-

2006

1.

.... ....- Cauchy

.... -

2.

3.

i

4. LAPLACE

Laplace

Laplace ( )

ii

1 -

1.1

, (...). .

Laplace ( )

2F (x) = 0 (1)

{ (, - , ).

{ ( ).

{ ( - ).

..

Poisson

2F (x) = q (2)

{ Poisson Laplace ( ..)

2F (x, t) = 1a2

F (x, t))

t(3)

{ .

1

{ ()

{ ( ).

{ .

{ Maxwell ( )

.

k 2F (x, t) +k F (x, t) = cF (x, t)t

(4)

k , c .

2F (x, t) = 1c2

2F (x, t)

t2(5)

{ c ,

{ ,( , ) c2 = T

.

Helmholtz

(2 + k2)F (x) = 0 (6)

Helmholtz .

( ,, ..)

2F (x, t) = 1c2

2F (x, t)

t2+ G(x, t) (7)

2

- , , .

:

LF (x, t) + kF (x, t) = G(x, t) (8)

L (,).G(x, t) F (x, t) . (8)

( F (x, t) )

.

G(x, t) = 0 ..

1.2 ....

....

F (u(xi), xi,u(xi)

xi,2u(xi)

xixj, ...,

Nu(xi)

xl11 xl22 ....x

lnn

) = q(xi) (9)

xi ,u(xi) , , l1 + l2 + l3 + ...ln = N .. F u(xi) - .. . q(xi) = 0 . , . ,

3

, ( -). ..(...) ... . .... (n) (m) - (n) . (m 1) . - , . -, , ( ...). .... Cauchy-Kovalevski , . , , , u(xi) - Taylor. (-Cauchy) . .... :

A (x, y)2F (x, y)

x2+ 2 B(x, y)

2F (x, y)

xy+ C(x, y)

2F (x, y)

y2=

G (x, y, F (x, y),F (x, y)

x,F (x, y)

y) (10)

A(x, y) , B(x, y) C(x, y)

4

x,y G(x, y, F (x, y), F (x,y)

x, F (x,y)

y)

F (x, y) - .

1.3 ....

.... .... = B(x, y)2 A(x, y)C(x, y) (x, y) :

1. = B(x, y)2 A(x, y) C(x, y) > 0

2. = B(x, y)2 A(x, y) C(x, y) = 0

3. = B(x, y)2 A(x, y) C(x, y) < 0

( 10 ) (x, y), B(x, y)2 A(x, y) C(x, y) (x, y) , . Ticomi, yuxx + uyy = 0, y < 0 y > 0 .

.... .

5

(. 10 ) , Cauchy-Kovalevski, - Cauchy (x, y) .

Cauchy

Cauchy (. 10 ) -, s. u(s) -

0.5 1 1.5 2 2.5 3

1.6

1.8

2

2.2

2.4

2.6

2.8

3

tn s

y

1: Chauchy

N(s) .

x = x(s), y = y(s), u(s) N(s)

6

N(s) = u. (11)

= dy(s)ds

ex +dx(s)

dsey (12)

N(s) = (dy(s)ds

ex +dx(s)

dsey).(

u

xex +

u

yey)

= ux

dy

ds|s +u

y

dx

ds|s (13)

t

t =dx(s)

dsex +

dy(s)

dsey (14)

du(s)

ds=

u

x

dx

ds|s +u

y

dy

ds|s (15)

(13) (15) ( Cauchy) - u ,

p(s) = ux|s

q(s) = uy|s .

2ux2

, 2uy2

, 2uxy

- ( 10 )

dp(s) = (p

xdx +

p

ydy) = (

2u

x2dx +

2u

xydy) |s (16)

dq(s) = (q

xdx +

q

ydy) = (

2u

y2dx +

2u

xydy) |s (17)

7

, (10 ,16 ,17 ),

A(x, y) 2B(x, y) C(x, y)dx dy 00 dx dy

s

.

A(x, y)(dy

dx)2 2B(x, y)dy

dx+ C(x, y) |s = 0 (18)

, - , - - , s. (. 18). Taylor s

u(x, y) = u(s) + (x x(s))ux|s +(y y(s))u

y|s

+(x x(s))2u2

x2|s +(y y(s))2u

2

y2|s +(x x(s))(y y(s)) u

2

xy|s +...(19)

-

..

(x, y) .. - :

A(x, y)(dy

dx)2 2B(x, y)dy

dx+ C(x, y) = 0 (20)

8

.... ( > 0) - , (x, y) =. (x, y) =. , , (x, t) = x + ct (x, t) = x ct. (=0) (x, y) = (x, y) =, - ( < 0) . . - ( ), , - - . Caushy . ( ) - , .

:

Cauchy : - , , - . Cauchy .

Dirichlet : -.

Neumann : .

9

. . .. .

Cauchy .

. Dirichlet Neu-mann .

.. Dirichlet Neu-mann .

, , Laplace , , 1 ....

1.

2F (x,t)x2

= 1c2

2F (x,t)t2

:

B(x, t) = 0, A(x, t) = 1 C(x, t) = 1c2

> 0. .. Cauchy Cauchy Dirichlet Neu-mann 2

2. Laplace

1 2U-

10

2F (x,y)x2

+ 2F (x,y)y2

= 0

:

B(x, y) = 0, A(x, y) = 1 C(x, y) = 1 < 0.

Laplace .. - Dirichlet .

3.

2F (x,t)x2

= 1c2

F (x,t)t

:

B(x, t) = 0, A(x, t) = 1 C(x, t) = 0 = 0. .. - Dirichlet Neu-mann .

( 1 , 3 , 5) (x, t).

1.4 ....

.... , .

, ,

11

= (x, y) = (x, y).

2u

= F (u, , ,

u

,u

) (21)

.

, ,

= (x, y) = x.

2u

2= F (u, , ,

u

,u

) (22)

-

= (x,y)+(x,y)2i

= (x,y)(x,y)2i

.

2u

2+

2u

2= F (u, , ,

u

,u

) (23)

(Laplace) . ,

12

A(x, t) = 1 , B(x, t) = 0 C(x, t) = 1c2

(. 20) (dx

dt)2 1

c2= 0 (24)

(x, t) = x + ct =.

(x, t) = x ct =.

-3 -2 -1 1 2 3

-2

2

4

6xctxct

x

t

2:

, (x, t) = x + ct (x, t) = x ct,

2u

= 0 (25)

(x, t) 1

c 1

c.

.

13

1.5 - -

- . , . :

1. , ,

2.

3. ,

4.

... , . :

. Laplace,Fourier

Green

Fourier

, .

14

L . -, , - (.. ) . L - . , .

(x1, x2, x3) x1 = c1 x2 = c2 x3 = c3 , c1, c2, c3, F (x1 = c1, x2, x3) = a a = 0 x1 = c1 nodal nodes F . x1 = c1 nodal F (x1, x2, x3) :

F (x1, x2, x3) = F1(x1)F2(x1, x2) (26)

nodal .. x2 = c2 :

F (x1, x2, x3) = F1(x1)F2(x1)F3(x3) (27)

F1(x1)F2(x1)F3(x3) . . .

15

a = 0

F (x1, x2, x3) = Fp(x1, x2, x3) + F (x1, x2, x3) (28)

Fp(x1, x2, x3) Fp(x1 = c1, x2, x3) = a F (x1, x2, x3) nodes . (. 28) .. . , , Poisson ..

.. - ( , ).

16

2

2.1

().

0.5 1 1.5 20.6

0.8

1

1.2

1.4

1.6

1.8

TA

A

B

TB

TAy

TBy

w1

w2

x xdx

ds

y

3:

A(x, y) B(x+ dx, y) - ds. TA( ) TB, y

TAy = TAcosw1 = TAdyds T dy

dx, dx ds (29)

TBy = TBsinw2 = TB dy

ds T dy

dx|x

17

dy= y + dy, dx|x = x + dx

TAy + TBy = T (dy

dx|x

dy

dx) = Tdx

d2y

dx2= Tdx

2y(x, t)

x2(30)

dmd2y

dt2= dx

2y(x, t)

t2= Tdx

2y

x2(31)

2y(x,t)x2

= 1c2

2y(x,t)t2

c2 =

T, ( = dm

dx).

