

)J.JIH

de;

N=I

.l{JlH

OL

I

;·

I

a •

}(1I)

t=O,I,

...

,Nl'

Y:.+1

= Y::N-I

{

N

=1

.l{JIH OR;

I

i=N

2

••••

, N

I

•

.•

{

i=O,

...

, N

a

•

{ N = Ns

)J.JIH

be.

N = N

)J.JIH

eLl'

e

'1

B

HatIaJlhHhlH

MOMeHT

BpeMeHH

t =

0.

HMeeT

MeCTO CJIe,ltYlOmee

COOTHOllIeHHe:

Yfs =

Y~il

=

To

=

canst.

i=O

....

, N

I

;

5=0,

...

, N

e

•

(12)

nJIH

HaXO.H<)J.eHHH

Yr.

Y

z

Ha

OTpeSKe

0:;:;;;

t

~

tn.Mhl

nOJIy'lHJIH

ypaBHeHHH

(7)

H (8) C

KpaeBhlMH

YCJiOBHHMH

(9)-(11),

HalJaJIhHhIM

i

YCJIOBHeM

(12)

H yCJIOBHHMH

D

k

_yk

Dk

_yk+~.

k

-'1

2'

k*

..

r, i -

.'

'z. $ -

r,

- , ,

•••

,

...

I

•

3i'a

paSHOCTHaH

3a)J.i:mi

peilIaeTc5i'

'lpe3BM'laihio

9<pq,eKTifBHhIM

MeTO,ltOM

nporoHKH,

Y'lHThIBaIOUfHM

CneUHaJ1hHhH';j

BH.ll.

MaTpHUbl

CHCTeM

ypaBHeHHH

(7) H (8) =-:-ee.

.rpeX,ltHarOHaJIbHOCTh.

npH

9TOM

II

'I

I.

. §

3J

522

~

-..;

~

~~

l~

~

C'i

CSI

,

~

~400

~

~.

~

s"

t~

~s::;

<l5

li5

!'::l'

'<S

~

<::;

"J-

~

If?

~OO

MO,lJ.EJII1POBAHI1E

TEnJIOBOrO

QJI1JI5I

BAJIKOB

erJI.

10

(1J

f.2J

(JJ

!

'~i

!

~"~\J

32,00

2*,00

fo,oO

8,00

tJ,0()

---.-

8.oel'fR,t:eIr

-

PHC.

10.2.2

m-f,/{-f

m-t.K~2

m-2,K-2

m-'1,K-f

\

m-2,K-f

m-J,K-2

"-~

n:f§2St>1(~1

J@g::

I,

34,00

33,00

32,00'

31,00

JO.OQ

PaOl/fle,l'H

Puc. 10.2.3.

.-:",

MO,lJ.EJIHPOBAHI1E

TEPMOynpyrl1X

nOJIEABAJIKA

lJHCJIO

apHlpMeTHlJeCKHX

;lleikTBHH

HMeeT

nopSl;lloKO«N

l-

I

)(N

a-"J)+

+

(N

6

-N

a

-I)

(Na-l))·

PerneHHe npoBo;llHTCSI. CHaqaJIa B;llOJlb-"

CTOJIOU.OB

s = 1, 2,

•..

, N 6

-1,

a 3aTeM'

B;llOJIb

CTpOK

i

==

1,

2.

...•

N

1-1

;llJlSl

k = 1

•.•••

k*

no

cJle;llYIOIIJ.eMY

aJirOpHTMy:

B

j

Aj~j+Dk

..

~.

_ A _ I

1/+1-

C/-y/A/

'

....

/+1-

C/+v/

A

/ •

A 0 r

k

+1 r A . r

'Vo+XO~,

1

'\'1

=,

....

1

=.

i

='\'/+1

/+1

+

....

/+1'

,=

l-xoV,

•

.

{i=I,

2

•...•

Ni-I.

{ N

1

,

N

a

,

eCJIH

i=l.

/=

.

6=

.

s=I,2,

....

N

6

-I.

N

a

,N

6

,

eCJIH

/=S,

v~,

v

S'

eCJIH

i =

I.

{

x~.

X.

St

eCJIH

i =

j.

v = x =

{

a

0,

eCJIH

j = s. a

1.

eCJIJ:f

j = s.

Be3YCJIOBHaSl

YCTOHlIHBOCTb

JIOKaJIbHO-O;llHoMepHoH cxeMbI 06ec-

nellHBaeTCSI

YCTOHlIMBOCTblO

nporoHollHbIX

¢OpMyJI (npH

9TOM

I

I'\'/+11

~ul.

j = i. s).

;llJISI

lIero

;llOJI>KHbI

ObITb

BbIIIOJIHeHbI

YCJIOB~SI

HX

YCTOHlIHBOCTH:

.

d

.'

'

:l

Aj>O.

B/>O,

C/~A/+B/,

O~xo~I,

j=i.s.

I

JIerKo

BH;lleTb

H3

(7),

(8)

M

(9)-(

11),

liTO

B'

HarneM CJIyllae

9TH

yCJIOBHSI

BWnOJIHeHw.

H3

YCTOHlJHBOCTH

M cyMMapHoH annpOKCHMa-

UHH

ee

CJIe;llyeT. liTO cxeMa

(7).(8)-(9)-(12)

paBHoMepHo

CXO;llMTCSI:

]

maxlyk-Tk

I

=0

(h+-r).

lIl

hST

Ha

pHC.

10.2.2

H

10.2.3

npHBe;lleHbI pe3YJIbTaTbI pactIeTa

no

H3JIO>KeHHOH

cxeMe TeMnepaTypbI

nOBep){HOCTH

ueHTpaJIbHOH lIaCT"

OOlJKH

BaJiKa

;llJISI

yCJIOBHH

lJHCTOBOH

KJIeTKH

CTaHa

ropHlIeH

npo-

KaTKH.

Ha

pHC.

10.2.2

npe;llCTaBJIeH KatiecTBeHHbIH xapaKTep TeMne-

paTypHoro

nOJiSI

T

(r,

0,

t)

Ha nbBepxHocTH BaJIKa (1) H

Ha

rJIyoHHe

1,

3,

7.

IO.M.M

OT

ero

nOBepxHocTH (cooTBeTcTBeHHo

TOllKM

(2). (3).

.(4),

(5)

Ha

rpa¢HKe).

Ha

pMC.

10.2.3

npe;llCTaBJIeHbI KpHBble

pac-

npe;lleJIeHHSI

T

(r,

0,

t)

Ha

;llBYX

ooopoTax

BaJIKa, BbIBe;lleHHweB

MOMeHTbI

BpeMeHH, cooTBeTcTBylOIIJ.He

KOHUY

30H TenJIo06MeHa

(m

-1-MeTaJIJI,

m:..-

2,

4-oKpy>KalOm.HH

B03;llyx, m

=3-oxJIa>K-

;lleHHe. k =

I-nepBbIHoOOPOT,

k

=2~BTOpoii

060pOT).

§ 3.

MO;lleJlHpOBaHHe

TepMoynpyrHx

nOJleH

BaJiKa

B ;llByMepHoMcnyqae

Hcxo;llHaSl

CHCTeMa

ypaBHeHHH

TepMoynpyro~THBblpa>KaeT

HCKOMoe

TepMOHanpSl>KeHHOe

COCTOSlHHe

BaJiKa 0'

(r,

z,

t)(a

rr

• O'zz.O"ea. Ott'

O',a,

0'9z) M W

(r.

z.

t)

==

(W,.W

z.

We).

BbI3BaHHoe

cHMMeTpHlJHbIJiI'

0

y

o

/

5Z4

~.

MO~EJlHPOBAHHE

TEnJlOBOro

npOcJ>HJISJ

BAJlkOB

[rJl.

10

OTHOCHTeJlbHO

OCH

TeMnepaTypHbIM

nbJleM,T

(r,'

Z,'

t)

B

KlnK~bIii

!PHK-

c.upOBaHHbIH

MOMeHT

BpeMeHH,

lJepea

!PYHK~HIO

TepMoynpyroro

nOTea-

nHaJlanepeMeuteHHH

FH

<PYHK~HIO

nepeMeuteI.IHH

JISIB3

(CM.

[272]):

:, ,

[iJ2F

I+Jt

Q3,

a"

J : )

orr(r,

Z,

t)=20

ar2-I-Jt~T+ar2az-J..tazL'1

,

[iJ

2

F I+/l

,as

a ] I

oz;(r.,z,t)=20

aZ2-1-/l~T+azs-(2-J..t)azL'1

,

oeo(r

z

t)=20

[

I

aF

I +

/l

'I

a

2

a

].

l

(1

)

----~T+---J..t-L'1<D

f

. " r

ar

I -

/l

r

ar az

az

'

2

.' [ a

F

as

. .

a,

J'

J

-'

."

°r:z(r,

z,

t)=20

araz+az

2

ar-(l-J..t)a,L'1

°rO=OOz=O"

0 0

,

aF

a

2

'.

