Алексеев Г.В. Классические методы математической физики. Часть 2
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v(x)T
′′
(t) = [Lv](x)T (t) ≡
"
n
X
i,j=1
∂
∂x
i
a
ij
(x)
∂v
∂x
j
− a(x)v
#
T (t).
T
′′
(t)
T (t)
=
[Lv](x)
v(x)
= −λ.
λ
T
T
′′
(t) + λT (t) = 0,
v
Lv + λv = 0.
v
v|
Γ
= 0.
λ
λ
a
ij
a
Ω
0 ≤ λ
1
≤ λ
2
≤ ... ≤ λ
k
≤ ... , lim
k→∞
λ
k
= ∞.
v
k
Z
Ω
v
2
k
(x)dx = 1,
Z
Ω
v
k
(x)v
m
(x)dx = 0, λ
k
6= λ
m
,
λ
k
{v
k
}
λ
k
Lv
k
= −λ
k
v
k
v
k
Ω
λ
k
= −
Z
Ω
v
k
(x)Lv
k
(x)dx = −
Z
Ω
v
k
(x)
"
n
X
i,j=1
∂
∂x
i
a
ij
(x)
∂v
k
∂x
j
− a(x)v
k
(x)
#
dx.
R
n
Z
Ω
∂u
∂x
i
vdx = −
Z
Ω
u
∂v
∂x
i
dx +
Z
Γ
uv cos(n, x
i
)ds, i = 1, 2, ..., n,
n Γ
λ
k
=
Z
Ω
"
n
X
i,j=1
a
ij
(x)
∂v
k
∂x
i
∂v
k
∂x
j
+ a(x)v
2
k
(x)
#
dx.
Γ Ω
v
k
|
Γ
= 0
λ
k
≥
Z
Ω
"
β
n
X
i=1
∂v
k
∂x
i
2
+ a(x)v
2
k
(x)
#
dx.
λ
1
> 0
λ
k
v
k
λ λ
k
T
k
(t) = a
k
cos
√
λ
k
t + b
k
sin
√
λ
k
t,
a
k
b
k
u
k
u
k
(x, t) = T
k
(t)v
k
(x) = (a
k
cos
√
λ
k
t + b
k
sin
√
λ
k
t)v
k
(x).
λ
k
u(x, t) =
∞
X
k=1
(a
k
cos
√
λ
k
t + b
k
sin
√
λ
k
t)v
k
(x)
a
k
b
k
ϕ
0
(x) =
∞
X
k=1
a
k
v
k
(x), ϕ
1
(x) =
∞
X
k=1
b
k
p
λ
k
v
k
(x).
ϕ
0
ϕ
1
{v
k
} {v
k
}
a
k
b
k
{v
k
}
λ
k
> 0
a
k
=
Z
Ω
ϕ
0
(x)v
k
(x)dx, b
k
=
1
√
λ
k
Z
Ω
ϕ
1
(x)v
k
(x)dx, k = 1, 2, ... .
a
k
b
k
u
x
i
t
ϕ
0
ϕ
1
ρ
∂
2
u
∂t
2
= div(pgradu) − qu + f,
u|
t=0
= ϕ
0
(x),
∂u
∂t
t=0
= ϕ
1
(x)
Ω, αu + β
∂u
∂n
Γ
= 0
(0, T ]
Ω R
2
Ω = (0 <
x < l, 0 < y < h) x, y l h
∂
2
u
∂t
2
= a
2
∂
2
u
∂x
2
+
∂
2
u
∂y
2
Q
T
= Ω × (0, T ] ,
u|
x=0
= 0, u|
x=l
= 0, u|
y=0
= 0, u|
y=h
= 0 (0, T ]
u|
t=0
= ϕ
0
(x, y),
∂u
∂t
t=0
= ϕ
1
(x, y)
Ω.
l h
ϕ
0
ϕ
1
u(x, y, t) = v(x, y)T (t).
T
′′
(t)
a
2
T (t)
=
v
xx
+ v
yy
v
= −λ
2
,
λ
2
T
′′
(t) + a
2
λ
2
T (t) = 0
T
∂
2
v
∂x
2
+
∂
2
v
∂y
2
+ λ
2
v = 0
Ω,
v|
x=0
= 0, v|
x=l
= 0, v|
y=0
= 0, v|
y=h
= 0
λ
2
v
Ω
v(x, y) = X(x)Y (y).
Y
′′
Y
+ λ
2
= −
X
′′
X
= λ
2
1
,
λ
2
1
X
′′
(x) + λ
2
1
X(x) = 0, Y
′′
(y) + λ
2
2
Y ( y) = 0
X Y λ
2
1
λ
2
2
λ
2
−λ
2
2
+ λ
2
= λ
2
1
λ
2
= λ
2
1
+ λ
2
2
.
X Y
X(0) = 0, X(l) = 0, Y (0) = 0, Y (h) = 0.
X
′′
+ λ
2
1
X = 0 (0, l), X(0) = X(l) = 0 ,
Y
′′
+ λ
2
2
Y = 0 (0, h), Y (0) = Y (h) = 0.
λ
2
1,m
=
mπ
l
2
, X
m
(x) = sin
mπ
l
x, m = 1, 2, 3 , ...,
λ
2
2,n
=
nπ
h
2
, Y
n
(y) = sin
nπ
h
y, n = 1, 2, 3, ... .
