class="bi x0 y0 w1 h1"
J
R T
J T
−1
. v(ξ) T v(x).
T

pT
−1
[A(ξ)u(ξ)](x) = f(x), x ∈ Ω,
p

u(x) = g(x), x ∈ Ω

, u(ξ) = A
2
(ξ)v(ξ),
Ω, Ω

J, Ω∪Ω

= J, p
p

Ω Ω

, A(ξ) =
A
1
(ξ)

[u(ξ)](x) = 0, x ∈ R \(a, b).
u(ξ) ∈ S

∩ C
∞
(R) f(x)
H
−p/2
(a, b), A(ξ)

pF
−1
[A(ξ)

u(ξ)](x) = f(x), x ∈ Ω,
p

F
−1
[

u(ξ)](x) = g(x), x ∈ Ω

= R \ Ω,
Ω

u
f, g A(ξ)
p p



u(ξ)](x). A(ξ)
A.
Σ
α
(R), Σ
α
+
(R), Σ
α
◦
(R)

A(ξ) ∈ Σ
α
◦
(R), A(ξ) ξ ∈ R,
f(x) ∈ H
−α/2
(Ω), g(x) ∈ H
α/2
(Ω

),

u(ξ)

u(ξ) = F [u](ξ), u ∈ H
α/2
(R).

(R), H
s
◦
(Ω), H
s
◦,◦
(Ω), H
s
(Ω).
X n
X X
n
. X
n
u
u = (u
1
, u
2
, . . . , u
n
),
S
n
= S × S . . . × S, S

n
= S

× S

)
T
, H
s
(R) = H
s
1
(R) × H
s
2
(R) × . . . × H
s
n
(R),
H
s
◦
(Ω) = H
s
1
◦
(Ω) × H
s
2
◦
(Ω) × . . . × H
s
n
◦
(Ω),

H
s
◦
(Ω)
(u, v)
s
=
n

j=1
(u
j
, v
j
)
s
j
, ||u||
s
=

n

j=1
||u
j
||
2
s
j

:=

n

j=1
||u
j
||
2
H
s
j
(Ω)

1/2
.
(H
s
(R))
∗
H
s
(R). (H
s
(R))
∗
H
−s
(R).
f ∈ H

](ξ).
Ω ⊂ R, u = (u
1
, u
2
, . . . , u
n
)
T
∈ H
s
(Ω), f ∈ H
−s
(Ω)
f = (
1
f
1
, 
2
f
2
, . . . , 
n
f
n
)
T
f Ω R
H

Φ(u) = [f , u] ||Φ|| = ||f||
H
−s
(Ω)
.
A(ξ) ∈ Σ
α
(R), u ∈ H
s
,

u(ξ) = F [u](ξ).
A
(Au)(x) := F
−1
[A(ξ)

u(ξ)](x), x ∈ R
n
,
A(ξ) = ||a
ij
(ξ)||
n×n
n, u = (u
1
, u
2
, . . . , u
n

α = (α
1
, α
2
, . . . , α
n
)
T
. A(ξ) = ||a
ij
(ξ)||
n×n
,
Σ
α
(R)
a
ii
(ξ) ∈ σ
α
i
(R), a
ij
(ξ) ∈ σ
β
ij
(R), β
ij
≤
1

1
, w
2
, . . . , w
n
)
T
∈ C
n
,
C
1
w w.
A(ξ) ∈ Σ
α
(R) Σ
α
◦
(R),
Rew
T
Aw ≥ 0, w = (w
1
, w
2
, . . . , w
n
)
T
∈ C

,α/2
=


∞
−∞
F [u
T
](ξ)A
+
(ξ)F [u](ξ)dξ

1/2
,
Ω R.
H
s
(Ω) H
s−ε
(Ω)
ε = (ε
1
, ε
2
, . . . , ε
n
)
T
> 0(⇔ ε
j

2
(x), . . . ,
f
n
(x))
T
∈ (D

(Ω))
n
, g(x) = (g
1
(x), g
2
(x), . . . , g
n
(x))
T
∈ (D

(Ω

))
n
Ω Ω

A(ξ) = ||a
ij
(ξ)||
n×n

v(ξ)](x) = f (x) − pF
−1
[A(ξ)



g(ξ)](x), x ∈ Ω,
v = F
−1
[ˆv] ∈ H
α/2
◦
(Ω)
v + 

g = u ∈ H
α/2
(R)


g g Ω

R
h(x) = f(x) − pF
−1
[A(ξ)



g(ξ)](x),

C
A(ξ) ∈ Σ
α/2
+
(R), f ∈ H
−α/2
(Ω),
g ∈ H
−α/2
(Ω

).
u = F
−1
[ˆu] ∈ H
α/2
(R)
A(ξ) ∈ Σ
α
◦
(R)
Ω R
A
+
(ξ) ∈ Σ
α
+
(R) A(ξ),
B(ξ) := A(ξ) −A
+