Wr(r, z,

t)=ar

+araz'

aF

a

2

,

Wz (r,

z,

t):

az

+

az2

-2(1-J..t)L'1<D,

Wo(r,

z,

i)-O.

'0

••

, ,

0

,

}

0

(2)

npH

3TOM

<PYHK~HSI

F

BHyTpH

~HJlHH~pa

y~OBJreTBopHeT

ypaB-

HeHHlOnyaccoHa

L'1F=

;+~'~T

(r,

z,

t),

(3)

a

if>YHK~HH

<D

BHyTpH

TOH

)Ke

06J1aCTH

y~oBJleTBopHeT

6HrapMo-

HHlJeCKoMY

ypaBHeHHIO

L'1L'1<D

=

O.

(4)

Ha

rpaHH~e

06J1aCTH

(pHC.

10.1.1)

~O.n)KHbI

Bbmo.nHHTbCSI

c.ne-

.nYIO~He

rpaHHlJHble

YC.nOBHSI

aa~alJH:

°rr(R,

z,

t)=orz(R,

z,

t)=O,

}

ozz(r,

±L,

t)=orz(r,

+L,

t)=O.

(5)

3~eCb

a

2

I a a

2

L'1--+-_+_

-

ar

2

r

or

az

2

- onepaTOp JIan.naca,

~

-K03<p<pH~HeHT

.nHijeHHoro

pac'Ur!"peHHH,

J..t-K03<p<pH~HeHT

nyaccoHa.'

,;

3a~alJa

TepMoynpyrocTH

paccMaTpHBaeTCSI

B

KBaaHCTaTHlJeCKo.ii'

nOCTaHOBKe,

npH

KOTOpOH

°

H,

W

aaBHCH'r

01'

BpeMe,HH'

t

KaK

01'

napaMeTpa

lJepea

<PYHK~HIO

T

(r,

z,

t),

0

nOCKO.nbKY

aa~aHHoe

TeMnepaTypHoe

no.ne

T

(r,

z,

t)

CHMMeTpHlJHO

OTHOCHTe.nbHO

n.nOCKOCTH

Z = °

.If

ocecHMMeTpHlJHO,.

6y,lteM,

'Mo.!ie.nH-

ponaTb

1/4'

oceBoro

CelJeHHH

~Jol.nHH,ltpa

(pHC.

10.2.1

).:

)lJIH.n.pOCTOTbI

!!~nO)KeHHe,

KOHelJHOpa3HoCTHoroccnoqj6~

Mo~.n.Hp()l3aHHH,

n0JJei!'

f~;"~

\

('2.

\::~.:.

§

3]

MO~EJlHPOBAHHE

TEPMOV-npy,fHX

nOJlER

BAJlI<Ao

,5250

npOBe,lteM

~ml

CJJYlJaH

paBHOMepHOHCeTKH,

HaHeCeHHOH.

Ha

paC~

CMaTpHBaeMylO

.06.naCTb.

,

;:

nYCTb U=

{O

<.r

<.

R,

0

<.

z'<.L

1

}

~cTyneHlJaTbIHnpHMOYfO.nb-

HHK

BbI,lte.neHHOH

06.naCTH,

a

f-ero'

rpaHH~a.

Ha

oTpeaKe

[0,

R]

nOCTpOl-IM

paBHoMepHylO

ceTKy

wh'={ri=ih,

h>O,

i=O,

...

; N

1

h

a Ra

oTpeaKe

[0, L

1

]-:-paBHoMepHYIO ceTKy

ro~

oj

{z/=fh,

h>

0',

f = 0,

...

, N

z}.

MHO)l{eCTBO

BHyTpeHllHx

ya.noB

060aHalJHM

u

i

!>

a

MHO)l{eCTBO

rpaHHlJHblX

ya.noB

fl!'

.

3a~aHHoe

TeMnepaTypHoe none

Ha

3TOH

ceTKe

B

Ka)K)I;bIH

epHK-

CHpOBaHHbIH

MOMeHT

BpeMeHH

t",

k=O,

...

, N

s

,

HMeeTBH,lt

MaTpH~bI

(T~i)=(T,,(ri'

z/»,

i=O,

.....

"N

1

,

1=0,

...

, N

z

.

nOCTpOHM

paaHOCTHblH

aHaJIOr

onepaTopa JIanJIaca

f!F

=

L'1

1

F

+L'1

s

F,

r~e

iJ2

1

iJ

f!1

=

iJr2+,

iJr'

a

iJ2

f!1

=

iJz"

B

TOlJKe

p = (ri, z

/)

Ka)l{~blH

Ha

onepaTopoB annpoKcHMHpyeM Tpex-

TOlJeQHbIM

onepaTopoM:

L'1

1

F

'"

A

1

F

=alhF~+1

+b}hF~

+ck

F

[_l.

(6)

L'1

2

F

'"

AzF

=

dhF~+l

+

bhF[

+

lhF~-l,

(7)

r~e

'"

eCTb

SHaK

annpoKcHMau,HH.

F~

= F (ri, z

t>

-CeTOQHaH

epyHK-

U,HH,

cooTBeTcTBylOmaH

HCKOMOH

<PYHK~HH

F(r,

z);

1 12rj+h b

1

2r{

2r{-h

a{h

=,

2rjhl'

til = - r.h2'

Cia

=

2r.h

2 ,

I I

1 2

dl=h"

bh=-h

2

' lh=dh·

'Hcno.nbsyH (6) H (7),

saMeHHM

onepaTop JIanJIaca

paaHOCTHb!M

onepaTopoM

AF",

AF=A

1

F

+A,F=

=

athF~+l

+

b1hF~

+

C}h

F

/-1

+

dhF~+1

+

lhF~-l,

"

.(8)

r,ne

b~h

=

b}h

+

bh,

,.I)OTOpblH

onpe:AeneH

Ha

peryJIHpHOM

nHTHTOlJelJHOM

1lI~6noHe

«Kpe~p

us

.

yaJIOB

<..

(r/:&1,

z/), (r{, z/),

,<,/,

%/:u).

~oq)ellII-!-0ST.~

:aJ;Inp.o~~H~~ti~

.....

527

····

.

/

,

:

:,:.:,;

c

526

MO,nEJlHPOBAHHE TEnJlOBOro

npOcJ>HJl5I

BAJlI(OB

[rJl.

10

OnepaTopa

JIanJlaca

paaHOCTHblM Onepl:I.TOpOM (8) COCTaBJl5IeT BeJlH-

'IHHy

0

(hI).

nO,ncTaB.iuISI (8) B .(3),

nOJIy'lHM

paaHoCTHbIH

aHaJlOr

ypaBHeHHSI

TIyaccoHa

,nJISI

BH'YTpeHHHx TOtIeK 06JIaCTH B

BH,ne

CHCTeMbI

(N

1

-1)

(N

2-

1

) JlHHeHHbIX

aJIre6paHtIeCKHX

ypaBHeHHH

(HH,neKck

B

0603HatIeHHH

CeTOtIHOH

<PYHK~HH

F~i

YKaabIBaeT Ha COOTBeTCTBHe

ee

3HatIeHHH

MOMeHTy BpeMeHH til aa,naHHSI

TeMnepaTypHorb

nOJISI):

abaF~.

i+l

+blh

F

'i

+clhF~.i_l

+d~FLtl

+IAF~11

= }

= TLI

(:+~)~.

(9)

p=(ri.

Zi)Euil.

i=I.2

•

•••

,N

1

-1;

1=1,2

•...•

N

-1.

2

llJISI

HanHcaHHSI

paaHocTHQro

aHaJIOra

6HrapMOHHtIeCKOro

ypaB-

HeHHSI (4) npe,ncTaBHM

ero

B BHAe

02

1 0 0

2

)2

AA<D

=

(

or2

+

7"

or+

OZ2

$ =

04

2

i)3

1

02i)4

04

2

lJ3

)

=

(

or

4

+-':-or

8

+/=2

orl

+

az

4

+20z20r2+-':-

OZ2

$.

(10)

0r

llaJIee

aaMeHHM AH<p<pepeHI.{HaJIbHble

onepaTopbI

pa3HocTHbIMH

Ha

peryJISlpHOM

13-TOtIe1JHOM Wa6JIOHe. HCnOJIb3YSl I.{eHTpaJIbHylO

paa-

HOCTHylO

npOH3BoAHylO

AJISI

06eCnetIeHHSI BeJIHtIHHbI norpeWHOCTH:

. annpOKCHM3I.{HH 0

(hI).

npH

3TOM

nOJIytIHM

cJIeAylOlI.{He Bblpa)l{eHHSI

pa3HocTHbIX

onepaTopoB:

.

04lI>

1

~I

4~f

6'

~f

4t'h1

~I)

4

or

''''''

F(Wi+2-

'Vi+!

