Воеводин Вл. В. Численные методы, параллельные вычисления и информационные технологии
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d = 0.5
d = 1
d = 5
Re(h)
151050
−5−10−15
0.018
0.016
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
−0.002
−0.004
d = 0.5
d = 1
d = 5
Im(h)
151050
−5−10−15
0.01
0
−0.01
−0.02
−0.03
−0.04
−0.05
−0.06
h(N)
p
Q
(M)
p
Q
(N) = 2ϕ
R
(N), N ∈ σ.
h(N)
(a), p
Q
(N)
(b) p
Q
(N)
g(N),
g(N), h(N), OZ
I(M) =
1
4π
Z
σ
p
Q
(N)
∂
∂n
M
∂F(M, N)
∂n
N
dσ
N
,
F(M, N,
I(M) =
~
n
M
·
~
Φ(M),
~
Φ(M) = Φ
x
(M)
~
i + Φ
y
(M)
~
j + Φ
z
(M)
~
k,
~
n
M
=
α
M
~
i +
β
M
~
j+
+
γ
M
~
k.
α
M
, β
M
, γ
M
~
n
M
,
Φ
x
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂x
M
∂n
N
dσ
N
;
Φ
y
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂y
M
∂n
N
dσ
N
;
Φ
z
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂z
M
∂n
N
dσ
N
.
~
n
N
=
α
~
i + β
~
j + γ
~
k, α, β, γ
~
n
N
.
F(M, N) ∇
M
F(M, N) = −∇
N
F(M, N) Φ
x
(M) = −
R
σ
p
Q
(N)×
×
α
∂
2
∂x
2
N
dσ
N
−
R
σ
p
Q
(N)
β
∂
2
∂x
N
∂y
N
dσ
N
−
R
σ
p
Q
(N)
γ
∂
2
∂x
N
∂z
N
dσ
N
.
Φ
x
(M)
Φ
x
(M) = −I
1
(M) − I
2
(M) − I
3
(M),
I
1
(M), I
2
(M), I
3
(M)
I
1
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂x
2
N
αdσ
N
;
I
2
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂x
N
∂y
N
βdσ
N
;
I
3
(M) =
Z
σ
p
Q
(N)
∂
2
F(M, N)
∂x
N
∂z
N
γdσ
N
.
F(M, N)
N 6= M
∂
2
F(M, N)
∂x
2
N
= −
∂
2
F(M, N)
∂y
2
N
−
∂
2
F(M, N)
∂z
2
N
− κ
2
F(M, N)
p
Q
(N),
p
Q
(N)
∂
2
F(M, N)
∂x
2
N
= −p
Q
(N)
∂
2
F(M, N)
∂y
2
N
−
−p
Q
(N)
∂
2
F(M, N)
∂z
2
N
− κ
2
p
Q
(N)F(M, N).
A(x, y)
B(x, y)
A(x, y)
∂B(x, y)
∂x
2
=
∂
∂x
A(x, y)
∂B(x, y)
∂x
−
∂A(x, y)
∂x
∂B(x, y)
∂x
;
A(x, y)
∂B(x, y)
∂x∂y
=
∂
∂x
A(x, y)
∂B(x, y)
∂y
−
∂A(x, y)
∂x
∂B(x, y)
∂y
.
A(x, y) = p
Q
(N) B(x, y) =
= F(M, N),
∂
∂y
N
,
p
Q
(N)
∂
2
F(M, N)
∂y
2
N
=
∂
∂y
N
p
Q
(N)
∂F(M, N)
∂y
N
−
−
∂p
Q
(N)
∂y
N
∂F(M, N)
∂y
N
,
∂
∂x
∂
∂z
N
,
p
Q
(N)
∂
2
F(M, N)
∂z
2
N
=
∂
∂z
N
p
Q
(N)
∂F(M, N)
∂z
N
−
−
∂p
Q
(N)
∂z
N
∂F(M, N)
∂z
N
.
I
1
(M)
I
1
(M) =
Z
σ
∂p
Q
(N)
∂y
N
∂F(M, N)
∂y
N
+
∂p
Q
(N)
∂z
N
∂F(M, N)
∂z
N
−
− κ
2
p
Q
(N)F(M, N)
αdσ −
Z
σ
∂
∂y
N
−p
Q
(N)
∂F(M, N)
∂y
N
−
−
∂
∂z
N
p
Q
(N)
∂F(M, N)
∂z
N
αdσ.
p
Q
∂
2
F(M, N)
∂x
N
∂y
N
,
A(x, y) = p
Q
(N) B(x, y) = F(M, N),
∂
∂x
∂
∂x
N
,
∂
∂y
∂
∂y
N
p
Q
(N)
∂
2
F(M, N)
∂x
N
∂y
N
=
∂
∂x
N
p
Q
(N)
∂F(M, N)
∂y
N
−
−
∂p
Q
(N)
∂x
N
∂F(M, N)
∂y
N
.