, f(x, t), ,

2y(x, t)

x2 1

c22y(x, t)

t2=

1

c2f(x, t) (32)

(2 1c2

2

t2)u(x, t) 1

c2f(x, t) (33)

. P, v . t0 , (P,, v) .

(2 1c2

2

t2)(x, t) =

1

c2. f(x, t) (34)

18

f(x, t) . c2 c2 = P

|t0 .

2.2

- u(x, t = 0) = a(x) u(x,t)

t|t=0= b(x) .

.

- ( y < x < ). (. 5) u(x, t) . - Cauchy t = 0 (Cauchy) .., x ct =. - .

2 u(x, t)

x2=

1

c22 u(x, t)

t2(35)

. (, ) : = x + ct = x ct3. (. 35) :

2 u(, )

= 0 (36)

3 (x, t) = = x + ct (x, t) = = x ct . , . .

19

. (36) :

u(, ) = f1() + f2() = f1(x + ct) + f2(x ct) (37) f1(), f2() .

:

u(x, t = 0) = a(x)

u(x,t)t

|t=0= b(x)

x1 x x2. t = 0 :

a(x) = f1(x) + f2(x)

b(x) = u

t|t=0 + u t |t=0 = cdf1d cdf2d

df1(x)dx

= 12[da(x)

dx+ b(x)

c]

df2(x)dx

= 12[da(x)

dx b(x)

c]

x1 x x :

f1(x) =1

2[a(x) +

1

c

xx1

b(x)dx + A1] (38)

f2(x) =1

2[a(x) 1

c

xx1

b(x)dx + A2] (39)

20

A1, A2 . f1(x) + f2(x) = a(x)

A1 + A2 = 0. (38) (39) x

x1 x x2 ( )

(x, t) xct : x1 xct x2 :

f1(x + ct) =1

2[a(x + ct) +

1

c

x+ctx1

b(x)dx + A1] (40)

f2(x ct) = 12[a(x ct) 1

c

xctx1

b(x)dx + A2] (41)

u (x, t) = f1(x + ct) + f2(x ct) ==

1

2[a(x + ct) + a(x ct) + 1

c

x+ctx1

b(x)dx 1c

xctx1

b(x)dx + A1 + A2]

=1

2[a(x + ct) + a(x ct) + 1

c

x+ctxct

b(x)dx] (42)

:

u(x, t) = 12[a(x + ct) + a(x ct) + 1

c

x+ctxct b(x

)dx]

(42) DAlembert - ( x, t) (x ct) ( ) (x1, x2). (x1, x2) . . x1 , x2 DAlembert (x, t). : f1(x+ct)

21

c, f2(x ct) .

1 - u(x, t = 0) = Exp[4x2]. t. DAlembert

u(x, t) = 12(Exp[4(x ct)2] + Exp[4(x + ct)2]).

- .. c = 4 . ( 4)

-20 -10 10 20

0.5

1t1

xctxct

-20 -10 10 20

0.5

1t1

xcxct

-20 -10 10 20

0.5

1t0

-20 -10 10 20

0.5

1t1

xctxct

4: Exp[4x2] (t1 = 0 < t2 < t3 < t4)

22

. - . , .

2

u(x, t = 0) = f(x) =

0 : < x < L

a(1 x2L2

) : L < x < L0 : L < x 0 u(x, t)

u(x, t) =1

2f(x + ct) + f(x ct)) (43)

f(xct) f(x) x xct.

f(xct) =

0 : < xct < La(1 (xct)2

L2) : L < xct < L0 : L < xct

-4 -2 2 4

1

2

3

4

5

6

I II III

IV VI

V

xctxct

xL L

t

to

xoxo

5:

t < to =. t > L

c

3L.

2.3

0 x < , .

u(x, t = 0) = f(x) , ut|t=0 = g(x) 0 x 0 (

24

). , - Dirichlet Neumann t .. u(x, t = 0) = 0, (x, t) ( ) 4. 5

-. DAlembert . f(x) g(x) u(x, t) x.

f(x) = f(x) , g(x) = g(x)

u(x, t) = 12[f(x + ct) + f(x ct) + 1

c

x+ctxct

g(x)dx] (44)

x y

u(x, t) = 12[f(x ct) + f(x + ct) 1

c

xctx+ct

g(y)dy]

=1

2[f(x ct) + f(x + ct) = 1

c

x+ctxct

g(y)dy]

= u(x, t) (45)

f(x) g(x) u(x, t) u(x, t) =u(x, t) . u(x = 0, t) = 0 u(x, t) x < 0 f(x) g(x) x < 0

4 U Cauchy Dirichlet

5 x = 0

25

-4 -2 2 4

-4

-2

2

4 xctxct

x

III

III

IV

6: x, t x ct = 0

(x, t) x ct ( ) .

x ct > 0

u(x, t) =1

2[f(x + ct) + f(x ct) + 1

c

x+ctxct

g(x)dx] (46)

x ct < 0

u(x, t) =1

2[f(x ct) f(x + ct) 1

c

x+ctxct

g(x)dx] (47)

x ct < 0 x + ct > 0

u(x, t) =1

2[f(x+ ct) f(x+ ct) 1

c

0xct

g(x)dx+x+ct0

g(x)dx]

(48)

26

IV x ct > 0 x + ct < 0

u(x, t) =1

2(f(xct)+f(xct)+ 1

c

0xct

g(x)dxx+ct0

g(x)dx)(49)

(0 < x 0 . f(x) g(x) , u(x, t) .

u(x, t) = U(x, t) + (x, t)

U(x, t),(x, t)

U(x, t = 0) = f(x), Ut(x, t) |t=0= g(x) ,U(x = 0, t) = 0

(x, t = 0) = 0, t(x, t) |t=0= 0 ,(x = 0, t) = (t) U(x, t) 1 (x, t) ( DAlembert

(x, t) = f1(x + ct) + f2(x ct) (50)

f1(x + ct) f2(x ct) x > 0 t > 0

) f1(x) + f2(x) = 0) cf 1(x) cf 2(x) = 0

) f1(ct) + f2(ct) = (t) (t) = 0 t < 0

27

() ()

f1(x) = f2(x) = C = x > 0.

()

f1(z) + f2(z) = (z/c)

z z

f1(z) + f2(z) = (z/c) = 0 f1(z) = f2(z) =

f1(x) f2(x)

f2(z) = (z/c), z > 0

z = x + ct > 0

(x, t) =

{f2(x ct) = (t xc ) : x < ct

0 : ct < x

(x ) .

2.4

. , . Fourier. -

28

, , .

) L .

- u(x, t) :

2u(x, t)

x2=

1

c22u(x, t)

t2 f(x, t) (51)

f(x, t) = 0 , f(x, t) = 0 f(x, t). , Cauchy , -

u(x, t = 0) = f(x)

u(x,t)t

|t=0= g(x)

Cauchy 0xL. - (xct) (0, L). Dirichlet x = 0 x = L , u(x = 0, t) = u(x =L, t) = 0. (. 51) f(x, t) = 0, x=0 x=L nodes u(x, t) -. 6 u(x, t) = X(x).T (t).

6

29

(. 51) :

T (t)d2X(x)

dx2=

X(x)

c2d2T (t)

dt2(52)

X(x)T (t) = 01

X(x)

d2X(x)

dx2=

1

c2T (t)

d2T (t)

dt2(53)

(. 53) x t ,

1

X(x)

d2X(x)

dx2= (54)

1

c2T (t)

d2T (t)

dt2= (55)

. d

2X(x)dx2

=

X(x):

1. = k2 > 0

X(x) = c1ekx + c2e

kx (56)

X(x = 0) = 0 X(x = L) = 0 c1 + c2 = 0 c1ekL + c2ekL = 0. c1 = c2 = 0, .

30

2. = k2 = 0

X(x) = c1x + c2 (57)

X(0) = 0 X(L) = 0 c1 = c2 = 0

3. = k2 < 0

X(x) = c1 sin kx + c2 cos kx (58)

X(0) = 0=c2 = 0 (59)

X(L) = 0 :

c1 sin(kL) = 0 (60)

c1 = 0 , kL =n k = n

L7.