+

wi-

'V1_l

+""'1_2

•

(11)

i)31I1

1

(~I

2~1

2h-.f .t'hl )

orS

,...,

2h

8

'Vi+2-

""'i+l +

Wi-l-'Vi_I'

(12)

02lI>

1

~I

2t'hf

<D

)

orl

,...,

h2

('VI+

I

-

'VI

+

1-1'

'

(13)

i)4lI>

,...,

J.-

($1.+2-4$1+1

+

6$1

-4$1-1

+

$1-2)

OZ4

h

4

'

Iii.

i •

(14)

04lI>

1 (t'hf+1

2~f

+

~f-I

2t'hf+1 +4t'fif

2~f-l

+

.

azlor

l

"'"

7i4

wi+1

- ""'i+!

.....

i+I-""'i

""'i-

""'i

+

<Df.!1-2<D~_1

+

<D~::D.

(15)

0SiIl

1

(~f+!'

~f+!

2~f

'+2~f

+~f-l

~f-l)

az

2

0r

,...,

'2h

s

.....

I+I-""'i-l-

.....

1+1

""'i-l

wi+l

-""'i-I'

(16)

TIoAc11aBJISISI

(11

)-(16)

B (10).

nOJly'lHM

pa3HocTHblA

aHaJIQr

tlHrapMOHHtIeCKOrO

ypaBHeHHSI

BBHAe

CHCTeMbI

(N

1-1)

(N

2

-1)

13-'IJIeHHblX

aJIre6paHtIeCKHX

ypaBHeHH~

(HHAeKc

k,

RaK'

H paHe.e,

.,

··..

. "

.'

\

..

::::'(

c

§

3]

MO,nEJlI1POBAHI1E TEPMOy.npyrI1X nOJlEfI BAJlI(A

YKa3bIBaeT Ha 3Ha4eHHe

ceT04HoA

<PYHKI.{HH

B MOMeHT BpeMeHH t,.):

mth<D~i

+nth<Dt

1+1

+

pth<Dt

i-I

+

sA<Dtt

l

+

ql

h

<Dki

1

+

+uth<Dft,II+l +

g}h<Dt'-I/-t

+'\'}h<Dk

l

+I +

lp}h<D'1

+2+

2

+

TA<DLt

2

+

"'th<Dt

1+2

+'I1ktD'i = 0

(17)

.AJISI

Bcex

i=I,2,

....

NI-I.

f=l,

....

Nt-I.

rAe

2h2

mth=

20rl+ri

h

-

nth

=

-8,f-3rih+h2

h

4

rl

rlh

4

I _

-8rl+2rih+h2

1_

8

1 _ -

8r

i-

r

i

h

Pih

-

2h

4

•

Sh

- -

h4

•

qih -

2h

4

rl

.

ri

I

2ri+h

I

2ri-h

1

2ri+h

I _

ri+

h

Uih

= ---,-:ji4 ,

Vih

= ---,-:ji4 = gih.

'\'ih

=

~h4

•

lpih

- r.h4 ,

I . I

I I

1

ri-

h

Th='I1h

=

h4'

'¢}h

= ---r{iT .

npH

HanHcaHHH

ypaBHeHHH

(9).

(17)

AJISI

npHKoHTypHblx

TO'leK;

KOTopble

cocraBJISlIOT Y3JIbI. OTCTalOlI.{He

OT

rpaHHI.{bI r

Ha

war

h'

HCnOJIb30BaJIHCb

TaK)l{e

3aKoHTypHble

TOtIKH.

3HatIeHHe

epyHKI.{HH

B

3THJC

TOtIK3X

AJISI

Y3JIOB

PI=(rN.-I'

z/),

f=l,

...•

N

6

-1.

H

Y3J10BPs=(ri.

ZN.-I),

i=l,

...

, N

I

-2,

BblpaaHM

3KCTpanOJISI-

I.{HOHHbIMH epopMyJIaMH

BJ:lJJ:a·

<D

Sk

=

<D'

nk

+

2h

(

~~

),.

•

rJJ:e

<D

Sk

'

<D

nk

-3HaqeHHe

CeTOtIHOH epyHKLI.HH

COOTBeTCTBeHHO

B

aa-

.

(alI»

KOHTypHOH H npeJJ:KoHTypHoH

TOqKaX,

an,.

-npoH3BoJJ:Hafl

no

HOpMaJIH K

KOHTypy

B TOtIKe

Ha

KOHType.

TorJJ:a

JJ:JISI

Y3JIOB

PI

B

Bblpa)l{eHHe

(17)

nOJJ:CT3BHM

<Df+2

=

2<Df+1

-ll>~.

3

JJ:JISJ

Y3J10B

P2

npH-

MeM

<Df+2=2<Df+1-<D~.

.

llJISI

OCTaJIbHbI~

npHKoHTypHblX

Y3JIOB

Ps

=

(r

I.

ZI), f =

I,

....

•..•

N

6

-1,

H

P4=(ri.

ZI).

i=2

•

...

,.NI-I.

no

C006pa:>KeHHSlM

CHMMeTpHH

npHMeM

<Df-I

~

<Df-t

=

ll>f.

,Ii

(.".

B

yp3BHeHHe

(17),

KpoMe

BHyTpeHHHx.

BOWJIH

TaK:>Ke

3HaqeH;H5I

KOHTypHbIX H-eH3BeCTHbIX CeTOtIHOH <PyHKI.{HH

tD.

KOTopble

(~SJ3a~bl

KOHTypHWMH ypaBHeHH5IMH.

COCTaBJIeHHe KOHTypHbIX

ypaBHeHHH.

CBSl3bIBalOUI.HX 3HaqeHHSI

ce-

TOtIHbIX epyHKUHH F H ll> B TOtIKaX

Ha

KOHType.

npoBoJJ:HTCF.

t'orJIac·

I

HO

aMaHHbIM

rpaHHtIHbIM

yCJIOBH5IM (5).

npH

3TOM

Ha

BHyTpeHHHx

528

c."

MO).{

EJlliPOBAHliE

TEnJlOBOro

nt>J.-iliJl51 BAJlI(OB

[r

JI,

JlHHH5JX

KOHTypa

no

YCJlOBH5IM CHMMeTpHH

,rrpHMeM

rpaHHtI!I!>Ie

YCJlO-

BH5I B

BH~e

a(Jzz

_ Q }

az

- ,

OOrz

Ha

JlHHHH

OR

(I

8)

-a:z-

=0

'

a(Jrr

'0'

,}

-

-a;-=

,

"Ha

JlHHHH

OL

l

.,

a

(Jrz

= 0 ' ,

(19)

ar

' '

CTpyKTypa

rpaHijtIHbIX

COOTHOllIeHHH

~Jl5J

Y3JIOB P

-.-

(r,o.

z,),

Pe~(rN"

z,),

f=I,

..

"

Ne-I;

P7=(ri'

zo),

ps=(r[,

ZN.),

5

t=I"

..

1

..

,

,N

1-

,

rrOJIytIeHHa5J rrOCJIe

3aMeHbI

~H4>tPepeHUHaJIbHbIX

orrepa-

TOpOB

'pa3HOCTHbIMH,

rrpe~ctaBJI5JeTC5J

B

BH~eaJIreOpaHtIeCKHX

ypaB-

HeHHH

Ha

I3-TotIetIHOM

llIaOJIOHe

C

rrpaBOH

'tIaCTbIO,

orrpe~eJI5JeMOH'

H3BeCTHbIMH

3HatIeHHiIMH

TeMrrepaTypHoro

rrOJI5J

B COQTBeTCTBYIOlUHX

KOHTypHbIX

TOtIKax.

DpH

3TOM 3HatIeHH5J <l>yHKUHii B

3a'KOHTYPHbIX

TOtI~aX

orrpe~eJISJ:IOTC5J

orrHcaHHbIMH

BbIllIe

3~CTparrOJI5JUHOHHbIMH

<l>opMyJIaMH, a 3HatIeHH5J <l>yHKUHH B

TOtIKaX,

y~aJIeHHbIX

3a

KOHTyp

Ha

2h,

orrpe~eJI5JIOTC,5J

H3 COOTBeTcTByIOlUHX

OHrapMOHHtIeCKHX

ypaB-

H~HHH

B

KOHTypHbIX

TOtIKax.

,

'

TaKHM

oopa30M,

KOHTypHhIe

COOTHOllieHIUI rrOaBOJI5JT rrOJIYtIHTb'

k~

ypaBHeHHH,

CB5JabIBaIOlUHX 3HatieHH5J CeTOtIHbIX <l>yHKUHH F H

<D

Ha

KOHType.

CJJe~OBaTeJIbHO,

~JI5J

rrOJIytIeHH5J HeHaBeCTHbIX HMeeTC5J'

BMeCTe C

k

s

ypaBHeHH5J~H

~JI5J

BHyTpeHHHX

Y3JIOB

Bcero

k = k +k

r

B

ypaBHeHHH,

tITO OOeCrretIHBaeT B03MO)KHOCTb

O~HOSHatIHOrO

orrpe~e

JIeHH5J HCKOMbIX HeH3BeCTHbIX H3 CHCTeMbI k JIHHeHHbIX

aJlreOpaHtIeCKHX

y~aBHeHHH.

