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,
.;sin;cos zzryrx
=
=
=
ϕ
ϕ
,
2
2
2
2
2
2
2
2
2
2
11
ϕ∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
+
∂
∂ u
rr
u
rr
u
z
u
y
u
x
u
.
0
11
2
2
2
2
=
∂
∂
+
∂
∂
+
∂
∂
ϕ
u
rr
u
rr
u
. (81)
.
(81), (80)
z
,
z
. ,
–
D
,
2
1
22
1
: RyxK =+
2
2
22
2
: RyxK =+
:
( ) ( )
2211
;,; uRuuRu ==ϕϕ
. (82)
.
ϕ
, ,
ϕ
.
(81) :
0
1
2
2
=
∂
∂
+
∂
∂
r
u
r
r
u
.
, :
21
ln Cru +=
. (83)
1
2
(82):
22122111
ln,ln CRuCRu +=+=
.
:
.
ln
lnln
,
ln
1
2
1122
2
1
2
12
1
R
R
RuRu
R
R
uu
−
=
−
=
1
2
(83), :
.
ln
lnln
1
2
2
1
1
2
R
R
R
r
u
R
r
u
u
−
=
,
x = r cos ϕ ; y = r sin ϕ ; z = z.
,
∂ 2u ∂ 2 u ∂ 2 u ∂ 2u 1 ∂ u 1 ∂ 2 u
+ + = + +
∂ x 2 ∂ y 2 ∂ z 2 ∂r 2 r ∂r r ∂ϕ 2 .
∂ 2u 1 ∂ u 1 ∂ 2 u
+ + = 0. (81)
∂r 2 r ∂r r ∂ϕ 2
.
(81), (80) z,
z. ,
– D,
K1 : x 2 + y 2 = R12 K 2 : x 2 + y 2 = R22
:
u(R1;ϕ ) = u1, u(R2 ;ϕ ) = u2 . (82)
.
ϕ, , ϕ.
(81) :
∂ 2u 1 ∂ u
+ = 0.
∂r 2 r ∂r
, :
u= 1 ln r + C2 . (83)
1 2 (82):
u1 = 1 ln R1 + C2 , u 2 = 1 ln R2 + C2 .
:
u 2 − u1 u 2 ln R2 − u1 ln R1
1 = , 2 = .
R2 R2
ln ln
R1 R1
1 2 (83), :
r r
u 2 ln − u1 ln
R1 R2
u= .
R2
ln
R1