(. 2.4) = k2 :

Tn(t) = ansinnc

Lt + bncos

nc

Lt (61)

Sturm-Liouville

u(x, t) =

n=0

sin(n

Lx) (an sin(

nc

Lt) + bn cos(

nc

Lt)) (62)

( ). an, bn - , -. t = 0,

7. L = d2dx2 X(0) = X(L) = 0 , k2 = n22 1L2 sin kx . Sturm-Liouville

31

u(x, t = 0) = f(x) u(x,t)t

|t=0= g(x)

u(x, t = 0) =

n=0

sin(n

Lx) bn = f(x) (63)

bn Fourier f(x) sin(n

Lx) :

bn =2

L

L0

f(x)sin(n

Lx)dx (64)

:

u(x, t)

t|t=0= g(x) =

n=0

nc

Lsin(

n

Lx) an (65)

an Fourier g(x) :

nc

Lan =

2

L

L0

g(x) sin(n

Lx)dx (66)

u(x, t) =

n=0

an sin(n

Lx) sin(

nc

Lt) +

n=0

bn sin(n

Lx) cos(

nc

Lt) =

=

n=0

1n(x, t) +

n=0

2n(x, t) (67)

1n(x, t) 2n(x, t) - . - Tn = 2Lnc sin(

nLx).

n = 2Ln sin(

nLct) cos(n

Lct).

1 = c2L

32

, , , . 1n(x, t) 2n(x, t) . (. 4) - , t = 0 , t = 3 . - Fourier, . bn = 0 ,

0.5 1 1.5 2 2.5 3

-1

-0.5

0.51

Sin3x

0.5 1 1.5 2 2.5 3

-1

-0.5

0.51

Sin4x

0.5 1 1.5 2 2.5 3

0.20.40.60.81

Sinx0.5 1 1.5 2 2.5 3

-1

-0.5

0.51

Sin2x

7: ( - )

an = 0. - . (. 67) .

33

(.67)

u(x, t) =

n=0

an2

[ cos(n

L(x ct)) cos(n

L(x + ct))] +

n=0

bn2

[ sin(n

L(x ct)) + sin(n

L(x + ct))]

= G(x ct) + F (x + ct) (68)

G(x ct) =

n=0

(an2

cos(nL

(x ct)) + bn2

sin(nL

(x ct))) (69)

F (x + ct) =

n=0

(an2

cos(nL

(x + ct)) + bn2

sin(nL

(x + ct))) (70)

DAlembert. G(x ct) F (x + ct) t = 0

f(x) = G(x) + F (x) (71)

g(x) = c(F (x)G(x)) (72)

f(x), g(x) . DAlembert.

u(x, t) = G(x ct) + F (x + ct) ==

1

2[f(x + ct) + f(x ct) + 1

c

x+ctxct

g(x)dx] (73)

u(x, t) , (. 68 ) ,

34

f(x) g(x) Fou-rier 0xL.

L . - . ()

u(x, t = 0) = f(x) =

{ axh

: 0 x ha(Lx)(Lh) : h x L

. .

x

ux,t0

LL2

( 62) . , Fourier an = 0 , bn

bn =2

L

L0

f(x)sin(n

Lx)dx

=2aL2

h(L h)2n2 sin(n

Lh) (74)

35

n |bn|2 |bn|21 1 12 0.125 03 0.0234 0.014 0.0 05 0.0016 0.001176 0.0015 07 0.00041 0.00038 0 09 0.00015 0.000110 0.0002 0

1: h = L2( 2)

h = L2( 3).

( 62)

u(x, t) =2aL2

h(L h)2

n=1

1

n2sin(

n

Lh)sin(

n

Lx) cos(

nc

Lt) (75)

:

|bn|2 = | 2aL2h(Lh)2 1n2 sin(nL h)|2

, 1n2, ( )

h = L2,

. , .. , , , = c

L

. - , -. 10 .

36

, .

)

L . - .

2u(x, t)

x2=

1

c22u(x, t)

t2(76)

u(x, t = 0) = f(x) ,u(x,t)

t|t=0= g(x) .

u(x,t)x

|x=0= 0 .u(x,t)

x|x=L= 0 .

u(x, t) = (A sin(kx) + B cos(kz))(C sin(kct) + D cos(kct)) (77)

x = 0 A = 0, x = L k = n

L n = 0, 1, 2, 3, ...

u(x, t) =

n=0

an cos(n

Lx) sin(

nc

Lt) +

n=0

bn cos(n

Lx) cos(

nc

Lt) (78)

an bn .

37

u(x, t = 0) =

n=0

cos(n

Lx) bn = f(x) (79)

bn Fourier f(x) cos(n

Lx) :

bn =2

L

L0

f(x)cos(n

Lx)dx (80)

:u(x, t)

t|t=0= g(x) =

n=0

nc

Lcos(

n

Lx) an (81)

an Fourier g(x) :

nc

Lan =

2

L

L0

g(x) cos(n

Lx)dx (82)

:

g(x) = Asin(5n

Lx)sin(

3n

Lx) (83)

.

- . bn = 0. an Fourier 8 .

nc

Lan =

2A

L

L0

sin(5n

Lx)sin(

3n

Lx)cos(

n

Lx)dx

8 - , cos(nL x) - (sin(a) sin(b0 = 12 (cos(a b) cos(a + b))

38

=2A

L

L0

(1

2(cos(

2

Lx) cos(8

Lx)) cos(

n

Lx)dx

=A

2(2n n8) (84)

a22 = (

AL4c

)2 2 = 2cL a28 = (

AL16c

)2 8 = 8cL .

2.5

, T , :

E =1

2

L0

(|u(x, t)t

|2 + T |u(x, t)x

|2)dx (85)

u(x, t) . (.85) . :

u(x, t) =

n=0

an sin(n

Lx) sin(nt) +

n=0

bn sin(n

Lx) cos(nt) (86)

n = nL

T . (. 86)

(. 85) sin(n

Lx) [0, L]

:E =

2T

4L

n=0

n2(|an|2 + |bn|2) (87)

. dE

dt= 0. ,

(. ), .

39

2.6

- f(x, t). u(x, t = 0) = a(x) u(x,t)

t|t=0 = b(x)

.

:

2u(x, t)

x2 1

c22u(x, t))

t2= f(x, t) (88)

f(x, t) (x, t). (. 88) .. . :

u(x, t) =

n=0

Tn(t) sin(n

Lx) (89)

- Sturm-Liouvill - . Fourier f(x, t) .

f(x, t) =

n=0

fn(t) sin(n

Lx) (90)

fn(t) Fourier f(x, t). Fourier u(x, t) f(x, t) - . (.88) Tn(t).

Tn (t) +

c2n22

L2T (t) = c2fn(t) (91)

fn(t) =2

L

L0

f(x, t) sin(n

Lx)dx (92)

40

(91) -

Tn(t) = A1T1n(t) + A2T2n(t) + Tp(t) (93)

T1n(t) , T2n(t) Tp(t) .

T1n(t) = sin(nc

Lt) (94)

T2n(t) = cos(nc

Lt) (95)

Tp(t) = T2n(t)

t0

c2fn(t)T1n(t)

W [T1n(t), T2n(t)]dt

T1n(t)t

0

c2fn(t)T2n(t)

W [T1n(t), T2n(t)]dt (96)

W [T1n(t), T2n(t)] Wronsky T1n(t) , T2n(t). T1n, T2n Tp(t) (. 93) (. 88).

u(x, t) =0

[An sin(nc

Lt) + Bn cos(

nc

Lt)

+Lc

n

t0

fn(t) sin(

nc

L(t t))dt] sin(n

Lx) (97)

An, Bn -, .

An =2

cn

L0

b(x) sin(n

Lx)dx (98)

Bn =2

L

L0

a(x) cos(n

Lx)dx (99)

41

An, Bn Fourier.

u(x, t) =

n=0

(An sin(nc

Lt) + Bn cos(

nc

Lt)) sin(

n

Lx) +

+

n=1

Lc

n

t0

fn(t) sin(

nc

L(t t))dt sin(n

Lx) (100)

. , .

u(x, t) =

n=1

Lc

n

t0

fn(t) sin(

nc

L(t t))dt sin(n

Lx) (101)

(. 101)

u(x, t = 0) = 0

Leibnitz -

u(x,t)t

|t=0= 0

- ( ). . ( ) (. 85) (. 100). (. 100) - .