OTJIHtIHTeJI6HOH

tIepTOH

rrOJIytIeHHOH

CHCTeM~

5JBJI5JeTC5J

HaJIHtIHe

OOJIblllOro

tIMCJIa

HyJIeBbIX

3JIeMeHTOB

KaK

B M-aTpHue

CH.

CTeMbI, OOyCJIOBJIeHHOe

rrpHMeHeHHeM

5-TottetIHOrO

H

I3-TotIetIHoro

IIiaOJIOHOB,

TaK

H B

rrpaBOH

tIaCTH

ee,

BbI3BaHHoe

O~HOPO~'HOCTbIO

(N

l

-I)(N

2

-I)

ypaBHeHHH

(5).

PellIeHHe

CHCTeMbI rr03BOJIHT

rrOJIytIHTb

HCKOMbre aHatIeHH5J BeJIH-

tIHHbI

TepMOHarrp5J)KeHHH

B KaKOH-JIHOO

TOqKe

KaKHa

KqHType,

TaK

H:

:BHyTpH

3a~aHHoH

OOJIaCTH

no

pa3HoCTHbIM

aHaJIOraM

oPMyJI (1)

npM

rrO~CTaHOBKe

B HHX

3HatIeHHH

CeTOtIHbIX <l>yHKUHH F M

<D

B 3TOH

TOtIKe,

'

I1cKoMble

)Ke

3HatIeHHH

rrOJIH

TepMorrepeMelUeHHH

Ha

rroBepxHoCTH

l.{,HJIHH~pa

W

r

(R,

z,

t)

~JIH

~aHHoro

MOMeHTa

BpeMeHH

t rrOJIytIaIOTCH

npif

9TOM B

BH~e

CeTOtIHOH <l>yHKUHH WtkNI = W,k(rN" z/)

rro

<PoPMyJIe

' ,

WtkNl

-:~

(F'llN,-F~N,)+

+

'

~

(Cl)'+1

-<D

f

-<D'+1. . +

<D"

) f - I 2 N ,

_Q'

','h

2

,

k

N,

'kN,

k,

N,~I

kN,-I,

-,

,

...

, 1

'/;",

';')

'-:>:'

C

""

MOJJ.BJlHPOBAHliE

nJlOCl(li,X

nOJlEA

BAJlI(A

HJ

1I~

YtIHTbrB'iUI;1

tITO

'no

YCJIOBH5JM CHMMeTpUH

W;

(R,

0,

t)

=0,

nOJ;ly-

neM

W~kN,

=

W~kN,

. .ll.JIH

3HatIeHHliW

r B

yaJIax

(rN"

ZN."':l)

H

'(rN'

z}l)

crrpaBe~JIHBbl

opMyJIbI

~".'

I

; , W

N

.-

1

2W

N

.-.,

W

N

•.

,...

W

N

, 3W

N

.-.

2W

N

.-

1

~'

rll

N,=

rlt.

N.-

rlt.N.,

rkNl-

rlt.

N.-

rlt.

N.·

§ 4.

MO~.JIHpOBaHHe

BJIOCKHX

TepMoynpyrHx

Bond

BaJIKa

,

npoBe~eM

orrpeAeneHHe

repMoynpyrHx

nOJIeA,

npHMeHHH

rHno-

~~

nJIOCKOH

S8A8tIH

TepMoynpyrocT.H.

STa

3a~atIa

~JIH

cnytIaH

,MJlHHOrO

UHJIHHApHtIeCKOrO

TeJIS HOCHT

H83BaHHe

3aaallu 0

nAoClCOa

let/JopMal{UU

H

~onycKaeT,

8HatIHTeJI'bHoe

ynporueHHe

MareMaTHtIeCKo.A

eropOHbI

pellIeHHH.,

'

YnporueHHe

SaKJIlOtIaeTCH'

B TOM, tITO B 9TOM

cnytIae

Koop~HHaT·

HyJO OCb Z MO)KHO

OT6poCHTb

'H

Bce

HBJIeHHe

paCCMaTpHBaTb

KaK

6b1

npOHeXO~HruHM

B

O~HOH

nJIOCKOCTH

(r,

e).

OCHOBaHHeM

~JIH

npHMeHeHHH

9Toro

cnytI8H

KH8weA

3a~-a'le

MO~eT

cnY)KHTb

HeOOJIb-

UlaH

BemitIHH8

TeMrrepaTypHoro

nepen8.l1.8

B~OJIb

OCH

Z

MH

paOOtIHX

BaJIKOB B

rrpouecce

npOK8TKH

(rropH-.ztKa

1O-30°C

Me)K~y

cepe~HHOH'

H

KpaeM

OOtIKH) OTHOCHTeJIbHO AJIHHbI

60tIKH

paOOtIero

BaJIKa

H;

UIHpHHbI

rrOJIOCbI.

3to

rrOaBOJIHeT

paCCMaTpHBaTb

TOHKHe

~HCKH

BaJIKa

KaK

rrJIOC:\<OCTH

H

HCrrOJIb30BaTb

MaTeM8THtIeCKHH

arrnapaT

aa~atIH,

o rrJIOCKOH

~e<l>opMauHH

[272,

346].'-

.

KBa3HCTaTHQeCKaH

rrOCTaHOBKa

3a~atIHTepMoynpyrocTH

B 3TOM

cnytIae

~JIH

ocecHMMeTpHtIHOrO

TeMrrepaTypHoro

rrOJIH T

(r,

t)

B

pa~H

',aJIbHOM CetIeHHH rrOJIOfO

II.HJIHH~pa:

(c;,

BHY'FpeHHHM

pa~HycoM

r

0)

'rqmBo~HT

K

CJIe~YIOrueM(Y

pellIeHHIO

,nJIH

Ko~rrOHe~T

TepMoyrrpyrHx,

nOJIe"

(CM. [272]):

r

PE

0rr(r,

t

r

=1

[

('-r:

2

)SR

T

(r,'.f)fdr-S

T(r,t)rdr],

(1)

/l.

r

l

RI-

ro

"

'0

, R r .

pE

rl+r:

.'

1 '

oee

(r,

t)

=-1-

(

,)ST

(r,

t)

rdr+

2

ST

(r,

t)rdr-T(r,

t)]

,

/l.

[ r

l

R2_

ro

r,

.'

r,

'.

(2)

C1

zz

(r,

t)=IPE

[ 2 2 SRT(r,

t);dr~T'(r,

t)].

(3).

/l.

R2_

ro

'

"

r.,

'

..

',

.

R

4PR

Wr(R,

t)

=i'~

ST

(r,

t)

rdr,

'V1

...

R2_

ro

t t

(4)'

r.

qe

P-K03<1><1>HII.HeHT

JIHHeAHOro

paCIlIHpeltHH.

IJ.-Ko9<1x1>ifIlHeHT

nyaccoHa,

a",

O'llll,

'O',u~cooTBeTcTBeHHo,

pa,nI1aJIbHOe,

TaHreHII.H-

c

"

····

.......

530

MO.L\EJIHPOBAHHE

TEnJIOBOro

npOI1JI~

BAJIKOB

[rJI.

10:'

cMhHoe

H OCeBOe

HOpMaJIbHble

Hanp5DKeHH5l

B

BaJIKe,

E

-MO.L\YJIb

np0.L\OJIbHOH

ynpyroCTH

MaTepHaJIa

BaJIKa.

.

TeMnepaTYPHblM

nOJIeM T

(r,

t)

.L\JI5l

paCqeTHOro

pa.L\HaJlhHoro

CeqeHH5l

Zs

=

const

H MOMeHTa

BpeMeHH

t,.

CJIy}f{HT

BeKTOp-CTOJi6eu

np5lMoyroJibHOH

MaTpHUbl

ut

s

'

nOJIyqeliHOH

H3 .L\ByMepHoH Mo.z(eJIH

TeMnepaTypHoro

noJi5l

BaJIKa.

TaKHM

06pa30M,

BXO.L\HhlMH .L\aHHhlMH

aJIrOpllTMa

p~CqeTa

no

npHBe.L\eHHhlM <PoPMyJIaM 5lBJl5lIOTC5l

Ha60p

MaTpHU

M

'

i

=z

0, ,

Nt,

s

s=O,

...

, N

a

,

B nOCJIe.L\OBaTeJIhHhle MOMeHThl

BpeMeHH

k=l,

2,

,k',

HepaBHoMepHhle

ceTKH

no

pa.L\HYCY

OOh

=

fr

j

E

[0,

R], i =

0,

, N

l'

~~

r

N

,

= R} H .L\JIHHe

60qKH.

