# РЕШЕНИЕ ПРЯМОЙ ЗАДАЧИ ГЕОТЕРМИИ ДЛЯ ТРЕХМЕРНОЙ НЕОДНОРОДНОЙ СРЕДЫ

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• 112

, , .

, () [1, 2].

. 1. . .

,

, (, ).

D, [3, 4], N Dn (. 1), .

, , , : ( ) . .

.

:

(1)

(t,x) x t; an=n/(ncn) ; n ; cn ; n ; fn(t,x) Dn.

t=0 :

(2)

:

(0)(0, ) ( ), .n nD x x x

2( , ) / ( , ) ( , ), ,n n nt t a t f t D x x x x

. 2014. . 324. 1

550.831.01

,. .. ,

,

, 394036, . , . , . 19. Email: pyatakovjv@mail.ru

. . , , , . : , . . . .

: , , , , , , , , .

• 113

, .

, [2], (. 2, ).

D=S1S2, S1

(3)

(1)(x) , , . (3) [2]. S2

q(t,x):

(4)

n=n(x) D x; (x)=p, xDpS2 (. 2, ),p=1,2,N. , , , [5].

Dn Dk :

(5)

(6)

x'Dn, x"Dk; n' n"=n"(x) , Dp Dk x'Snk; Snk=DnDk (. 2, ).

(t,x), (1)(6).

. D

Dn S1,

,

Si1

Dm (Si1=S1Dm) Dm (. 3, a). Si1 Dm i(1)=m, ci(1)=cm, ai(1)=am,i(1)=m. S2

Si2

i(2), ci(2), ai(2), i(2).

S3

Si3

Dn Dk: Si3=DnDk, nk. Si3 Si3 ni(3) (. 3, ). Dn , ni(3)

Dn. : 'i=n, "i =k, c'i=cn, c"i =ck, 'i=n,"i =k, a"i =an, a"i=ak. L(a,(t,x))=(t,x)/ta2(t,x). (1) :

(7)

. 2. : ) ; ) ; ) .

:

(8)

n=n() Dn;n(0)() Dn t=0; qn(,)=n(,)/n dS

0

(0)

1

0

0

( , ( , )) ( , , )( , ( , , )) ( , )

( ) ( , , )

( ) ( , ) ( , , )

( , , ) / ( , ) .

n

n

n

n

tn n

n nD

n nD

t

n n n nD

t

n nD

L a G a tL a G a t d Vd

G a t d V

c q G a t d Sd

a G a t d Sd

xx

x

x

x n

( , ( , )) ( , ), .n n nL a t f t D x x x

33

31

( ),N

ii

S S

22

21

( )N

ii

S S

11

11

( ),N

ii

S S

1

( ).N

nn

D D

lim ( , ) / lim ( , ) / 0,n kt t x xx x x n x n

lim ( , ) lim ( , ),t t

x x x x

x x

2( ) ( , ) / ( , ), ;t q t S x x n x x

(1)1( , ) ( ), ;t S x x x

• 114

Dn Dn ;G(an,t,x) [6]:

(9)

(t,x) (7), , (9), (8) :

(10)

kn(x)=1, xDn kn(x)=0, xDn. (10)

Dn, n=1,2,N, (t,x) , :

(11)

, ,

(12)

q(t,x) xS1, (t,x), xS2 (t,x), q(t,x) xSi3 , (11) (3)(6) [7].

(12) 1(t,x), . , ( ) [8].

(12) ( , ).

0

( , ) ( , , )i

t

i iD

f G a t d Vd x

(0) ( ) ( , , ) ,i

i iD

G a t d V x

3

3

2

( )2( ) ( )

( ) ( )1 1 0

1 0

3

( )( ) 2

( , )

( , )( , , )

( , , )

( , ) ,( , , )

( , )

( , , )( )

k

ki

i

tN kk k i

i i k kk i i iS

i itN

i i

i i iS

i i

kk i

i

t

qa G a t d Sdc

a G a tc

d Sda G a t

ct

G a ta

x

x

x

x

x

xn

3

3

2( )

1 1 0

2

(3)

21 0

(3)

( , )

( ) ( , , )

( , ) .( ) ( , , )

k

ki

i

tNk

ik i S

i itN

i

i S i i

i

d Sd

a G a t

q d Sda G a t

xn

xn

11 0

(0)

( , ) ( , ) ( , , )