42

:

L . ( f(x)). .

u(x, t)

2u(x,t)x2

1c2

2u(x,t)t2

= gc2

g . Fourier g.

g

c2=

n=0

fn sin(n

Lx) (102)

fn =2

L

L0

g

c2sin(

n

Lx)dx =

2g(1 cosn)nc2

(103)

u(x, t) = U(x, t) + u(x, t) U(x, t) . (. 101)

u(x, t) =1

Lc

n

t0

fn sin(nc

L(t t)) dt sin(n

Lx)

=

n=1

2gL2

c2n33(1 cos(n)) (1 cos(nct

L)) sin(

n

Lx)

=

n=1

2gL2

c2n33(1 + (1)n)) cos(nct

L)) sin(

n

Lx)

+

n=1

2gL2

c2n33(1 + (1)n)) sin(n

Lx)

43

=

n=1

2gL2

c2n33(1 + (1)n)) cos(nct

L)) sin(

n

Lx)

+gx

2c2(L x) (104)

(. 104) Fourier

gx

2c2(L x) =

n=1

2gL2

c2n33(1 + (1)n)) sin(n

Lx) (105)

(. 104) , u(x, t) = u1(x) + u2(x, t)

( . 100)

u(x, t) =0

(An sin(nc

Lt)) + Bn cos(

nc

Lt)) sin(

n

Lx) +

+

n=1

2gL2

c2n33(1 + (1)n)) cos(nct

L)) sin(

n

Lx)

+gx2

(L x) (106)

Fourier An Bn - . u(x, t) = u1(x) + u2(x, t) , - u(x, t) = u1(x) + U(x, t) , U(x, t) u1(x) . f(x) u1(x) , . , . (.5) ( )

44

0.20.40.60.8 1 x

-1.2-1

-0.8-0.6-0.4-0.2

ux

00.20.40.60.8 1x 0246810

t-2-1.5-1-0.50ux,t

0.20.40.60 8x

8:

.9

2.7

L . u(x, y, t) - . .. ( -) .

2u(x, y, t)

x2+

2u(x, y, t)

y2 1

c22u(x, y, t)

t2= 0 (107)

u(x = 0, y, t) = u(x = L, y, t) = u(x, y = 0, t) = u(x, y = L, t) = 0

u(x, y, t = 0) = f(x, y) ut(x, y, t) |t=0= g(x, y)

Cauchy Cauchy 0 < x < L 0 < y < L (x, y).

9. u(x, t) = us(x) + U(x, t)

45

u(x, y, t) = X(x)Y (y)T (t)

(. 107)

1

X(x)

d2X(x)

dx2+

1

Y (y)

d2Y (y)

dy2=

1

T (t)

1

c2d2T (t)

dt2= 0 (108)

(. 108) t, (x, y). ..

1

X(x)

d2X(x)

dx2= 1 (109)

1

Y (y)

d2Y (y)

dy2= 2 (110)

1

T (t)

1

c2d2T (t)

dt2= (111)

1 + 2 = . X(0) = X(L) = 0 Y (0) = Y (L) = 0 Sturm-Liouville (x, y) . . Sturm-Liouville ,

Xn(x) = sin(nxL

) Ym(y) = sin(myL )

n,m = 0, 1, 2, 3... Xn(x) Ym(y)

u(x, y, t)) =

n,m=0

sin(n

Lx) sin(

m

Ly) (anm sin(nmt) + bnm cos(nmt))

=

n,m=0

unm(x, y, t) (112)

46

nm = cLn2 + m2 . 11 -

,12 - 21. , - ( ) , u12(x, y, t) = u21(x, y, t), (), . (9) - . u11(x, y, t = 0) ,u12(x, y, t = 0) , u13(x, y, t = 0) , u32(x, y, t = 0) , - u11(x, y, t = 1), u12(x, y, t = 1) ,u13(x, y, t = 1) ,u32(x, y, t = 1). nodal lines - . - . u12(x, y, t = 0) - x , u21(x, y, t = 0) y. - . - ,.. u12(x, y, t) u21(x, y, t) = 0 x = y. - (). (. 112) - anm ,bnm Fourier .

u(x, y, t = 0) = f(x, y) =

n,m=0

bnm sin(n

Lx) sin(

m

Ly) (113)

bnm =4

L2

L0

sin(n

Lx) dx

L0

f(x, y) sin(m

Ly) dy (114)

47

ut(x, y, t) |t=0= g(x, y) =

n,m=0

anm nm sin(n

Lx) sin(

m

Ly) (115)

anm =4

L21

nm

L0

sin(n

Lx) dx

L0

g(x, y) sin(m

Ly) dy (116)

9:

(. 112) . Fou-rier .

48

2.8

R . f(, ) - g(, ) (, ) -, u(, , t), .

u(, , t) ( ) :

2u(, , t) = 1c2

2u(, , t))

t2(117)

10 11

2 = 1

+

1

22

2(118)

(1

+

1

22

2 1

c22

t2)u(, , t) = 0 (119)

u(, , t) = X(, )T (t).

u(, , t) = X(, )T (t) (. 117 ) X(, )

(2 + k2)X(, ) = 0 (120)

Helmholtz :

(1

+

1

22

2+ k2)X(, ) = 0 (121)

10 :2 = 1 + 12

2

2 +2

z2 . z = 0 .

11 . - .

49

T (t) ,

T (t) + k2c2T (t) = 0 (122)

k2 12. (. 122) :

T (t) = C1 sin(kct) + C2 cos(kct) (123)

(. 121) (, ). = R =. - ( - ) nodal . 2 X(, ) = R()(). (. 121)

2

R()

d2R()

d2+

R()

dR()

d+

1

()

d2()

d2+ k22 = 0 (124)

(. 124) , . 2 ..

d2R()

d2+

1

dR()

d+ (k2

2

2)R() = 0 (125)

d2()

d2+ 2() = 0 (126)

- 2, (. 126) 2.

12 k2 > 0 k2 < 0 . k2 > 0

50

() = ( + 2),(0) = (2) () = ( + 2) .

( ) 2 = m2 m = 0,1,2,3,4... (. 126) :

m() = A1 sin(m) + B1 cos(m) (127)

m() Fourier sin(m) cos(m) [0,2]. (. 125) =k, (k = 0), :

d2R()d2

+1

dR()d

+ (1 m2

2)R() = 0 (128)

(. 128) Bessel m Bessel Jm() Neumann Nm() . Neumann = 0.

5 10 15 20x-0.2

0.20.40.6

J

5 10 15 20x

-1-0.75-0.5-0.25

0.25

Y

10: BesselJm(x) Neumann Ym(x)

, ( - ). (128) AJm() + BNm().

51

, B = 0, R1 < < R2 A,B = 0. Bessel - . (128) -, R()|=R = 0 R() |=R = 0 , R , Sturm-Liouville 13 - Jm(kmnR ). kmn Jm(kR) = 0, kR = kmn( kmn n Bessel, Jm(kR)). - Bessel Jm(kmnR ), , [0, R] . X(, ) umn

umn(, , t) = J m(kmn

R)[Amn cos(m) + Bmn sin(m)]

[ A cos(kmnct

R) + B sin(

kmnct

R)] (129)

mn = kmncR

u (, , t) =

m,n

Jm(kmn

R)[Amn cos(m) + Bmn sin(m)] cos(

kmnct

R)

+mn

Jm(kmn

R)[A

mn cos(m) + B

mn sin(m)] sin(

kmnct

R) (130)

Amn , Bmn , Amn , B

mn

.

u(, , t = 0) = f(, ) =

=

m,n

Jm(kmn

R)[Amn cos(m) + Bmn sin(m)] (131)

u

t|t=0 = g(, ) =

13 Sturm-Liouville - = 0 . Bessel Legendre Sturm-Liouville -

52

=

m,n

Jm(kmn

R)[A

mn cos(m) + B

mn sin(m)]

kmn

R(132)

(. 131) (. 132) f(, ) g(, ) (, ) Fourier-Bessel (sin(m), Jm(kmnR )), (cos(m), Jm(

kmnR

)). Fourier-Bessel Bessel.

Amn =2

R2(Jm+1(kmn))2

20

d

R0

f(, )Jm(kmn

R) sin(m)d (133)

Bmn =2

R2(Jm+1(kmn))2

20

d

R0

f(, )Jm(kmn

R) cos(m)d (134)

kmnc

RA

mn =

2

R2(Jm+1(kmn))2

20

d

R0

g(, )Jm(kmn

R) sin(m)d

(135)kmnc

RB

mn =

2

R2(Jm+1(kmn))2

20

d

R0

g(, )Jm(kmn

R) cos(m)d

(136)

Fourier-Bessel - Bessel(). R = 1cm .