BaJIKH

00

,

= fZs E[0,

La],

s

=J,

, N

,

i

t

N

=

haL

umpHHa

nOJIocbI

2L

2

,

epH3HqeCKHe nOCT05lHHhle

MaTepHaJIhf

BaRKa

/l-,

~,

E. .

Tor.L\a

aJIrOpHTM

MO.L\eJIHpOBaHH5l KOMnOHeHT

TepMoynpyrHx

tIOJIeH

BaJIKa

B

npOCTeHlIleM

CJIyqae,

Kor.L\a

.L\JI5l

annpOKCHMaUHH

onpe.L\e-

JIeHHhlX

HHTerpaJIOB,

BXO.L\siLUHX

B (1)-':(4), HCnOJIh3YIOTC5l

epoP~YJIl>l

rrp5lMoyrOJIbHHKOB,

npe.L\CTaBHTC5l

rpynnoH

KOMnaKTHhlX

aJIre6paH-

qeCKHx

Bblpa}f{eHHH

CJIe.L\YIOLUeH

CTpyKTyphl:

N,-I

W,(R,

zS'

t")'=I'I~dk,

S)=1'1

~

[!Asrj(rj+l-rj)],

(=0

f

(/,.)

= W,

(R,

zS,

t,.)

/zs=o-W,

(R

1

,

zS'

t,.)

/Zs=L

'

g

u,,(rj,

zs'

f")=1'2(r,)~dk,

f)+l'a(rj)~2(i,

s,

k)=

.

N,-f

=I'~

(r,)

~l

(k, f)

+I'a

(ri)

~

(Y~sr,

(r'+I-

r

i)],

(=0

i=0,1,

...

,N

1

; s==0,1,

...

,N

, .

3

r.L\e

~E

1'3

(r,)=-

,f-,~

~E

1'1

=1-p.'

1'2

(r

i) = - 2

1'3

(rt>.

1-,: '

(1-p.)r(

.1laJiee,

HMeeM

(Jee

(r"

Z;s,

f")=1'5(ri)~dk,

S)+1'6

(rt>~2(i,

s,

k)+1'10

,

(.L\e

s

~E'

. ( ) 2 •

1'6

()

r,

=..

-p.

)

,(

I'

1'1

=

-1'6

r

s

i

_

r

(,

1'5

=

-I'dritri+r~

i=O,I,

...

,N

1

;

-0,1,

...

,N

,

l-r~

H,

HaKOHeu,

3

ozz(r"

ZI'

f")=l's~dk,

s)+I'.M

s

,

r.L\e

I's

=

-I'g,

1'.

=

~I's,

i

=0,

.,

" N

1

;

s

=0,

...

, N •

~

l-:-ro

.

a

B

KaqeCTBe

KOHTpOJIbHOrO

COOTHOlIleHH5l AJI5l TeKYLUeH

npOBepKH

npaBHJIhHOCTH

BhlqHCJIeHHH

ueJIeC006pa3HO

npHH5lTb

paBeHcTBO

(J" (r

i

,

zs,

fir)

+

(Jee

(r

i

,

zs,

fir)

= (Ju

(r

i

,

zs' f

).

k

.

'\

cr·,

r

/1

A B A

11

PAcnPE)J.EJIEHHbIE nAPAMETPbl 8

3KOHOMJ1l-lECKHX

MO)J.EJIflX

§

1.

Mo.n.eJIbMHoronpo.n.yKToBoro

npoH3Bo.n.cTBa

OllH<CaHHe COCT05lHHH

06beKToB

C nOMOIUblO <!JYHKUHH pacnpe.L\e.-

JIeHH5l B03MO}f{HO

He

TOJIbKO

.L\JI5l

epH3HqeCKHX BeJIHqHH

(TaKHX,

KaK.

CKa}f{eM,

TeMnepaTypa,

KOHueHTpaUH5l

H T• .n..),

HO

.L\JISI

BeJIHqHH

.L\pyroH,

TaK

CKa3aTb,

HeqJH3HqeCKOH

npHpOAbl,

BCTpeqalOLUHXCSI B

3KOHOMHqeCKHX, 6HOJIOr'HqeCKHX H

.L\pyrHx

npou.eccax

no.n.06Horb

po.n.a.

'BC5lKOe

onHcaHHe

C· nOMolu,bIO

HenpepbtBHblX

MO.L\eJIeH

eCTb,

H.L\eaJIH3au,H5l

(npe.L\eJIbHblH

CJIyqaH),

KOTOpa5l,

O.L\HaKO, XOpOlUO

(T.

e.

t

60JIblIlOH,

a HHOr.L\a H C

nO.L\aBJISI,IOLUeH

CTeneHhlO

TOqHOCTl:()i

annpOKcHMHpyeT

peaJIhHOe

pacnpe.L\eJIeHHe.

TaK,

HanpHMep,

eCJIl:(

paCCMaTpHBaTh

npouecc

.L\H<!J<!JY3HH

Bew.eCTB,

TO

pacnpe.L\eJIeHHe

KOH-

ueHTpaUHH,

CTporo

rOBOp5l.,

He

HenpepblBHO,

TaK

KaK

BeLUeCTBO

COCTOHT

H3 OT.L\eJIhHhlX

qaCTHU'

(aTOMOB,

MOJIeKyJI),

KOTopble

3aHH-

MaIOT

KOHeqHhlH

06beM

npOCTpaHCTBa,

H Me}f{.L\y HHMH HMeIOTC5l

«UyCTOThl». O.L\HaKO

OnbIT

nOKa3bIBaeT,

qTO

TaKoe

.L\HCKpeTHoe

pac-

npe.L\eJIeHHe C

orpOMHOH

CTeneHblO

TOqHOCTH,

,BnOJIHe

y.L\OBJIeTBOpH-

teJIbHOH

.L\JISI

OqeHh

MHorHX

npaKTHqeCKHX

H

TeOpeTHqeCKHX

3a.L\aQ,

MO}J<eT

6bITb

annpOKCHMHpOBaHO

HenpephtBHblMH

pacnpe.L\eJIeHH5lMl:(,

.L\JISI

KOTOpblX

MO}f{HO

nHcaTh

npe.L\eJIhHhle

ypaBHeHH5l

B

.L\H<!J<!JepeH-

UHaJIaX,

npOH3BO.L\HblX,

HHTerpaJIax.

C aHHJIOrHqHOH

cHTyaUHeH

iIo.L\qac

MO}f{HO CTOJIKHYThC5l H B .L\PY-

rHX

npHJIO}f{eHHSIX,

HanpFiMep

npH

nOCTpoeHHH

MO.L\eJIeH

3KOHOMH-

qeCKHX

H

6HOJIOfHqeCKHX

npou,eCCOB H

06beKTOB.

tIaCTo

HMee-T CMblCJI

OT .

.L\HCKpeTHblxnepeMeHHbtX

nepeHTH

K

HenpepbtBHhlM.

HanpHMep,

eCJIH B 3KOHOMHqeCKOH

MO.L\eJIH

paCCMaTpHB.aeTCSI CHCTeMa C MHO>Ke-

CTBOM npOH3BO.L\HTeJIeH HJIH

nOTpe6HTeJIeH,

TO

HX

OKa3btBaeTC5l

Ra·

CTOJIbKO

MHoro,

qTO y.L\06Hee

nepeHTH

K

onHcaHHIO

HX

pacnpe.L\€JIe-

HHH C nOM0I.IJ.blO <!JYHKUHH

pacnpe.L\eJIeHHH

OT

HerrpephlBHblX

iIepe-

MeHHblX.

AHaJIOfHqHaSI

cHTyaUH5I

MO}f{eT BCTpeTHTbC5l

npH

onHcaHHH

pac-

npe.L\eJIeHHH

nonyJI5lUHH

>KHBbIX

opraHH3MOB

pa3Horo

ypOBHW

npH

paCCMoTpeHHH

6HOJIOrHqeCKHX

CHcTeM.

B

3ToH

r,1IaBe

Ha

npOCTOM

npHMepe

onHCaHH5l

paBHoBeCHprx

COCT05lHHH B 3KOHOMHqeCKOH

MQ.L\eJIH

npOCToro

TOBapHoro

npOH3-

,:,.

1

I

.-

·1

i

j

1

SI<()HOMHqECKHE MOlJ.EC:>

[f

n.

II

632

B0Jl.CTBa paCCMaTpHBaeTCH

npHMeHeHHe

HenpepbIBHbIX<PYHKl.\HA

pac-

.

npeJl.eJJeHHH.

3J1.eCb

6YJl.eT

rrOKa3aHo,

kaK

MHorHe

3KOHOMHlJ:eCKHe

nOHHTHH

MoryT

6bITb

onHcaHbl

B

TepMHHax

HenpepblBHbIX

<PYHKl.\HA

pacnpeJl.eJIeHHSI

[222].

I1TaK,

HalJ:HeM C

onHcaHHH

.

