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,
ϕ
.
,
ϕ
,
E
, (65) .
7. .
0
r
( )
ϕf
,
ϕ
–
(. .3).
( )
ϕ;ru
, ,
,
0
2
2
2
2
=
∂
∂
+
∂
∂
y
u
x
u
( ) ( )
ϕϕ fru =;
0
. (67)
(67) .
0
11
2
2
22
2
=
∂
∂
+
∂
∂
+
∂
∂
ϕ
u
rr
u
rr
u
(68)
( )
ϕ;ru
.
( ) ( ) ( )
rRru ϕϕ Φ=;
. (69)
(68) (67), :
( ) () ( ) () ( ) ()
0
2
=Φ
′′
+
′
Φ+
′′
Φ rRrRrrRr ϕϕϕ
( )
( )
( ) ( )
( )
2
2
k
rR
rRrrRr
−=
′
+
′′
−=
Φ
Φ
′′
ϕ
ϕ
. (70)
(70) :
( ) ( )
0
2
=Φ+Φ
′′
ϕϕ k
,
( ) ( ) ( )
0
22
=−
′
+
′′
rRkrRrrRr
(71)
(71):
( )
ϕϕϕ kBkA sincos +=Φ
. (72)
(71)
( )
m
rrR =
.
(71)
( )
01
2122
=−+−
−− mmm
rkrmrrmmr
0
22
=− km
.
(71)
( )
kk
DrrrR
−
+=
. (73)
(72) (73) (69):
( ) (
)
(
)
,....3,2,1,sincos; =++=
−
krDrkBkAru
k
k
k
kkkk
ϕϕϕ
(74)
0
=
k
:
, ϕ .
,
ϕ, E, (65) .
7. .
r0
f (ϕ ) , ϕ –
( . .3). u (r ; ϕ ) , ,
,
∂ 2u ∂ 2u
+ =0
∂ x2 ∂ y 2
u (r0 ; ϕ ) = f (ϕ ) . (67)
(67) .
∂ u 1 ∂u 1 ∂ u
2 2
+ + =0 (68)
∂ r2 r ∂ r r2 ∂ϕ 2
u (r ; ϕ ) .
u (r ; ϕ ) = Φ (ϕ )R(r ) . (69)
(68) (67), :
r Φ(ϕ )R′′(r ) + r Φ(ϕ )R′(r ) + Φ′′(ϕ )R(r ) = 0
2
Φ′′(ϕ ) r 2 R′′(r ) + r R′(r )
=− = −k 2 .
Φ(ϕ ) R(r )
(70)
(70) :
Φ′′(ϕ ) + k Φ(ϕ ) = 0 ,
2
r R′′(r ) + r R′(r ) − k 2 R(r ) = 0
2
(71)
(71):
Φ (ϕ ) = A cos kϕ + B sin kϕ . (72)
(71) R(r ) = r . m
(71)
r m(m − 1)r
2 m− 2
+ rmr m−1
−k r =0 2 m
m2 − k 2 = 0 .
(71)
R(r ) = r + Dr − k . k
(73)
(72) (73) (69):
uk (r; ϕ ) = ( Ak cos kϕ + Bk sin kϕ ) ( k )
r k + Dk r − k , k = 1,2,3,.... (74)
k =0 :