- m = 0 . (nodal lines) Bessel. m = 0

53

-1-0.5 00.5

1

-1-0.5

00.51

00.51

1-0.5 00.5

1-0.5

00.5

-1-0.5 00.5 1-1

-0.500.51

-0.5-0.250

0.250.5

10.5 00.5

-1-0.5 00.5

1-1-0.500.51

00.250.5

0.751

1-0.5 00.5

-1-0.5 00.5

1

-1-0.5

00.51

00.51

1-0.5 00.5

1-0.5

00.5

11:

n = 1 . .

- u(, , t = 0) = f() . u(, , t = 0) = f() = 2(R ) t = 0

54

(. 130). Amn Bmn

. , , - , 14, m = 0. -

u(, t) =n

J0(k0n

R)A0n cos(

k0nct

R) (137)

J0(k0nR ) Bessel k0n J0(k0nR ). Fourier-Bessel A0n

A0n =1

R2(J1(k0n))2

R0

f()J0(k0n

R)d (138)

A0n . - Bessel k01 = 2.4 , k02 = 5.22 , k03 = 8.65 .... R = c = 1 A01 = 0.68 , A0,1 =0.06 , A03 = 0.05 .....

2.9 ( )

(. 5) :

[1

r2

rr2

r+

1

r2 sin

(sin

) +

1

r2sin2

2

2

14 20

sin(m)d = 0m

55

1c2

2

t2]u(r, , , t) = 0 (139)

-. - (r, t). (139) :

[1

r2

rr2

r 1

c22

t2]u(r, t) = 0 (140)

u(r, t) = R(r,t)r

(.140) R(r, t) :

2R(r, t)

r2 1

c22R(r, t)

t2= 0 (141)

(. 141) 15 R(r, t) = F1(r+ct)+F2(rct) F1(r + ct) F2(r ct) , :

u(r, t) = F1(r+ct)r

+ F2(r+ct)r

(. 140) . r =.

a - . :

u(r, , , t) = R(r)()()T (t) (142)

(. 139). - () ()

15

56

(k, , ) . - k2 > 0 . .. :

d2T (t)

dt2= k2c2T (t) (143)

d2()

d2= () (144)

(1

sin()

d

dsin()

d

d

sin2())() = () (145)

d2R(r)

dr2+

2

r

dR(r)

dr+ (k2

r2)R(r) = 0 (146)

-. (. 143) k = kc:

T (t) = Asinkct + Bcoskct

(. 144) = m2 (m = 0,1,2,3, ...), () 2.

() = Asin(m) + B

cos(m)

(. 145) , x = cos d()

d= sin d(x)

dx, 1 x 1, :

(1 x2)d2(x)

dx2 2xd(x)

dx+ ( m

2

1 x2 )(x) = 0 (147)

57

(. 147) m = 0 Legendre. Legendre -. = l(l + 1), l = 0, 1, 2, 3, ..., Legen-dre16 Pl(x) Ql(x), Legendre. x = 1 Ql(x) - . Ql(x) . . m = 0 (. 147) - Legendre Pml (x) . 17 Legendre m - |m| l. (. 147) :

(cos ) = APml (cos )

(. 146) R(r) = u(r)r

- Bessel

d2u(r)dr2

+ 1r

du(r)dr

+ (k2 (l+ 12 )2r2

)u(r) = 0

Bessel (l + 12) , Jl+ 1

2(kr)

Nl+ 12(kr).

R(r) = AJ

l+12(kr)

r

+ BN

l+12(kr)

r

16Pl(x) = 12nn!dn

dxn (x2 1)l

17Pml (x) = (x2 1)m2 dmdxmPl(x)

58

R(r) = Ajl(kr) + Bnl(kr)

jl(kr) nl(kr) Bes-sel Neumann . Neumann - r 0 B . l,m ,|m| l:

u (r, , , t) = R(r)()()T (t) =

=m,l

(Asinkct + Bcoskct)(Asin(m) + B

cos(m))

Pml (cos )(Ajl(kr) + B

nl(kr)) =

=m,l

(Ajl(kr) + B

nl(kr))(Asinkct + Bcoskct)

Y ml (, ) (148)

Y ml (, )

Y ml (, ) =

2l+14

(lm)!(l+m)!

Pml ()eim, |m| l

. :

20d

0sin Y ml (, )Y

m

l (, )d = llmm

Fourier Y ml (, ).

59

f(, ) =l=0

lm=l

flmYml (, )

flm Fourier

flm =20d

0sin Y ml (, )f(, )d

(. 148) - .

:

a , , - u0 . t = 0 . .

(z = 0) t < 0. , (u = ) . . - . - .

.

- .

1. :

60

(r, , t = 0) = u0rcos

2. (r,,t)

t|t=0 = 0

3. - .(r,,,t)

r|r=a = 0

.

(. 148). B = 0 = 0 m = 0.

(r, , t) =l

(Asinkct + Bcoskct)Pl(cos )jl(kr) (149)

(r, , , t)

r|r=a = 0 =

=l

(Asinkct + Bcoskct)Pl(cos )d

Jl+1

2(kr)

r

dr|r=a (150)

(2) A = 0.

(r, , t) =l

(Bcoskct)Pl(cos )Jl+ 1

2(kr)r

(151)

(3)

Jl+ 1

2(ka)a

12a

32

Jl+ 12(ka) = 0 (152)

(r, , t) =l=0

n=1

BcoskctPlcos Jl+ 1

2(klnr

a)

r(153)

61

kln = ka (152).

(r, , t = 0) =l=0

n=1

BPl(cos )Jl+ 1

2(klnr

a)

r= u0rcos

u0r32 cos =

l=0

n=1

BlnPl(cos )Jl+ 12(klnr

a) (154)

(. 154) Bln Fourier u0r

32 cos

Legendre-Bessel. Legendre,(P0(cos ) = 1, P1(cos ) = cos()) l = 1, .

u0r32 =

n=1

BnJ 32(k1nr

a) (155)

Bn =2u0

r=ar=0

r52J 3

2(k1nr

a)dr

a2J 32(k1n)(1 2k21n )

(156)

(r, , t) =

n=1

Bncos J 3

2(k1nr

a)

rcos(

k1nct

a) (157)

u(r, , ) = (r,,)

r

2.10

1. .

62

2. . t = 0 x = 0 ut(x, t = 0) = a(x) a (x) Dirac. u(x, t) t > 0.

3. (xct) . t = 0 .

4. (x = 0) .

u(x, t = 0) = f(x) =

0 : 0 < x < L

hL(x L)(4 x+L

L) : L < x < 3L0 : 3L < x 0.(. ut (x, t = 0) = A (x x0))

6. x > 0, , t > 0

u(x, t = 0) = f(x) , u(x,t)t

|t=0 = g(x)

63

u(x,t)x

|x=0= 0

t = 0 f(x) g(x) x = 0.

7.

f(x, t) = 4(x ct)2(x + ct)

.

8. L > 2 , -.

u(x, t = 0) = f(x) =

{1 : L

2 1 < x < 1 + L

2

0 : 0 < x < L2 1, L

2+ 1 < x < L

.

9. 3a -

u(x, t = 0) = f(x) =

exa

: 0 < x < ae3a2x

a: a < x < 2a

ex3aa

: 2a < x < 3a

.

10.

(2

x2 1

c22

t2)u(x, t) = x sin(x) (158)

64

u(x, t = 0) = 0 ut(x, t) |t=0=0. ;

11. L g(x) = I(x L

2).

(. .. ( 2x2 1

c22

t2 h

c2t

)u(x, t) = 0)

12. L1 L2 - ut(x, t) |t=0= 1. .

13. L u(x, y, t = 0) = sin2( x/L) sin( y/L). .

14. L - - u(x, y, t = 0) = f(x, y) . - .

15. - - .

16. - L . . (. -

65

- ).

17. ) R = 1 u(, , t = 0) = g J0 () ut(, , t = 0) |t=0) = 0 , J0 () Bessel .

18. a - u(, , t = 0) = J3 (

k31a

) cos(3) ut(, , t = 0) |t=0) = 0 , k31 Bessel . ) .