MOJl.enH

MHOrOnpOJl.yKToBoro

npOHS-

BOJl.CTBa.

nYCTb

HMeeTCH 1

«oTpacneA»

npoHsBoJl.cTBa,

npHlJ:eM Ka>KJl.aH

X-H

oTpacnb

npOH3BOJl.CTBa npOH3BOJl.HT

OJl.HH

BHJI.

npOJl.YKTa

(ToBapa).

&rOT

npOJl.YKT TaK>Ke 6YJl.eM

0603HalJ:aTb

lJ:epe3 x. BYJl.eM ClJ:HTaTb,

lJ:TO

x

HenpepbIBHO

MeHHeTCH B

npeJl.enax

OT

HynSI

Jl.O

I,

T.

e.

0

~

x

~

1.

nYCTb;

.Jl.aJlee, B

x-A

OTpaCJlH

pa60TaeT

Y (x) npOH3BOJl.HTeJleA,

Ka>KJl.bIA

H3

KOTOpblX

npOH3BOJl.liT 3TOT

x-A

npoJl.YKT.

Ka>KJl.bIH

npo-

H3BOJl.HTeJIb 6YJl.eT 0603HalJ:aTbCH

«HoMepoM»

'y, KOTOpbIA TaK>Ke

HB-

JIHeTCH BeJIlilJ:HHOA,

HenpepblBHo

MeHHIOrneAcH

B

npeJl.eJIax

O~y~Y(x),

t t

'~.v

qJlJKCtl/JodOH6I

y

DOlle

npofiJdOQUmeRet!

~

•

J

"",,,

.

'"""'.

!

"II

' I , , , , J , I J

~mpacllu

o

v

PHC.

11.1.1. Puc. 11.1.

2.

T.

e.

npaBblA

npeJl.eJl Vex) H3MeHeHHH «HoMepOB» y npOH3BOJl.HTeJleA

Ka>KJl.oA Jl.aHHOH

x-A

CYfpaCJlH 3aBHCHT

OT

«HoMepa»

X 3TOH OTpaCJlH.

BBeJl.eHHble 0603HalJ:eHHH HJlJlIOCTPHPYIOTCH rpa<plilJ:eCKH

lia

pHC.

11.1.1.

06nacTb

G B nJlOCKOCTH

(x,

y)

MO>KHO

Ha3BaTb

«nOJleM

npo-

J13BoJl.HTeJleH»•

nYCTb

y-A

npOH3BOJl.HTeJlb

x-A

OTpaCJlH npOH3B0Jl.IiT q

(x,

y)

x-ro

rtpo.u:YKTa.

nYCTb

H3J1.ep>KKH

(HJlH

TPYJl.OBble

3aTpaTbI,

BbIlJ:IiCJIHeMble

J1J1H

B

Jl.eHbraX,

HJlH', HenOCpeJl.CTBeHHO B pa60lJ:eM

BpeMeHH)

y-ro

npOH3BOJl.HTeJlH B x-A

OTpaCJlIiXapai<TepH3YIOTCH

<PYHkl.\HeA t (q, x,

y).

3Ta

<PYHKUHH

onpeJl.eJleHa

B 06J1aCTH q

~

0, 0

~

x

~

I,

0

~

y

~

Y

(x).

31'y

<PYHKUHIO Mbl 6y.z(eM

Ha3blBaTb.

Uflou8uoyaAbflou npou38oocm8efl-

flOll

<PYHKUHeA.

TaKHM

06p'a30M,

eCJlH

y-A

npOH3BoJl.HTeJlb

x-A

OT-

paCJlH

npoH3BeJl.eT

q

(x,

y)

npoMKTa,

TO

ero

H3,nep1KKIi 6YJl.YT

paBHbl'

t

(x,

y)

= t (q

(x,

y),

x,

y).

THnHlJ:HbIA

rpa<pHK

<PYHKUHH t

(q,

x, y)

KaK

<PYHKl.\HH

OT

q

npH

4>HKCHpOBaHHb~X

x H y

H306pa>KeH

Ha

pHC.

11.1.2.

3Ta

<PYHKl.\HB

xapaKTepH3yeTcH

CJIeJl.YlOrnHMH CiCHOBH'blMH

cBoAcTBaMH:

1)

t(q,

x,

y)~O

npH

q~O,

O~x~l,

O:::;;y:::;;Y(x).

EYro YC-

JlOBHe

cOBeplIIeHHo

HCHO, eCJlH ylJ:eCTb, tiTO

npH

npOH3B0Jl.CTBe

MLI

BCerJl.a

npoH3BoAHM

saTpaTLI'pecypcoB.

I

533

CI~EJlb

MHoronpO,D.VI<TOBOrO npoH3BO,D.CTBA

\

S

I]

,2)::t

(q,tx,

y)~O

npH

q~O,

'O~x<l,

O~y<Y(x).·

3'1'0

1

q,

cJle,zJ.yeTH3

Toro,

lJ:TO

Ka1KJI.~HOBoe

yBeJl,HqeHHe

BblnycKa

npO,D.YK-

l.\HH

f!.q

Tpe6yeT

YBeJlHlJ:eHHSl H3,D.ep1KeK

M.

1

3)

1.IacTo

<PYHKUHSl t (q, x, y) o6JIa.n.aeT

erne

O,D.HHM

cBoikTBOMl

"(jUt

I

&i

(q,

x, y)

~O.

q.

3To

yCJIOBHe

a,D.eK~aTHo

TOMy

4>aKTy,

lJ:TO

C YBeJIHlJ:eHHeM

06'beMa

npOH3BO)1.CTBa q npHxoJl.HTCSl

HcnOJlb30BaTb

Bce

60Jlee

xy,D.lIIHe

pe-

~ypCbl,

Tpe6ylOrnHe

,ll,JlSl

CBoero

HC-

:c,!!

f/lliKcupodaHIJI

.

IfOJlb30BaHHSl

Bce

60.nblIIHX

H3)1.ep- t

.'

1KeK.

CHalJ:aJlaIiCnOJIb3yIDTCSlJlYlJ:-

trme

pecYPcbl'(JlYlJ:lIIHe'

3eMJlH,

JIylJ:lIIHe PYJl.bI H

T.'

JI..), a

3aTeM

60nee XYJl.lIIHe (6eJl.Hble

3eMJlH,

6eJl.-

HbIe py)1.bl H

T.

)1..), qTO, eCTeCTBeH-

HO, 06XOJl.I1TCSl )1.0p01Ke.

OTHOCHTeJlbHO

3Toro

nOCJleJl.He-

\

ro

yCJIOBMSl Hy1KHO CJl.eJlaTb

3aMe-

0

't'

lJ:aHHe. J].eJlo B TOM,

liTO,

nO-8M-

.

PHC.

11,1.3.

Jl.IiMOMy,

MoryT

6blTb

MCKJIlOlIeHHSI

H3

9Toro

npaBIiJla,

CBSI3aHHble,

Ha-

npHMep,

C MaCCOBOCTblO npOH3BO)1.CTBa, Kor,D.a

npH

KaKOM-TO

06'beMe

npOH3BOJl.CTBa q

nepBaSI

npoH3BoJl.HaSl ot/oq HalJ:HHaeT

YMeHblIIaTbcSl

(T.

e.

i:J

2

t/i:Jq'

<

0)

(pHC.

11.1.3). . .

CneAylOlu

HM

Ba1KHbIM nOHSlTHeM, CBSl3aHHbiM C 06'beMOM

npoH3-

BO,D.cTBa

q H iIP01i3BOJl.CTBeHHoH <pyHKUHeH. t (q, x,

y),

SlBJlSIeTCSl

nOHSlTHe cpe,D.HeA HH,D.HBH,D.yaJIbHoH CTOIiMOCTH w (q, x, y):

w(q;

x,

y)=t(q,

X,

y).

(1)

q

TaKHM

06pa30M,

eCJlH yAH npOH3BOJl.HTeJlb

x-ro

npo,ny'KTa

npo-

H3BeJl q

(x,

y)

npO)1.yKTa,

TO

cpeJl.HSlSl CTOHMOCTb w

(x,y)

3Toro

KOJlHlIeC1'Ba M01KeT

6bITb

BbllJ:HCJleHa

no

4>opMYJIe (1):

w(x,

y)=w(q(x,y),

x,.y)=t(q(~,(%.\~,

y).

(2)

OlJ:eBH)l.HO,

lJ:TO

eCJIH H3BeCTHa CpeJl.HSlSl CTOHMOCTb w(q, x, y) H

KOJlHlJ:ecTBO npOJl.YKTa

q,

TO

H3

<pOpMyJlbI (1) nOJlYlJ:aeM H3,1(ep1KKH,

HAyrn

He

Ha

npoH3BoAcTBO

3Toro

KOJlHlJ:eCTBa

npoJl.yKTa:

t (q, x,

y)

=w

(4,

x,

y)

q,(3)

H,

aHaJlOrHlJ:HO,

, .