∂
∂x
∂
∂x
N
,
∂
∂y
∂
∂z
N
p
Q
(N)
∂
2
F(M, N)
∂x
N
∂z
N
=
∂
∂x
N
p
Q
(N)
∂F(M, N)
∂z
N
−
−
∂p
Q
(N)
∂x
N
∂F(M, N)
∂z
N
.
I
2
(M) I
3
(M) :
I
2
(M) = −
Z
σ
∂p
Q
(N)
∂x
N
∂F(M, N)
∂y
N
βdσ
N
+
+
Z
σ
β
−
∂
∂x
N
−p
Q
(N)
∂F(M, N)
∂y
N
dσ
N
,
I
3
(M) = −
Z
σ
∂p
Q
(N)
∂x
N
∂F(M, N)
∂z
N
γdσ
N
+
+
Z
σ
γ
−
∂
∂x
N
−p
Q
(N)
∂F(M, N)
∂z
N
dσ
N
.
I
1
(M), I
2
(M) I
3
(M),
Φ
x
(M)
Φ
x
(M) = −
Z
σ
α
∂
∂y
N
−p
Q
(N)
∂F(M, N)
∂y
N
−
−
∂
∂z
N
p
Q
(N)
∂F(M, N)
∂z
N
+
−
∂
∂x
N
−p
Q
(N)
∂F(M, N)
∂y
N
×
×
β+ γ
∂
∂x
N
p
Q
(N)
∂F(M, N)
∂z
N
dσ−
Z
σ
∂p
Q
(N)
∂y
N
×
×
∂F(M, N)
∂y
N
+
∂p
Q
(N)
∂z
N
∂F(M, N)
∂z
N
− κ
2
p
Q
(N)F(M, N)
α−
−
β
∂p
Q
(N)
∂x
N
∂F(M, N)
∂y
N
−
γ
∂p
Q
(N)
∂x
N
∂F(M, N)
∂z
N
dσ.
Z
σ
∂R
∂y
−
∂Q
∂z
α +
∂P
∂y
−
∂R
∂x
β+
+
∂Q
∂x
−
∂P
∂y
γ
dσ =
Z
L
σ
Pdx + Qdy + Rdz,
L
σ
σ. P = 0,
Q = p
Q
(N)
∂F(M, N)
∂z
N
R = −p
Q
(N)
∂F(M, N)
∂y
N
Z
σ
α
∂
∂y
N
−p
Q
(N)
∂F(M, N)
∂y
N
−
∂
∂z
N
p
Q
(N)
∂F(M, N)
∂z
N
+
+
β
−
∂
∂x
N
−p
Q
(N)
∂F(M, N)
∂y
N
+
γ
∂
∂x
N
p
Q
(N) ×
×
∂F(M, N)
∂y
N
dσ =
Z
L
σ
p
Q
(N)
∂F(M, N)
∂z
N
dy
N
−
∂F(M, N)
∂y
N
dz
N
.
Φ
x
(M) = −
Z
L
σ
p
Q
(N)
∂F(M, N)
∂z
N
dy
N
−
∂F(M, N)
∂y
N
dz
N
−
−
Z
σ
∂p
Q
(N)
∂y
N
∂F(M, N)
∂y
N
+
∂p
Q
(N)
∂z
N
∂F(M, N)
∂z
N
−
− κ
2
p
Q
(N)F(M, N)
α −
∂p
Q
(N)
∂x
N
∂F(M, N)
∂y
N
β−
−
∂p
Q
(N)
∂x
N
∂F(M, N)
∂z
N
γ
dσ.
Φ
x
(M)
σ σ
1
σ
2
, σ = σ
1
∪σ
2
.
σ
1
Φ
x
(M)
Z
L
σ
1
f
1
(N)dl
N1
,
σ
2
Z
L
σ
2
f
2
(N)dl
N2
.
L
σ
1
= L
σ
2
,
f
1
(N) = f
2
(N), dl
N1
= −dl
N2
.
σ
Z
L
σ
p
Q
(N)
∂F(M, N)
∂z
N
dy
N
−
∂F(M, N)
∂y
N
dz
N
= 0.
∂p
Q
(N)
∂y
N
∂F(M, N)
∂y
N
=
∂
∂y
N
F(M, N)
∂p
Q
(N)
∂y
N
−
−F(M, N)
∂
2
p
Q
(N)
∂y
2
N
;
∂p
Q
(N)
∂z
N
∂F(M, N)
∂z
N
=
∂
∂z
N
F(M, N)
∂p
Q
(N)
∂z
N
−
−F(M, N)
∂
2
p
Q
(N)
∂z
2
N
.
