Post on 08-Jan-2016
description
-
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