(4)

x,

y)=w(x,

y).q(x,

y).

t (x, y)

=!

t

(q

(X,

y),

'-

-I

534

3KOHOMH'lECKHE

MO.n{J:

[rJI.

II

.

THlIHllHbIH

rpa<pHK

W (q, x, y)

K8K

<PYijKl.I,HH

O:rq

lIpH

<pHKCH-

pOBaHHbIX

x, y

6y.l(eT

HMeTb

BH.l(,

H306piDKeHHbIH

Ha

pHC.

1

I.

1.4.

<tlyHKQHH

w (q, x, y)

o6JIa.l(aeT

CJIe.l(yIOlQHMH

CBoHcTBaMH,

Heno-

Cpe.l(CTBeHHo

BblTeKaIOlQHMH

H3

cBoHcTB

<PYHKQHH t (q, x, y);

1)

w(q,

x,

y)"~o

npH

q)30,

O~x~/,

O~y~Y(x);

2) lIaCTO

w(q,

x, y)

06JIa.l(aeT

cBOHGTBOM:

Ow

.

aq

(q, x, y)

~

0;

a

2

w

3)

a

q

2

~

0.

nOCJIe.l(Hee yCJIOBHe TeCHO

CBH3aHO

C yCJIOBHeM 3) )l;JIH

<pyHKUHH

f.

.

ECJIH

t

HMeeT

rpa<p

HK

,

H306pa1KeHHbIH

Ha

pHC.

11.1.3,

TO

COOTBeT-

CTByIOlQHH

rpa<p

HK

w6y.l(eT

HMeTb

BH.l(,

H306pa1KeHHblH

Ha

pHC.

1

I.

1.5.

<1>yHKUHHW

6y.l(eT

HMeTb

B 3TOM

CJIylIae

MHHHMyM.

lO

z,!/

!/JlI/(CUpO!tlHol

l/J

Il~'

---------'-_~

0

1

q

;

PHC.

I

I.I.4.

'

PHC.

11.1.5.

<1>YHKQHIO

w(q,

X,

y)

KaK

<PYHKUHIO

OT

q--M01KHO

fIa3BaTb

¢YH,K-

[{ueu

UHou8uoya/lbHOZO

npeO/lO:JICeHUJl

y;fO

npOH3BO.l(HTeJIH

X-H

OT-

PCl:CJIH.

3aMeTHM,

lITO Cpe.l(HHH CTOHMOCTb W

(q,

X,

y) .l(eHCTBHTeJIbHO

j!ijJIHeTCH

Cpe.l(HeH

(apH<pMeTHlIeCKOH)

CTOHMOCTbIO

OT

.l(H<p<pepeHUH-

aJIbHOH

HJIH JI<?KaJIbHOH CTOHMOCTH V (q,

X,

y),

KOTOpyIO

M01KHO

OlIpe.l(eJIHTb

TaK:

v(q,

X,

y)=at(q'a

X

'

y)

q

(5)

B CaMOM .l(eJIe, eCTeCTBeHHO B 3TOM

CJIylIae

Cpe.l(HIOIO CTOHMOCTp

w(q,

X,

y)

Onpe.l(eJIHTb

KaK

Cpe.l(Hee

(apH<pMeTHlIeCKoe)

.l(H<p<pepeH-

l{HBJIbHOH

CTOHMOCTH V (q,

X,

y)

Ha

HHTepBaJIe

(0, q):

q q

X,

y)=q

IS'

I

Jat

aq(q,

X,

y)dq.

(6)

W(q,

v(q,

X,

y)dq=q

'.

0

:.-.::~:'

~

:.

...

2:1

..

"'"

......

',

'.,::'.'.'.

MOAEJIb

MHOrOTIPd,nYKTOBOro

n?0H3Bo.nCTBA

C

535

I1HTerpaJi

B (6) BblqHCJIHeTCH JIetKO. B

pe3YJIbTaTe

nOJlYllHM

t

(q.

X

•.

g)

w(q,

x,y)==

~

S

dt(q,

X;

y)=t(q,

x,

y).

o

nl

no

C1'BO,

OT

3T0

COBIIa.n.aeT C

oIIpe.n.eJIeHHeM,

.n.aHHblM <pOpMyJlOH (1).

nOJlb3YHCb

OIIpe.n.eJIeHHbIMH

BbIllie

IIOHHTHHMH

H

BeJIHlIHHaMH,

ijBe.n.eM B

paCCMOTpeHHe

HOBbIe IIOHHTHSI,

xapaKTepH3YIOIUHe

06w.e-

OTpaCJIeBOe npoH3Bo.n.CTBO Ka1K.n.oH

x-ii

OTpaCJIH

npH

13cex x E

[0,

I].

npOH3BO.n.trBO

x-HOTpaCJIH

paBHO

cCyMMe» 06'beMOB npOH3Bo.n.CTB

BceM

IIpOH3Bo.n.HTeJIHM,

T.

e.

Y(x)

q(x)=

~

q(x,

y)dg,

O~x~l.

CO

o

06IUeOTpaCJIeBble

H3.n.ep1KKH t (x) OTpaCJlH x

Ha

npOH3Bo.n.CTBO

KonHlIeCTBa

npo.nyKTa

q(x)

COOTBeTCTBeHHO

6y.l(yT

paBHbI

Y(x)

~

t (x) =

~

t

(q

(x, g),

X,

g)dy

=

~

t (x, y) dg, °

~

X

~

l. (8)

o 0

Ha

OCHOBaHHH

(7)

H (8) M01KHO

onpe.n.eJIHTb

cpe.n.HeoTpacmmyIO

CTOHMOCTb B

x-ii

OTpaCJIH:

•

t (x)

w (x) = q

(x)'

°

~

X

~

l. (9)

Ba>KHO OTMeTHTb,

'lTO

onpe.l(eJlellHbIe

<popMyJIaMH(7),

(8) H (9)

06tu.eoTpaCJIeBble

BeJlHlIHHbI

HBJIHIOTCH

y>Ke

He

<PYHKQHHMH, a <PYHK-

UHOHaJI8MH

OT

<pyHKUHH

q (x, g)

KaK

<pyHKUHH

OT

g

npH

<pHKCH-

POB8HHOM

X.

4T06bI

no.n.lIepKHyTb

3TO

CyIII,eCTBeHHoe

06CTOHTeJlb-'

<PYHKUHIO q (x,

y)

MbI

6y.n.eM

eme

0603HalIaTb

KaK

q"

(y),

liTO

BbI.n.eJIHeT TOT <paJ(T,

'ITO

q

(X,

g).

paCCM~TpHBaeTCH

KaK

<pyHKUHH

Y

H8

OTpe3Ke

[0, Y (x)], a X

iPHKCHpOBaHO

Ha

[0,

1].

B

9THX

0603HalIeHHHx

p8BeHcTBa

(7),

(8) H (9)

MO>KHO

nepenHcaTb

TaR:

Y(x)

q(x)=

~

q"(y)dg,,

O~x~/,

(10)

o

Y(x)

t(x)=

~

t(q,,(g),

X,

g)dg,

O~x~l,

(II)

o

Y(X)

W(x)=t(x)

_

~t(qx(Y)X,

y)dy

(12)

q (x) -

Y(x)

~

q"

(V)

dy

0.,

I

1·'/

!

8KOHOMHIJECI<HE

MoLiH

[fn.

II

536

§ 2.

MOAenb

nOTpe(SJleHHJI

npH

npocTOM

TOBapHoM

npOH3BO,ltCTBenOTpe6JIeHHe

OCYl11eCT-

BJIHeTCH

lIepea

phIlroK,

Ha

KOTOPhIA npOH3BO,ltHTeJIH

Bcex

OTpaCJIeA

BhIHOCHT

CBOH

TOBaphl,

a

nOTpe6HTeJIH

IioKynalOT

3TH

TOBaphl.

TaKHM

06pa30M,

CBH3h npOHSBO,ltCTBa H

nOTpe6JIeHHH

OCYl11eCTBJIHeTCH

lIepes,

pbIHOK,

lIepes

MexaHHSM

neH,:

Onpe,lteJIHeMhIA cnpOCOM H Iipe.ztJIOlKe-

HHeM.

,

UpH

paCCMOTpeHHH

HaweA,

MO,lteJIH

nOTpe6JIeHHH

IJepes

PhlHQH

MhI'

6y,lteM

,HCXO,ltHTbHS

Ha~lHqH5I

yKalK,ltOro'

nOTpe6HTeJI5I CBoeH

i-.

onpe,lteJIeHHOH <!>YHKnHH nOJIeaHOCTH H HaJIHIIH5I y

Hero

onpe,lteJIeH-

Horo

,ltOXO,lta.