19. R g J1 (k01R ) . (. - Bessel 0 < < R)

20. a < b . - f()

21. a - . u(, , t = 0) = AJ0(k01

a) + g

4c2(2 a2)

.

66

3

3.1

V S. dV dQ = cdV , c (x, t) .

V

n

dS

12:

Q S t

Q1 = K(x)(S)(t)(x, t). (159)

. . K(x) - S. -

q =Q

(t)(S)(160)

67

(t1, t2)

Q1 = t2

t1

dtS

K(x).dS

= t2

t1

dtV

.(K(x))dV (161)

dV 1(x, t1) 2(x, t2), Q2 = c(dV)(2 1)

Q2 =V

c(2(x, t2)1(x, t1))dV

= t2

t1

dtV

tcdV

= t2

t1

dtV

tcdV (162)

-

Q3 =

t2t1

dtV

F (x, t)dV (163)

F(x, t) .

Q3 = Q1 + Q2 (164)

(164) (162) (163)

68

(x, t)

t=

1

c[K(x).(x, t) + K(x)2(x, t) + F (x, t) (165)

- ,

(x, t)

t=

1

cK2(x, t) (166)

(166) - (.. ), .

3.2

- :

2T (x, t) = 1a2

T(x, t)

t(167)

(167) Dirichlet ( - - ) Neumann( . - ) Robin( ) -18 T0(x, t = 0) = f(x). t > 019.

18

19 - . ,

69

(167) T (x, t) = X(x)T (t) .

1

X(x)

d2X(x)

dx2=

1

a2T (t)

dT (t)

dt(168)

T (t)

dT (t)

dt= k2a2T (t) (169)

k2 . (169) T (t) = ek

2a2t. k2 > 0 T (t) 0 X(x) = Aeikx + Beikx. k Fourier .

T (x, t) =

A(k)eikxk2a2tdk (170)

A(k) Fourier 20

T (x, t = 0) = f(x)

=

A(k)eikxdk (171)

A(k) =1

2

f(x)eikx

dx

(172)

t > 0 , t > 0. . .

20

ei(kk)xdx = 2(k k) (k k)

Dirac.

70

T (x, t) =1

2

dk

f(x)eik(xx

)k2a2tdx

=1

2

a2tf(x

)e

(xx )24a2t dx

=

=

f(x)T0(x, t)dx

(173)

T0(x x , t) . (.173) . - t 0

limt0T (x, t) =

limt0

1

4a2tf(x

)e

(xx )24a2t dx

= f(x) (174)

Dirac21

0.5 1 1.5 2

0.25

0.5

0.75

1

1.25

1.5

1.75t1

t2

t3

t4

13: t1 < t2 < t3 < t4

. Q

21 limt0

1

4a2te (xx

)2

4a2t = (x x)

71

x0 , . 2 Q = 2cT0, T0 . Q t :

T (x, t) =

T0e(xx )2

4a2t

2at

dx

=Q

2c2at

x0+x0

e(xx )2

4a2t dx (175)

0 - x0 t = 0 .

lim0

1

2

x0+x0

e(xx )2

4a2t dx= e

(x0x)24a2t (176)

lim0T (x, t) T0(x, t) (177)

. t = 0 x0 Q. -

T (x, t)dx =Q

pc

T0(x, t)dx =Q

pc(178)

. 22

22

e(x0x)2

4a2t dx = 2a

t

72

3.3

Dirichlet L T0(x) . t - . T (x, t) (167), - T (x = 0, t) = T (x = L, t) = 0 T (x, t = 0) = T0(x). (167) T (x, t) = X(x)T (t) . T (t) T (t) = ek2a2t ,k2 , - t > . X(x)

d2X(x)

dx2+ k2X(x) = 0 (179)

X(0) = X(L) = 0 ( ) - Sturm-Liouville sin(n

Lx)

kn =nL

. (. 167) - ( )

T (x, t) =n

An sin(n

Lx)ek

2na

2t (180)

An Fourier .

T (x, t = 0) = T0(x) =n

An sin(n

Lx)

73

An =2

L

L0

T0(x) sin(n

Lx)dx (181)

100 Fourier (. 181) :

An =

{400n

: n = 2k + 10 : n = 2k

T (x, t) =

k=0

400

(2k + 1)sin(

(2k + 1)

Lx)e(

(2k+1)aL

)2t (182)

t e( (2k+1)aL )2t -.

T (x, t) 400

sin(

Lx)e(

aL

)2t (183)

t (182 ) 23, t T (x, t) 0 ( ).. 1

e -

. 23

.

74

- :

1

= lim

t1

tln|T (x, t)| (184)

.

: - . (. 180) :

T (x, t) = A1 sin(

Lx)e(

aL

)2t +

n=2

An sin(n

Lx)e(

naL

)2t

= A1sin(

Lx)e(

aL

)2t[1 + O(e(3aL

)2t)]

ln|T (x, t)| = lnA1 + lnsin(xL

) + lne(aL

)2t + ln[1 + O(e(3aL

)2t)]

1

= lim

t1

tln|T (x, t)| (a)

2

L2(185)

(.185) - , ( t) , ( ) . - .

Neumann

75

L T0(x) - . , - t .

:

T (x, t) = X(x)T (t)

T (t) = ek2a2t ( k2 > 0 ) X(x) = A sin(kx) + B cos(kx). . :

T (x,t)x

|x=0 = T (x,t)x |x=L = 0

A = 0 k = n

L. Sturm-

Liouville cos(nLx) kn = nL .

(167) -

T (x, t) =

n=0

Bn cos(n

Lx)ek

2na

2t (186)

Bn Fourier .

T (x, t = 0) = T0(x) =

n=0

Bn cos(n

Lx)

76

Bn =2

L

L0

T0(x) cos(n

Lx)dx (187)

n = 0

B0 =1

L

L0

T0(x)dx (188)

:

T (x, t) = B0 +

n=1

Bn cos(n

Lx)ek

2na

2t (189)

- :

limt T (x, t) = B0 =

1

L

L0

T0(x)dx (190)

( . 190) , - , , .

Robin ( )

L T (x, t = 0) = T0(x). ( ) - . .

77

T (x,t)x

hT (x, t)|x=0 = 0

T (x,t)x

+ hT (x, t)|x=L = 0

- - .

T (x, t) = X(x)ek2a2t

X(x)

d2X(x)

dx2+ k2

dX(x)

dt= 0 (191)

dX(x)dx

hX(x)|x=0 = 0dX(x)

dx+hX(x)|x=L = 0 (192)

(. 191) , X(x) = A sin(kx) + B cos(kx). A = hB

k

cot =(2 h2L2)

2hL(193)

= Lk . (. 193) n Sturm-Liouville

Xn(x) = Bn[hL

nsin(

nx

L) + cos(

nx

L)] (194)

Xn(x) (0, L).

78

24 Xn(x)

X(x) =

n=1

Bn(hL

nsin(

nx

L) + cos(

nx

L))e

2a2t

L2 (195)

2 4 6 8 10 12

-30

-20

-10

10

20

30

m

Cotm

14: (. 193)

3.4

L . : T (x = 0, t) = T1(t) , T (x =L, t) = T2(t) T (x, t = 0) =T0(x) . t > 0.

( ) .

24 Xn(x) (0, L) .

79

( ). ( , - .)

T (x, t) = U(x, t) + U(x, t)

U(x, t) U(x, t) . .

T (x = 0, t) = T1(t) T (x = L, t) = T2(t)

T (x, t) = T0(x)

:

T (x, t) = U(x, t) + U(x, t) (196)

U(x, t) - :

U(x = 0, t) = T1(t) U(x = L, t) = T2(t)

U(x, t) :

U(x = 0, t) = 0 U(x = L, t) = 0

U(x, t = 0) = T0(x) U(x, t = 0) = (x).

80

U(x, t) :

U(x, t) =L x

LT1(t) +

x

LT2(t) (197)

U(x, t)

2U(x, t)

x2 1

a2U(x, t)

t=

1

a2x

LT2(t) +

L xa2L

T1(t) F (x, t) (198)

(.198) .. - U(x, t) . (. 198) - F (x, t) . (. 198) (.88). sin(n

Lx) -

. Fourier F (x, t)

U(x, t) =n

An(t) sin(n

Lx) (199)

F (x, t) =n

bn(t) sin(n

Lx) (200)

bn(t) =2

L

L0

F (x, t) sin(n

Lx)dx (201)

(198). Fourier, An(t) :

n=1

[n22

L2An(t) 1

a2An(t) bn(t)] sin(n

Lx) = 0 (202)

81

(202) ..