α/2
(R).
•
H
s
(R), H
s
◦
(Ω), H
s
◦,◦
(Ω),
H
s
(Ω). H
−s
(R)
(H
s
(R))
∗
H
s
(R)
H
s
◦
(Ω)
(Au)(x) :=
F

−
∂Φ(x, 0)
∂y
= f
1
(x), x ∈ (a, b),
Φ(x, 0) = 0, x ∈ R \(a, b),
,



∂Φ(x, h)
∂y
= f
2
(x), x ∈ (a, b),
Φ(x, h) = 0, x ∈ R \(a, b),
f
1
, f
2
u
1
(ξ), u
2
(ξ) :

F
−1
[A(ξ)



|ξ|cosh(|ξ|h)
sinh(|ξ|h)
−
|ξ|
sinh(|ξ|h)
−
|ξ|
sinh(|ξ|h)
|ξ|cosh(|ξ|h)
sinh(|ξ|h)




.

pF
−1
[A(ξ)

u(ξ)](x) = f(x), x ∈ (a, b),
p

F
−1
[

u(ξ)](x) = 0, x ∈ R \(a, b),










1
πi

b
a
v
1
(t)dt
x − t
+

b
a
v
1
(t)
11
(x − t)dt +

b
a

(t)
22
(x − t)dt = −if
2
(x),
v
m
(t) ∈ L
2
ρ
(a, b), m = 1, 2; a < x < b,
v
m
∈ O
1
(a, b),

b
a
v
m
(x)dx = 0, (m = 1, 2),
u
m
(x) =
1
2

b
a

v
m
(t) =
φ
m
(t)
ρ(t)
, φ
m
(t) φ
m
(t) =
∞

j=1
A
(m)
j
T
j
[η(t)], A
(m)
j
{A
(m)
j
}
∞
j=1
∈

(mk)
nj
=

b
a
ρ(x)U
n
[η(x)]dx

b
a
T
j
[η(t)]
ρ(t)
l
mk
(x − t)dt,
F
(m)
n
= −i

b
a
ρ(x)U
n
[η(x)]f
m

2n−1,2j+1
= −
4i
b − a
C
(11)
nj
, C
2n,2j+1
= −
4i
b − a
C
(12)
nj
,
C
2n−1,2j+2
= −
4i
b − a
C
(21)
nj
, C
2n,2j+2
= −
4i
b − a
C

}
∞
n=1
∈ 
2
.
Φ(x, y)
Ψ(x, y)
∂
2
Φ
∂x
2
+
∂
2
Φ
∂y
2
= 0,
∂
2
Ψ
∂x
2
+
∂
2
Ψ
∂y

1
(ξ), u
2
(ξ)

F
−1
[|ξ|A
0
(ξ)

u(ξ)](x) = f(x), x ∈ (a, b),
F
−1
[

u(ξ)](x) = 0, x ∈ Ω

= R \ (a, b),
u
1
(x) = 2µu(x, 0), u
2
(x) = 2µv(x, 0),

u(ξ) = F [u(x)](ξ) =
(F [u
1
(x)], F [u
2

(ξ) a
22
(ξ)


,
a
11
(ξ) =
2(1 − ν)[cosh(|ξ|h) sinh(|ξ|h) + |ξ|h]
4(1 − ν)
2
+ |ξ|
2
h
2
+ (3 − 4ν) sinh
2
(|ξ|h)
,
a
21
(ξ) = a
12
(ξ) =
(1 − 2ν) sinh
2
(|ξ|h) + |ξ|
2
h

(ξ)

u(ξ)](x) = f(x), x ∈ (a, b),
p

F
−1
[

u(ξ)](x) = 0, x ∈ R \(a, b),
p p

(a, b) R \ (a, b).
f ∈ H
−α/2
(a, b).
u ∈ H
α/2
◦
(a, b)
τ
0
(x) σ
0
(x)
f(x) = (f
1
(x), f
2
(x))

