(- ), - , (, , ) - . - - , , . - , - :
u (x,y) =v (x,y) +w (x,y), v (x,y) w(x,y) - , -
- .
f (j)- .
u (r, j) = R (r) F(j) (89)
(87) . -
(90)
F(j) 2p. F(j),
,
(91)
(80), .
n ³ 1
r2R² + rR¢ n2R = 0,
R=r m. - m
m(m 1) r m+m r m n 2 r m=0Û m2 n 2=0,
m = n, Rn (r) = rn; r-n. (92)
n =0, , , rR² + R¢ =0
R 0(r) = 1; ln r.
(89) - ,
1, r cosj, r sin j, , rn cos n j, rn sin n j, .
,
(93)
r j, , . . (87). (93) (88),
(94)
(95)
(96) , (87 88) (93), (95-96).
1. , , (93) r < a. r-n ln r,
r =0.
, r > a, r-n ln r, -
(97)
a < r < b - (92 92¢)
, (93), (97 98) .
|
|
2. , f (j) -
, ..
An Bn (94) . (87 88) .
:
u (r, j,q) = R (r) Y (j,q), (99)
(101)
l. 0 £ r £ a, 0 £ q £ p,0 £ j £ 2p, j 2p , Y (j+2p,q)= Y (j,q).
(102)
,
(102)
(103)
F(j)
(91)
m= m, F m (j)=cos m j; sin m j; m = .
(73)
(104)
(105)
(104) ,
, sin2q= 1- x 2,
(104¢)
(105)
(105′)
(104¢) (105¢) , (., , [3], . 115)
q, - (104), (105):
(106)
(106) F m (j), (102)
(107)
, - .
1. - , .. m 1¹ m 2 n 1¹ n 2
(108)
(101), -
R=rm. ,
m= n; (n +1)
(109)
(109) (107), :
. -
(110)
, .
nm (110) (100),
(108)
(111)
m =0:
(112)
| m | ³ 1,
(113)
(99),(100), , (110), (111),(113).
3. , -
(114)
- , . .
(115)
n
(116)
n .
|
|
(107), (115) (116) :
n
(117)
4. (110)
(118)
a < r < b - (110) (118).
5. - - , .
248.
. , u=v (x,y) +w (x,y), v (x,y)
v (x,y) =X (x) Y (y),
X (x) -
( )
.
, -
w (x,y)
w (x,y) = X (x) Y (y).
Y (y)
Yj (y) Xj (x) j, - =0 =
y =0
w/y=b=0
,
v (x,y) w (x,y),
249.
W={(x, y): 0 £ x £ a, 0 £ y £ b }
u (x,y) = X (x) Y (y)
y =0 y=b
()
k =0 0()= 0 + 0. - W
, , , x =0 x=a.
k =2 n A 2 n =0, -
250. = =b , =0 =0 , Q.
. - (k )
,
,
, v (x,y):
, -
,
, :
()
W
- An Bn ,
,
251. u (r,z) - , u (r,h) =f (r).
. W, j, -
, u (r,z) =R (r) Z (z). -
|
|
u/r=a =0 , R (a)=0, k (r)= r r =0 :
r [0, ] (. 2 3).
- :
, , J 0(0)=1, N 0(0)=¥ J 0(x):
,
u/ z=0=0 ,
(. (81))
:
252. u (r,z) -
, q.
. -
(k ) W
W ,
. - u/z=h =0,
n Bn
253.
. u 0(r, j)= v (r)sin2j.
, v=cr 4,
,
w (r,j),
w (r, j) -
(93),
,
254.
, k
.
u (r, j,q,)= R (r) Y (j,q), :
(72), 2p- j. - l =n (n +1):
R (r) -
Z (r):
r= 0
, - :
(119)
, r=a
d = 4 m = 0 d =2
Anm (119) .
255. u (r, j,q ) 1 < r < 2,
. (80), (88) 4
(120)
(120) , -
|
|
,
256.
257. =0 q, u 1.
258. u (x,y) - =0- u 1, . - W .
259. u (x,y) - = =b , , - q.
260. u (0, y)=0, u (a,y)=0, u (x,0)= A (a x), u (x,¥)=0.
261. u (0, y)=0,
262. u (x,y,z) {0£ x £ a, 0£ y £ b, 0£ z £ c }, , u 1.
263. u (r,z) - h, u (r,z)/ r=a=f (z).
264. {0£ r £ a, 0£j£2p, 0£ z £ h }, u1, .
265. {0£ r £ a, 0£j£2p, 0£ z £ h }, - , , - 0. .
1. .. . . 1 5. .: , 1958-1960.
2. .., . . . .: , 1965.
3. .., .. . .: , 1965.
4. . ., . . . .: , 1967.
5. . . . .: , 1972.
6. . . . .: , 1969.
7. . . .- .: , 1966.
8. . . . .: , 1967.
9. . ., . ., . . . .: , 1962.
10. . . - . .: , 1961.
11. . ., . . - . .: , 1966.
12. . ., . ., . . . .:, 1956.
13. . . . .: , 1968.
14. ./ ... .: , 1974.
15. . ., . . . .: , 1977.
16. .. . .: , 1998.
......3
1. 充....4
2. 充11
3.
...21
4. ..27
5. ...39
6. 酅49
7.
ⅅ60
8. ..81
.103
=:
,
.
. .
. .
. .
|
|
00.00.2006. .
. . . . . . . . 4,0.
. .
.
02330/0056804 02.03.2004.
220050, , , 4.
= .
.
02330/0056850 30.04.2004.
220050, , . , 6.