An(t) = Cne(n2a22

L2t) + e(

n2a22

L2t)

t0

bn(t)e(

n2a22

L2t)dt

(203)

U(x, t) =n

[Cne(n2a22

L2t)

+ e(n2a22

L2t)

t0

bn(t)e(

n2a22

L2t)dt] sin(

n

Lx) =

=n

Cne(n2a22

L2t) sin(

n

Lx)

+n

e(n2a22

L2t)

t0

bn(t)e(

n2a22

L2t)dt

sin(

n

Lx)

= U(x, t) + U(x, t) (204)

Cn -

Cn =2

L

L0

(x) sin(n

Lx)dx

T (x, t) =L x

LT1(t) +

x

LT2(t)

+n

Cne(n2a22

L2t) sin(

n

Lx)

+n

e(n2a22

L2t)

t0

bn(t)e(

n2a22

L2t)dt

sin(

n

Lx)

(205)

1 -, T (x = 0, t) = C1 T (x = L, t) = C2 T (x, t = 0) = g(x).

82

(. 205) T1(t) = C1 , T2(t) = C2 .

T (x, t) =L x

LC1 +

x

LC2

+n

Cne(n2a22

L2t) sin(

n

Lx) (206)

Fourier , Cn

Cn =2

L

L0

(g(x) L xL

C1 xLC2) sin(

n

Lx)dx

, . ( ) . , , T(x) :

d2T(x)

dx2= 0 (207)

. (207)

T(x) = a1x + a2 (208)

a1, a2 - T(x = 0) = C1 T(x = L) = C2.

a2 = C1

a1 =C2 C1

L

83

T(x) =C2 C1

Lx + C1 (209)

:

T (x, t) =n

Cnen2a2

L2t sin(

n

Lx) +

C2 C1L

x + C1 (210)

T (x, t = 0) =n

Cn sin(n

Lx) +

C2 C1L

x + C1 = g(x)

g(x) C2 C1L

x C1 =n

Cn sin(n

Lx)

Fourier Cn :

Cn =2

L

L0

(g(x) C2 C1L

x C1) sin(nL

x)dx (211)

( 210) :

limtT (x, t) = limtT(x) + limtTo(x, t) = T(x) (212)

t limtTo(x, t) 0 :

limt[T (x, t) T(x)] = 0 (213)

T(x) . ( ) , .

T (x, t) - , T(x, t) .. t > 0 :

limt[T (x, t) T(x, t)] = 0 (214)

84

( 2) , . C1(t), C2(t)

T(x, t) =C2(t) C1(t)

Lx + C1(t) (215)

Fourier Cn

Cn =2

L

L0

(g(x) T(x, t = 0)) sin(nL

x)dx (216)

3.5

.. Dirichlet Neumann . L

T (x = 0, t) = a(t), T (x = L, t) = b(t) T (x, t = 0) = f(x)

T1(x, t) (167) . T2(x, t) W (x, t) = T1(x, t) T2(x, t), W (x, t) (167) :

W (x = 0, t) = 0, W (x = L, t) = 0 W (x, t = 0) = 0

85

:L

0

W (x, t)2W (x, t)

x2dx 1

a2

L0

W (x, t)W (x, t)

tdx = 0 =

=

L0

[

xW (x, t)

W (x, t)

x (W (x, t)

x)2]dx 1

2a2

L0

W (x, t)2

tdx =

= W (x, t)W (x, t)

x|L0

L0

(W (x, t)

x)2dx 1

2a2

L0

W (x, t)2

tdx = 0

(217)

L

0

(W (x, t)

x)2dx =

1

2a2d

dt

L0

W (x, t)2dx (218)

t

t

0

dt

L0

(W (x, t)

x)2dx =

1

2a2

L0

W (x, t)2dx (219)

(219) (). W (x, t) - T1(x, t) =T2(x, t). - .

3.6

L R - . T0 ,. .

:

(1

+

1

22

2+

2

z2 1

a2

t)T (, , z, t) = 0 (220)

86

:

T ( = R, , z, t) = T (, , z = 0, t) = T (, , z = L, t) = 0

(220) - .

T (, , z, t) = R()()Z(z)T (t) (221)

(221) (220) - . , k2 > 0 ( k2 > 0 ) :

dT (t)

dt= a2k2T (t) (222)

z, , ():

d2Z(z)

dz2+ 2Z(z) = 0 (223)

d2()

d2+ m2() = 0 (224)

d2R()

d2+

1

dR()

d+ (k2 2 m

2

2)R() = 0 (225)

-.

m 2 ( 2).

87

2 z, (Z(z = 0) = Z(z = L) = 0).

k2 - , - .

(.223) T (, , z = 0, t) = T (, , z = L, t) = 0

Sturm-Liouville sin(nzL

) - 2 = (n

L)2 . (225) Bessel, (128),

Bessel Neumann, m (m) ,

k2 2, :

R() = AJm(k2 2) + BNm(

k2 2)

= 0 Neumann (Nm(x 0) ). ( - ) B = 0

R() = AJm(k2 2)

, R( = R) = 0,

R( = R) = AJm(k2 2R) = 0 (226)

(.225) (.226) - Sturm-Liouville ( ) Jm(kmiR ) (kmi

R) , k

2mi

R2= k2 n22

L2. kmi i -

Bessel m , Jm(kmi) = 0.

88

(220) Sturm-Liouville :

T ( = R, , z, t) =

m,n,i

Jm(kmiR

)sin(nz

L)

[ Amnisin() + Bmnicos()]e(kmi

2

R2+n

22

L2)a2t (227)

t = 0 :

T ( = R, , z, t = 0) = T0 =

=

m,n,i

Jm(kmiR

)sin(nz

L)[Amnisin(m) + Bmnicos(m)] (228)

Amni Bmni Fourier - T0 Jm(

kmiR

), sin(nzL

) , sin(m) cos(m).

Amni = (2

L)(

1

)(

2

[RJm+1(kmi)]2)

L0

sin(nz

L)dz

20

sin(m)d

R0

T0Jm(kmiR

)d (229)

Bmni = (2

L)(

1

)(

2

[RJm+1(kmi)]2)

L0

sin(nz

L)dz

20

cos(m)d

R0

T0Jm(kmiR

)d (230)

Amni = 0 m Bmni = 0 m m = 0,

B0ni =16[1 (1)n]Ln2[RJ1(k0i)]

R0

T0J0(k0iR

)d (231)

B0ni = 0 n = 2l + 1.

T ( = R, z, t = 0) = T0

i,l=0

ea2[

k20i

R2+

(2l+1)22

L2]tFil(, z) =

= T0ea2[ k

201

R2+

2

L2]t(F10(, z) + O(t)) (232)

89

(184)

1

= lim

t1

tln|T (x, t)| a2(k

201

R2+

2

L2) (233)

: |t = 0.1729R2a2 . , , . L L < , L

3.7

1. - x > 0 x = 0 t .

2. T (x, t) L, ,

tT (x, t) a2

2

x2T (x, t) = b2T (x, t) (234)

x = 0 , x = L T (x, t =0) = T0 cos

2(xL

)

3. L T (x, t = 0) = 3 sin(x

L sin(3x

L

-

,

.

4. L T (x, t = 0) = T0 +T1 cos

xL

. - t > 0 -.

5. k, L . Q0

91

. - , ( T (x,t)

x|x=0= kQ0 ).

6. - L , H T (x =0, y, t) = T (x = L, y, t) = 0 , T

y|y=0= 0 T (x, y = H, t) = g(x)

T (x, y, t = 0) = f(x, y) . . ( -).

7. R1 =1cm R2 =

2cm L1 = 1cm L2 = 0.5cm -

. 500 . ( ).

8.

2(r, , , t) + b(r, , , t) = 1k

t(r, , , t)

k, b . 0 . . , ( ).

9. R . t = 0 - T0 T .

92

, ( -

tT (x, t) a2 2

x2T (x, t) = )

(: T (, , t) = U(, , t) + U())

93

4 Laplace

4.1

Laplace Poisson Helmholtz - .. . Laplace, , - , , ( ). , , - . - . - .

Laplace - .

: G Rn . G - Laplace G. - . ( ) .