β
∂
∂x
N
F(M, N)
∂p
Q
(N)
∂y
N
= F(M, N)
∂
2
p
Q
(N)
∂x
N
∂y
N
×
×
β + β
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
;
γ
∂
∂x
N
F(M, N)
∂p
Q
(N)
∂z
N
= F(M, N)
∂
2
p
Q
(N)
∂x
N
∂z
N
×
×
γ + γ
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
,
Φ
x
(M) = κ
2
Z
σ
p
Q
(N)F(M, N)
αdσ −
Z
σ
−
∂
2
p
Q
(N)
∂y
2
N
+
+
∂
2
p
Q
(N)
∂z
2
N
F(M, N)
α + β
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂y
N
+
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
+
γ
F(M, N) ×
×
∂
2
p
Q
(N)
∂x
N
∂z
N
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
dσ−
−
Z
σ
α
∂
∂y
N
F(M, N)
∂p
Q
(N)
∂y
N
−
∂
∂z
N
−F(M, N) ×
×
∂p
Q
(N)
∂z
N
+
−
∂
∂x
N
F(M, N)
∂p
Q
(N)
∂y
N
β + γ×
×
∂
∂x
N
F(M, N)
∂p
Q
(N)
∂z
N
dσ.
Φ
x
(M) = κ
2
Z
σ
p
Q
(N)F(M, N)
αdσ −
Z
σ
−
∂
2
p
Q
(N)
∂y
2
N
+
+
∂
2
p
Q
(N)
∂z
2
N
F(M, N)
α + β
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂y
N
+
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
+
γ
F(M, N) ×
×
∂
2
p
Q
(N)
∂x
N
∂z
N
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
dσ.
p
Q
(M),
p
Q
(M)
m = 1. p
Q
(M)
−
∂
2
p
Q
(N)
∂y
2
N
+
∂
2
p
Q
(N)
∂z
2
N
=
∂
2
p
Q
(N)
∂x
2
N
+ κ
2
p
Q
(N).
Φ
x
(M) =
Z
σ
F(M, N)
∂
2
p
Q
(N)
∂x
2
N
α +
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂y
N
+
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
β+
+ γ
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂z
N
+
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
−
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
dσ.
Φ
y
(M) =
Z
σ
F(M, N)
∂
2
p
Q
(N)
∂y
2
N
β +
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂y
N
+
+
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
−
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
α+
+ γ
F(M, N)
∂
2
p
Q
(N)
∂y
N
∂z
N
+
∂F(M, N)
∂y
N
∂p
Q
(N)
∂z
N
−
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂y
N
dσ.
Φ
z
(M) =
Z
σ
F(M, N)
∂
2
p
Q
(N)
∂z
2
N
γ +
F(M, N)
∂
2
p
Q
(N)
∂x
N
∂z
N
+
+
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
−
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
α+
+ β
F(M, N)
∂
2
p
Q
(N)
∂y
N
∂z
N
+
∂F(M, N)
∂z
N
∂p
Q
(N)
∂y
N
−
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂z
N
dσ.
I(M)
Z
σ
α
M
β
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
+
+
α
M
γ
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
+
+
α β
M
∂F(M, N)
∂y
N
∂p
Q
(N)
∂x
N
−
∂F(M, N)
∂x
N
∂p
Q
(N)
∂y
N
+
+
β
M
γ
∂F(M, N)
∂y
N
∂p
Q
(N)
∂z
N
−
∂F(M, N)
∂z
N
∂p
Q
(N)
∂y
N
+
+
α γ
M
∂F(M, N)
∂z
N
∂p
Q
(N)
∂x
N
−
∂F(M, N)
∂x
N
∂p
Q
(N)
∂z
N
+
+
β γ
M
∂F(M, N)
∂z
N
∂p
Q
(N)
∂y
N
−
∂F(M, N)
∂y
N
∂p
Q
(N)
∂z
N
dσ,
∇
N
F(M, N) =
= −∇
M
F(M, N), ∇
N
F(M, N) = −∇
M
1
p
Q
(N)
I(M) =
Z
σ
F(M, N)
∂
∂n
N
~
n
M
·∇
M1
p
Q
(N)
dσ
m
I(M) =
m
X
i=1
Z
σ
F(M, N)
∂
∂n
N
~
n
M
·∇
Mi
p
Q
(N)
dσ.
p
Q
(M)
p
Q
(M)
−
∂
2
p
Q
(N)
∂y
2
N
+
∂
2
p
Q
(N)
∂z
2
N
=
∂
2
p
Q
(N)
∂x
2
N
.
I(M) = κ
2
Z
σ
p
Q
(N)F(M, N)
~
n
M
·
~
n
M
dσ+
+
m
X
i=1
Z
σ
F(M, N)
∂
∂n
N
~
n
M
· ∇
Mi
p
Q
(N)
dσ.
M
i
, i = 1, 2, . . . , m σ,
M →N,
σ
M.
I(M)
(iκr
MN
),
//
//
//