Ha

OCHOBaHHH

!noro

MhI HaH,lteM,

KaKoe

MaKCHMaJIhHoe

y,ltOBJIeTBOpeHHe

CBoeH'

nOTpe6HOCTH

KalKAhlii

nOTpe6HTeJIh

MOlKeT

nonyIJHTh

• .1J.aJIee, MhI

nOJIYIlHM'

TaKHe

KpHBhIe

cnpoca

no

KalK,ltOMy

BH,lty

npOH3BO,ltHMOrO

TOBapa.

B

,ltaJIbHeRrneM,

B § 3, Mhl coe,ltHHHM

,ltBe .MO,lteJIH npOH3Bo,ltCTBa H nOTpe6JIeHH5I B

0.nHy3aMKHyTylO

Mo.neJIb

npocToro

TOBapHoro

npOH3BO,ltCTBa.

.1J.a.nHM

TOIlHYIO MaTeMaTHIJecKYIO

nocTaHoBKy

3a.naIlH

•

.:'

I1TaK,

np.e,ltnOJIOlKHM, liTO y

HaCHMeeTC5I

m

nOTpe6HTeJIeH.

l(alK-

.nhlH

nOTpe6HTeJIh

HMeeT HOMep

Z,

KOTOphiR Mhl

TaKlKe

6y.neM

ClIHTaTh

pacnpe.neJIeHHhlM

HenpephlBHO

Ha

OTpe3Ke

[9,

m],

T.

e.

0

~

z:::;;;

m.

!"

.1J.aJIee, JIpe.nnOJIOlKHM,

liTO

Ha

phIHKe

yCTaHOBHJIOCh

onpe,lteJIeHHoe

"

'pacnpe.neJIeHHe

neH.

3T0

pacnpe.neJIeHHe

neH

Mbl

0603HallHM

C

(X),

.

O,~

x

~

t.

Mbl

npe.nnOJIaraeM'

TeM

caMblM

COBeprneHHylO

KOHKypeH-

nHIO

nOKynaTeJIeH.

3T0

03HallaeT,

liTO BeJIHIIHHa

nOKynoK

KalK,ltOro

OT;lleJIhHOrO

z-ronoTpe6HTeJI5I

x-ro

TOBapa

He

BJIH5IeT

Ha

ueHy

3Toro

TOBapa,

KOTOPYIO

0603HallHM

C (x).

nyCTh,

.naJIee, Ka.>K.nbIR.

nOTpe6H-

T¢JIh Z HMeeT

.nOXO.n

D (z), 0

~

z

~

m,

(BblpameHHhIH,

HanpHMep,

B.

py6JI5Ix),

KOTOphlH

OH

nOJIHOCThlO

ynoTpe6HT

Ha

:ripH06peTeHHe

,

npO,ltyKTOB

no

yCTaHoBHBrnHMC5I

Ha

phIHKe

ueHaM

C

(i).

'0603HallHM

!

lIepe3

q (x,

z)

TO

KOJIHlIeCTBO

npo.nyKTa

x, KOTOpoe B .neHCTBHTIMh-

\'

HOCTH

KynHT

z-A

IlOTpe6HTeJIb.

HaM

npH.neTC5I

paccMaTpHBaTb

q (x,

z)

KaK

<!>YHKUHIO

OT

x

npH

<!>HKCHpoBaHHOM z E

[0,

m].

3TOT

<!>aKT

Mol

6y.neM

OTpalKaTb

CJIe.nYIOI11HM

06pa30M:

' ,

q (x,

z)

==

q. (x). (1)

.1J.JI5I

onpe.neJIeHH5I

SToro

pacnpe.neJIeHH5I

nOKynoK

z-ro

nOTpe6H-

TeJI5I Mbl

6y.neM

HCXO.nHTb H3

'fOrO,

liTO KalK.nbIH .Z-R,

nOTpe6HTeJIh

06JIa.naeT

onpe.neJIeHHhl~

<!>YHKnHOHaJIOM nOJIe3HOCTH n

(qz

(x».

3TOT

<!>YHKUHOHaJI n (q.

(x»

onpe,lteJIeH

Ha

<!>YHKUH5IX

qz

(x).

TaKHM

00-

pa30M,

npe.nnOJIaraeTCH,

liTO

z-A

nOTpe6HTeJIh

cTpeMHTC5I

TaK

c.neJIaTh

CBOH

nOKynKH,

T.

e.

HaHTH

TaKylO

<!>YHKUHIO

qz

(x),

lIT06hl

MaKCH-

MH3HpoBaTh

cBoH

<!>YHKUHOHaJI nOJIe3HOCTH n

(qz

(x»

B

npe.neJIax

cymecTBylOUI,ero

pacnpe.neJIeHHH

neB

C (x) H

HMelOUI,erOC51

y

Hero

.

.noxo.na

D (z).

..

,I

.

'·,'"

\

~\,

.:>.>

C

531

h'

MO;nEJIb nOTPEBJIEHHSI

P]

I1HbIMH

CJIOBaMH,

KalK.nblH

nOTpe6HTenh

z

pernaeT

cJIe.nYlOmylO

I \

I

BapHaUHoHHylO

3a.nallY

C OrpaHHIJeHH51MH.

HaHTH

'max

n

(qz

(x))

npH

ii'

qz

(x)

I

<!>HKcHpoBaHHoM z E

[0,

m]

H

npH

yCJIOBHH

"I

t

I

(2)

~c(x)qz(x)dx=D(Z),

O~z~m

•.

o

3Ty

npocTylO

BapHaUHoHHyiO

3a.na1lY MOlKHO

pernHTh,

HCnOJIh3yH

TeopeMy

J1arpaHlKa,

no

KOTOpOH

AJI51

cymecTBOBaHH5I·

OnTHMaJIbHOH

HH.D.HBHAyaJIhHOR q>yHKUHH

nOTpe6JIeHHH

q.(x)

He06xoAHMO

cym

ecT

-

BOBaHHe

TaKoR

q>YHKI.I:HH

t..

(z)

(MHolKHTeJI5I J1a

r

paHlKa),

lIT06hl

q>yHKI.I:HOHaJI

I

(3)

n

(qz

(x))

-t..

(z)

~

C (x)

qz

(x)

dx

o

AOCTHraJI

3KCTpeMyMa.

He06xoAHMbIM

yCJIOBHeM

9KcTpeMyMa,

KaK

H3BeCTHO,

.nJI5I

9Toro

cJIYlla51

51BJI5IeTC5I

yCJIOBHe

gradn

(qz(x))

=t..(z)c(x),

o~x~l,

O~z~m.

(4)

TaKHM

06pa30M,

yCJIOBHe (4) .naeT

ypaBHeHHe,

KOTOpOMy YAOB-

JIeTBOp5IeT OIITHMaJIbHa51 epyHKI.I:H5I

qz

(x). Perna51

9TO

ypaBHeHHe

OTHOCHTeJIbHO

qz

(x), MbI

onpe.neJIHM

qz

(x)

KaK

q>YHKI.I:HIO

OT

npoH3-

BeAeHH5I t..(z)c(x),

T.

e.

.

.Qz(x)

=F(t..

(z)c

(x»,

O~x~l,

O~z~m.

(5)

D.JI5I

on

pe.neJIeHH5I, MHOlKHTeJI5I

J1arpaHlKa

A.'(z) BOCnOJIb3yeMC5I

e

yCJIOBHeM (2). nO.ncTaBJI5I5I (5) a (2),

nOJIyllHM

ypaBHeHHe,

onpe.n

-

t·

JI5IlOmee

t..

(z): i

I

(6)

~

C (x) F

(t..

(z)

c (x))

dx

= D (z),

O~z~m.

o

\.

,

I·

Pa3perna51

9TO

ypaBHeHHe

OTHOCHTeJIbHO

t..

(z),

nOJIyllHM

t..

(z)

KaK

I

<!>YHKI.I:HIO

OT

C (x) H D (x),

T.

eo'

t..

(z)

= f

(c

(x), D (z)), 0

~

z

~

m.

(7)

;.-,

nOACTaBJI5I5I (7) B (5),

nOJIyllHM

OKOHlIaTeJIhHO BH.n OllTHMaJIbHOH

<!>YHKI.I:HH

HH.nHBH.nyaJIbHOrO nOTpe6JIeHH5I

KalKAoro

z-

ro

nOTpe6HTeJI5I:

qz(x)=F(f(c(x),

D (z))c (x)),

O~x~l,

O~z~m.

(8)

OTCIO.na

YlKe

JIerKO

nOJIYlIHTb

<!>YHKI.I:HIO

cnpoca,

T.

e.

saBHCHMOCTh

I.I:eHbl C (x) x-ro

npo;n,yKTa

OT

KOJIHlIeCTBa q (x)

9Toro

np°;ll.YKTa.

'rpaq>HlIeCKoe

H306palKeHHe

<!>YHKI.I:HH

cnpoca

Ha3bIBalOT

KpU80a

cnpoca.

. .

18 A.

r.

BYTKOBCKKA

Бутковский А.Г. Методы управления системами с распределенными параметрами

Подождите немного. Документ загружается.