94

- , . ( ). - ( ) .

Laplace - Dirichlet Neumann , Cauchy - .

: T1(x) 2T1(x) =0 V T1|S(V ) = T0 - Laplace, 2T2(x) = 0 - T2|S(V ) = T0.

(x) = T1(x) T2(x)

(x)|S(V ) = 0.

V

. ((x)(x)) dV =V

| (x) |2 dV +V

2(x) dV

=V

| (x) |2 dV (235)

Gauss V

. ((x)(x)) dV =

S(V )

(x) (x) . dS (236)

95

(235) (236) V

| (x) |2 dV = 0 (237)

(x) V , .

T1(x) = T2(x)

. , ,

- , , .

4.2 Laplace

, , . z. - z (0 x L (0 y L ) . () .

T (x, y) (x, y) Laplace .

T (x = 0, y) = g1(y)

96

T (x = L, y) = g2(y)

T (x, y = 0) = g3(x)

T (x, y = L) = g4(x)

g1,2(y) g3,4(x) . ( , Di-richlet, ) . - , - , . , : :

T (x, y) = T1(x, y) + T2(x, y) + T3(x, y) + T4(x, Y ) (238)

=2T = 0g4

g3

g1

g2 +2T1 = 00

0

g1

0 +2T2 = 00

0

0

g2 +2T3 = 00

g3

0

0 2T4 = 0g4

0

0

0

( )

Ti(x, y) Laplace - :

T1(x, y = 0) = g1(x) , T3(x = 0, y) = T3(x, y = L) = T3(x = L, y) = 0

(239)

97

T2(x = L, y) = g2(y) , T2(x = 0, y) = T2(x, y = 0) = T2(x, y = L) = 0

(240)T3(x, y = L) = g3(x) , T4(x = 0, y) = T4(x = L, y) = T4(x, y = 0) = 0

(241)T4(x = 0, y) = g4(y) , T1(x = L, y) = T1(x, y = 0) = T1(x, y = L) = 0

(242)

Ti(x, y). - . T4(x, y)

2T4(x, y)

x2+

2T4(x, y)

y2= 0 (243)

:

T4(x = 0, y) = g4(y) , T4(x = L, y) = T4(x, y = 0) = T4(x, y = L) = 0

(244) :

T4(x, y) = X4(x)Y4(y) (245)

(.243)

1

X4(x)

d2X4(x)

dx2= 1

Y4(y)

d2Y4(y)

dy2= 2 (246)

y = 0 y = L , Y4(y = 0) = Y4(y = L) =0, 25 Y4(y) :

Y4n(y) = sin(ny

L) (247)

25

98

2 = (nL

)2 . X4(x) (.243) :

X4n(x) = A sinh(nx

L) + B cosh(

nx

L) (248)

(247)

T4(x, y) =

n=1

[An sinh(nx

L) + Bn cosh(

nx

L)] sin(

ny

L) (249)

x = L

T4(x = L, y) =

n=1

(An sinh(n) + Bn cosh(n)) sin(ny

L) = 0 (250)

y An Bn,

An sinh(n) + Bn cosh(n) = 0 (251)

An = Bn cosh(n)sinh(n)

(252)

x = 0

T4(x = 0, y) = g4(y) =

n=1

Bn sin(ny

L) (253)

( 253) Fourier sin(ny

L) g4(y) . Bn :

Bn =2

L

L0

g4(y) sin(ny

L)dy (254)

T1(x, y) , T2(x, y) , T3(x, y) .

: - 75 100 .

99

. . , () :

=2T = 075

75

75

100 +2T1 = 075

75

75

75 2T2 = 00

0

0

25

T1(x, y) = 75 . T2(x, y) .

T2(x = 0, y) = T2(x, y = 0) = T2(x, y = L) = 0 T2(x = L, 0) = 25

T2(x, y) =

n=1

(An sinh(nx

L) ) sin(

ny

L) (255)

T2(x = L, y) =

n=1

(An sinh(n) ) sin(ny

L) = 25 (256)

An sinh(n) =2

L

L0

25 sin(ny

L)dy =

50

n(1 (1)n)

T2(x, y) =

k=1

(100

(2k + 1)sinh(

(2k + 1)x

L) ) sin(

(2k + 1)y

L) (257)

T (x, y) = 75 +

k=1

(100

(2k + 1)sinh(

(2k + 1)x

L) ) sin(

(2k + 1)y

L)

(258)

100

4.3 Laplace ( )

: R - 75 30026.: - Laplace (144) (145), - (146) k = 0 = l(l+1), l = 0, 1, 2, 3.. .

d2R(r)

dr2+

2

r

dR(r)

dr l(l + 1)

r2R(r) = 0 (259)

(259) rl r(l+1) 27 . Laplace :

T (r, , ) =l,m

(Arl + Br(l+1))Y ml (, ) (260)

Y ml (, ) . ( ) -.

T (r, , ) = T1(r, , ) + T2(r, , ) (261)

:

T1(r = R, , ) = 75 (262)26: -

( ) ( -).

27 (259) ra. - a,a1 = l a2 = (l + 1)

101

T2(r = R, , ) =

{0 : 1 < cos < 0

225 : 0 < cos < 1

T1(r, , ) T2(r, , ) Laplace T1(r, , ) = 75 . T2(r, , ) m = 0

T2(r, , ) =l,m

(Arl + Br(l+1))Y ml (, )

=l=0

(Arl + Br(l+1))Pl(cos ) (263)

, , =0,(Br(l+1) r 0) (Arl r ) A = 0 .

T2(r, , ) =l,m

ArlPl(cos ) (264)

:

ARl =2l + 1

2

11

Pl(cos )T2(r = R, , )d cos =

=2l + 1

2[

01

Pl(cos )T2(r = R, , )d cos

+

10

Pl(cos )T2(r = R, , )d cos ] =

=2l + 1

2225

10

Pl(cos )d cos (265)

102

28 , .

T2(r, , ) = 225[1

2+

3

4

r

RP1(cos ) 7

16(r

R)3P3(cos ) + ..] (266)

:

T (r, , ) = 75 + 225[1

2+

3

4

r

RP1(cos ) 7

16(r

R)3P3(cos ) + ..] (267)

: - S R, ,

V (R) = Q4R

l=0

Pl(cos )

Pl(cos ) Legendre Q . S .:

V (r, , ) Laplace . (260). . . r V (r , , ) 0. r A = 0 :

V (r, , ) =l,m

Blr(l+1)Y ml (, ) (268)

28 Legendre Pl(x) =

P l+1(x)P l1(x)2l+1

103

R m = 0 :

V (r, , ) =l=0

Blr(l+1)Pl(cos ) (269)

Bl .

V (r = R, , ) =Q

4R

m

Pm(cos ) =

=l=0

BlR(l+1)Pl(cos ) (270)

BlR(l+1) =

(2l + 1)Q

8R

11

m

Pm(cos )Pl(cos )d cos =

=(2l + 1)Q

8R

m

11

Pm(cos )Pl(cos )d cos =

=(2l + 1)Q

8R

m

ml2

2l + 1=

Q

4R(271)

:

V (r, ) =l

Q

4R(R

r)l+1Pl(cos ) (272)

. , ... R.

S

E.dS = Qtotal

.

En = V (r, )r

= l=0

Q

4R

Rl+1

rl+2Pl(cos )(l 1)

104

En .

Qtotal = 20

0

l=0

Q

4R

Rl+1

Rl+2Pl(cos )(l 1)R2 sin dd =

=l=0

Q(l + 1)

2

0

Pl(cos ) sin d =

=l=0

Q(l + 1)

2

2

2l + 10l = Q (273)

- Legendre 29

290

Pl(cos )sin d =11

Pl(cos )P0(cos )d cos = 22l+1l0

105

4.4

1. Laplace z = x + iy z = x iy

2

x2+

2

y2= 4

2

zz

2. L . T1 = 1000 T2 = 250 . T (x, y, z) .

3. Laplace -. .(2 = 1

rr

(r r

) + 1r2

2

2+

2

z2)

4. R . T (R, , ) = T0 cos() sin() ( ).

5. L R . - 1000 .

6. R . T [R, ] = 100(1 cos()) - T (, ) . ,

106

7. , R1 , R2 , ( R1 < R2 ) V (R1, ) = V0 V2 = V0 (1 + cos ) . .

107