(λ
2
1,m
, λ
2
2,n
)
λ
2
mn
= λ
2
1,m
+ λ
2
2,n
= π
2
m
2
l
2
+
n
2
h
2
v
mn
v
mn
(x, y) = sin
mπx
l
sin
nπy
h
.
(λ
2
mn
, v
mn
) m ∈ N
+
n ∈ N
+
u
λ
2
λ
2
mn
T
mn
(t) = a
mn
cosaλ
mn
t + b
mn
sinaλ
mn
t,
a
mn
b
mn
u
mn
(x, y, t) = (a
mn
cosaλ
mn
t + b
mn
sinaλ
mn
t)sin
mπx
l
sin
nπy
h
, m, n = 1, 2 , ... .
u(x, y, t) =
∞
X
m,n=1
(a
mn
cosaλ
mn
t + b
mn
sinaλ
mn
t)sin
mπx
l
sin
nπy
h
.
Q
T
x, y t
Q
T
u|
t=0
= ϕ
0
(x, y) =
∞
X
m,n=1
a
mn
sin
mπx
l
sin
nπy
h
,
∂u
∂t
|
t=0
= ϕ
1
(x, y) =
∞
X
m,n=1
aλ
mn
b
mn
sin
mπx
l
sin
nπy
h
.
ϕ
0
ϕ
1
l
Z
0
h
Z
0
v
2
mn
(x, y)dxdy =
l
Z
0
sin
2
mπx
l
dx
h
Z
0
sin
2
nπy
h
dy =
lh
4
.
a
mn
b
mn
ϕ
0
ϕ
1
a
mn
=
4
lh
l
Z
0
h
Z
0
ϕ
0
(x, y)sin
mπx
l
sin
nπy
h
dxdy,
b
mn
=
4
aλ
mn
lh
l
Z
0
h
Z
0
ϕ
1
(x, y)sin
mπx
l
sin
nπy
h
dxdy.
a
mn
b
mn
u
a
mn
= α
mn
sinϕ
mn
, b
mn
= α
mn
cosϕ
mn
,
u(x, y, t) =
∞
X
m,n=1
α
mn
sin
mπx
l
sin
nπy
h
sin(aλ
mn
t + ϕ
mn
).
u
mn
(x, y, t) = α
mn
sin
mπx
l
sin
nπy
h
sin(aλ
mn
t + ϕ
mn
).
u
mn
(m, n)
(x, y)
Ω
ω
mn
T
mn
ω
mn
= aλ
mn
= aπ
r
m
2
l
2
+
n
2
h
2
, T
mn
=
2π
ω
mn
=
2lh
a
√
m
2
h
2
+ n
2
l
2
,
ϕ
mn
A
mn
(x, y) = α
mn
sin
mπx
l
sin
nπy
h
.
λ
2
k
ω
k
= aλ
k
ϕ
k
(x) = sinkπx A
k
(x)
λ
2
mn
ω
mn
= aλ
mn
l = h = π λ
2
mn
ω
mn
λ
2
mn
= m
2
+ n
2
, ω
mn
= a
p
m
2
+ n
2
.
m = n = 1 ω
11
u
11
(x, y, t) = α
11
sinxsinysin(ω
11
t + ϕ
11
).
ω
12
=
ω
21
= a
√
5 (m = 1, n = 2 m = 2, n = 1)
v
12
(x, y) = sinxsin2y v
21
(x, y) = sin2xsiny,
ω
12
= ω
21
= a
√
5
u
12
= α
12
sinxsin2ysin(ω
12
t + ϕ
12
), u
21
= α
21
sin2xsinysin(ω
21
t + ϕ
21
).
u
12
u
21
ϕ
12
= ϕ
21
sinxsin2y = 0 sin2xsiny = 0
y = π/2 u
12
x = π/2 u
21
u
12
u
21
αu
12
+ βu
21
= αsinxsin2y + βsin2xsiny, α = const, β = const,
λ
2
13
= λ
2
31
= 10 λ
2
23
= λ
2
32
= 13 λ
2
14
= λ
2
41
= 17
m n
(m, n) ∈ N
2
+
√
m
2
+ n
2
= const
λ
2
= 801125
801125 = 895
2
+10
2
= 890
2
+95
2
= 886
2
+127
2
= ... = 10
2
+895
2
(32 ).
λ
2
=
89000000015 03676650 = ... = 2793642145
2
+ 1046691725
2
= ... =
= 1458737557
2
+2602322951
2
= ... = 1968253
2
+2983286129
2
(1536 ).
α = 0 β = 0 α = β α = −β
v
mn
Ω = {(x, y) : x
2
+ y
2
< R
2
}
R x, y Q
T
= Ω ×(0, T ]
∂
2
u
∂t
2
= a
2
∂
2
u
∂x
2
+
∂
2
u
∂y
2
.
x, y r θ
x = = rcosθ y = rsinθ a
2
1
a
2
∂
2
u
∂t
2
=
∂
2
u
∂r
2
+
1
r
∂u
∂r
+
1
r
2
∂
2
u
∂θ
2
.
u|
r=R
= 0 (0, T ]
u|
t=0
= ϕ
0
(r, θ),
∂u
∂t
t=0
= ϕ
1
(r, θ), 0 ≤ r < R, 0 ≤ θ < 2π.
R r = R
ϕ
0
ϕ
1
u(x, y, t)
t (x, y)