∂x
4
+ 2
∂
4
Φ
∂x
2
∂y
2
+
∂
4
Φ
∂y
4
= 0
Φ


y=0
= r
1
(x), x ∈ R, Φ


y=h
= r
2
(x), x ∈ R,

M[Φ]


y=h
= 0, x ∈ R \(−a, a),
M[Φ] = M[Φ](x, y) =
∂
2
Φ
∂y
2
+ ν
∂
2
Φ
∂x
2
, 0 < ν < 1.
u
1
(ξ), u
2
(ξ) :

F
−1
[A(ξ)

u(ξ)](x) =


(x) − F
−1
[a
1
(ξ)r
1
(ξ)](x) + F
−1
[a
2
(ξ)r
2
(ξ)](x),

f
2
(x) = f
2
(x) + F
−1
[a
2
(ξ)r
1
(ξ)](x) − F
−1
[a
1
(ξ)r
2

[A(ξ)

u(ξ)](x) =

f(x), x ∈ (−a, a),
p

u := p

F
−1
[

u(ξ)](x) = 0, x ∈ R \(−a, a).
p p

(−a, a) R \(−a, a).
r
1
(x) r
2
(x) ∈ H
3
2
(R), f
1
(x) f
2
(x) ∈ H
1

F
−1

a
∗
mn
(ξ)
|ξ|
u
n
(ξ)

(x) =

f
m
(x), x ∈ (−a, a),
u
m
(x) = F
−1
[u
m
](x) = 0, x ∈ R \(−a, a), (m = 1, 2),
a
∗
11
(ξ) = a
∗
22




1
2π

a
−a
ln



1
x − y



u
1
(t)dt +
2

m=1

a
−a
u
m
(t)k
1m

(t)k
2m
(x − t)dt =

f
2
(x), m = 1, 2.
k
12
(x) = k
21
(x) =
1
π

∞
0
a
∗
12
(ξ)
ξ
cos(ξx)dξ,
k
nn
(x) =
1
2π
ln


2
= 0, (−∞ < x < ∞, 0 < y < h)



−Φ(x, 0) = f
1
(x), x ∈ (a, b),
∂Φ
∂y
(x, 0) = 0, x ∈ R \(a, b),
,



∂Φ
∂y
(x, h) = f
2
(x), x ∈ (a, b),
Φ(x, h) = 0, x ∈ R \(a, b),
f
1
, f
2
u
1
(ξ), u
2
(ξ) :


tanh(|ξ|h)
|ξ|
−
1
cosh(|ξ|h)
1
cosh(|ξ|h)
|ξ|tanh(|ξ|h)




.

pF
−1
[A(ξ)

u(ξ)](x) = f(x), x ∈ (a, b),
p

F
−1
[

u(ξ)](x) = 0, x ∈ R \(a, b),
p p

(a, b) R \ (a, b)

2
ρ
−1
(a, b)
u
2
(x) =
1
2

b
a
v
2
(t) (x − t)dt, v
2
∈ L
2
ρ
(a, b) ⊂ H
−1/2
◦
(a, b)
(a, b)
•

2




(y),
−1 < y < 1, m = 1, 2,
K
∗
11
(y − τ) = K
∗
22
(y − τ) =
λ
π

∞
0
e
−z
sinh z
sin[zλ(y − τ )]dz,
K
∗
12
(y − τ) = K
∗
21
(y − τ) =
−λ
π

∞
0

−1
v
∗
1
(τ)dτ = 0,

1
−1
v
∗
2
(τ)dτ = 0.
K
∗
11
(y − τ) K
∗
12
(y − τ) n = 4,
f
∗
1
(y) = a
0
+ a
1
y + a
2
y
2

y
4
+ b
5
y
5
{a
j
}
5
j=0
, {b
j
}
5
j=0
N = 6 N = 7 λ =
1
10
{a
0
= 1, a
1
= −2, a
2
= 1, a
3
= 0, a
4
= −1, a

1 − τ
2

1.4994616τ
3
− 0.50012τ
4
− 0.999992τ
5
+ 1.000000τ
6

v
∗
2,6
(τ) =
1
√
1 − τ
2

0.896727 − 2.3814487τ + 0.706384τ
2

+
1
√
1 − τ
2


x − t
+
2

k=1

b
a
v
k
(t)
mk
(x − t)dt = −if
m
(x),
v
m
(t) ∈ L
2
ρ
(a, b), m = 1, 2; a < x < b,

11
(x) = 
22
(x) =
−i
π

∞

2
L
2
ρ
= ||φ
m
(t) − φ
m,N
(t)||
2
L
2
ρ
−1
= O

1
N
2k−1

, m = 1, 2.
•
N = 6 N = 7,
(Au)(x) := F
−1
[A(ξ)

u(ξ)](x)
Σ
α