( ) ( , , ) ,i

i

tN

i i ii D

i i iD

t a f G a t d Vd

a G a t d V

x x

x

11 2 3( , ) [ ( , ) ( , ) ( , )], ,n nt a t t t D

x x x x x

0

(0)

1

0

2

0

( ) ( , )

( , ) ( , , )

( ) ( , , )

( ) ( , ) ( , , )

( ) ( , , ) / ( , ) ,

n

n

n

n

n nt

n n nD

n n nD

t

n n n n nD

t

n nD

k a t

a f G a t d Vd

a G a t d V

c a q G a t d Sd

a G a t d Sd

x x

x

x

x

x n

2 2 21 1 2 2 3 3(B (9) ( ) ( ) ( ) ( ) ).R x x x x

( , ( , , )) ( ) ( ).n nL a G a t t x x

3

2

( , , ) (2 ( ))

exp( ( ) / (4 ( ))),n n

n

G a t a t

R a t

x

x

. 2014. . 324. 1

. 3. .

• 115

[6], , , . , [2].

, [9, 10].

.

.

:

(13)

(x) x;fn(x)=cnnfn(x); an, n, cn, n , (1); fn(x) Dn.

: S1

(14)

S2 q (x):

(15)

n=n(x) D x; (x)=p, xDpS2, p=1,2,,N.

Dn Dk :

(16)

(17)

x'Dn, x"Dk; n' n"=n"(x) , Dn Dk xSnk; Snk=DnDk.

(x), (13)(17).

. , (x) , :

(18)

(19)

(20)

(21)

(20)

, Dn Dk: Si3=DnDk, nk; '()=n,"()=k, Si3.

q(1)(), (2)(), (3)(), (20), (21) , (19)(21) (14)(17).

, , [4], , (x) (19), .

, (19), , , Di f

i

( Di):

(22)

(22) Di , [4].

( a, , c, ) x3>0, f=fn xDn f=0 xDn. , x3=0 .

, (18)(21), :

( ) ( ) ( ),u p x x x

1( ) / (4 ) ( ) .i

niD

f R d V x x

33

31

N

ii

S S

1

2

3

(1) 13

(2) 1

(3) 1

( ) ( ) ( ) / ( )

( ) ( ) / ( )

( ( ) ( )) / ( ( ).

( )) ( ) ( ) / ( )

S

S

S

R d S

R d S

R d S

x x n

x n

x n

1

2

(1) 12

1

( ) ( ) ( )

( ) ( ) ,

S

S

q R d S

q R d S

x x

x

11

1

( ) ( ) ( ) ,i

N

ii D

f R d V

x x

11 2 3(4 ) [ ( ) ( ) ( )], ,( )

0, ,n n

n

DD

x x x xx

x

lim ( ) / lim ( ) / 0,n k x xx x x n x n

lim ( ) lim ( ),

x x x x

x x

2( ) ( ) / ( ), ;q S x x n x x

(1)1( ) ( ), ;S x x x

2 ( ) ( ) 0, ,n nnf D x x x

• q(1)() x3=0. p(x1,x2,x3)

, (18)., (18) x3

• 117

2. s

1. ,

. .

2. (). , , .

3. , , .

, , X1 X2 X3

500 500 0 0.0500 500 0,2 6,16270833500 500 0,4 1,21990563101

500 500 0,6 1,81090046101

500 500 0,8 2,38925533101

500 500 1,0 2,95497026101

500 500 1,2 3,50804527101

500 500 1,4 4,04848036101

500 500 1,6 4,57627556101

500 500 1,8 5,09143088101

500 500 2,0 5,59394633101

, 2,5104

1. .., ..

// . 2004. . 26. 2. . 4661

2. , / .. ,.. , .. , .. . : , 2011. 384 .

3. .., .. // . 2012. . 320. 1. . 105110.

4. .., .., .. // . 2012. . 321. 1. . 7683.

5. .., .. . .: , 1986. 222 .

6. .., .. . .: , 1999. 799 .

7. .. // . 1967. . XXII. . 2 (134). . 59107.

8. / .. ,.. , .. , .. // . 1993. . 15. 4. . 312

9. .., .. // . 1987. 7. . 94100.

10. .., .. // . 1991. . 10. 2. . 3442.

22.07.2013 .

• 118

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8. Kutas R.I., Chekunov A.V., Lyalko V.I., Mitnik M.M. Termodinamicheskaya evolyutsiya astenolitov [Thermodynamic evolutionof asthenolith]. Geophysical journal, 1993, vol. 15, no. 4,pp