Алексеев Г.В. Классические методы математической физики. Часть 2
Подождите немного. Документ загружается.
|a
(2)
k
|
k
≤
1
2
1
k
2
+ |a
(2)
k
|
2
,
|b
(3)
k
|
k
≤
1
2
1
k
2
+ |b
(3)
k
|
2
.
P
∞
k=1
(1/k
2
)
P
∞
k=1
|a
(2)
k
|
2
P
∞
k=1
|b
(3)
k
|
2
u(x, t) = −
l
π
3
∞
X
k=1
1
k
3
b
(3)
k
cos
kπat
l
+
1
a
a
(2)
k
sin
kπat
l
sin
kπx
l
.
Q
T
= [0, l] × [ 0, T ]
l
π
3
∞
X
k=1
1
k
3
|b
(3)
k
| +
1
a
|a
(2)
k
|
,
Q
T
Q
T
Q
T
(x, t) ∈ Q
T
(x, t) ∈ Q
T
ϕ
0
ϕ
1
a
k
= α
k
sinϕ
k
, b
k
= α
k
cosϕ
k
,
α
k
=
q
a
2
k
+ b
2
k
,
u(x, t) =
∞
X
k=1
α
k
sin
kπx
l
sin
kπat
l
+ ϕ
k
.
α
k
sin
kπx
l
sin
kπat
l
+ ϕ
k
,
ϕ
k
ω
k
x A
k
(x)
ω
k
=
kπa
l
=
kπ
l
s
T
ρ
, A
k
(x) = α
k
sin
kπx
l
.
ρ T
x = 0,
l
k
,
2l
k
, ...,
(k − 1)l
k
, l
A
k
(x)
l
2k
,
3l
2k
, ...,
(2k −1)l
2k
A
k
(x)
sin(kπx/l)
k
k
k
ω
1
ω
2
ω
1
=
π
l
a =
π
l
s
T
ρ
, ω
2
=
2π
l
s
T
ρ
, ... .
α
k
k
k
ω
1
l ρ T
α
k
l, ρ T
ϕ
0
ϕ
1
f(x, t)
∂
2
u
∂t
2
= a
2
∂
2
u
∂x
2
+ f(x, t)
Q
T
= (0, l) × (0, T ],
u|
x=0
= 0, u|
x=l
= 0 (0, T ]
u|
t=0
= ϕ
0
(x),
∂u
∂t
t=0
= ϕ
1
(x)
(0, l).
u = v + w.
v
∂
2
v
∂t
2
= a
2
∂
2
v
∂x
2
+ f(x, t)
Q
T
,
v|
x=0
= 0, v|
x=l
= 0 (0, T ]
v|
t=0
= 0,
∂v
∂t
t=0
= 0
(0, l),
w
∂
2
w
∂t
2
= a
2
∂
2
w
∂x
2
Q
T
,
w|
x=0
= 0, w|
x=l
= 0 (0, T ]
w|
t=0
= ϕ
0
(x),
∂w
∂t
t=0
= ϕ
1
(x)
(0, l).
v
f
v
v
v(x, t) =
∞
X
k=1
T
k
(t)sin
kπx
l
.
Q
T
T
k
(t)
∞
X
k=1
[T
′′
k
(t) + ω
2
k
T
k
(t)]sin
kπx
l
= f(x, t),
ω
k
= kπa/l f
(0, l)
f(x, t) =
∞
X
k=1
f
k
(t)sin
kπx
l
.
f
k
(t) t
f
k
(t) =
2
l
l
Z
0
f(x, t)si n
kπx
l
dx.
f sin(kπx/l)
T
′′
k
(t) + ω
2
k
T
k
(t) = f
k
(t), k = 1 , 2, ... .
k = 1, 2, ...
T
k
T
k
T
k
(0) = 0, T
′
k
(0) = 0, k = 1, 2, ... .
T
k
(t) =
1
ω
k
t
Z
0
f
k
(τ)sinω
k
(t − τ )dτ
f
k
(τ)
T
k
(t) =
2
lω
k
t
Z
0
[sinω
k
(t − τ )
l
Z
0
f(x, τ)sin
kπx
l
dx]dτ.
T
k
v
t
Q
T
x t Q
T
f ∈ C
2
(Q
T
) f(0, t) = 0, f(l, t) = 0 ∀t ∈ [0, T ].
u(x, t) =
∞
X
k=1
T
k
(t)sin
kπx
l
+
∞
X
k=1
a
k
cos
kπat
l
+ b
k
sin
kπat
l
sin
kπx
l
.
T
k
a
k
b
k
f ϕ
0
ϕ
1
∂
2
u
∂t
2
= a
2
∂
2
u
∂x
2
+ f(x, t)
Q
T
,
u|
x=0
= g
1
(t), u|
x=l
= g
2
(t) (0, T ]
u|
t=0
= ϕ
0
(x),
∂u
∂t
t=o
= ϕ
1
(x)
(0, l).
w(x, t) = g
1
(t) + [g
2
(t) − g
1
(t)]
x
l
.
w|
x=0
= g
1
(t), w|
x=l
= g
2
(t).
u
u = v + w,
v
v
v|
x=0
= 0 v|
x=l
= 0
v|
t=0
= u|
t=0
− w|
t=0
= ϕ
0
(x) − g
1
(0) − [g
2
(0) − g
1
(0)]
x
l
≡
ϕ
0
(x),
∂v
∂t
t=0
=
∂u
∂t
t=0
−
∂w
∂t
t=0
= ϕ
1
(x) − g
′
1
(0) − [g
′
2
(0) − g
′
1
(0)]
x
l
≡
ϕ
1
(x).
∂
2
v
∂t
2
= a
2
∂
2
v
∂x
2
+
¯
f(x, t),
¯
f(x, t) = f(x, t) − g
′′
1
(t) − [g
′′
2
(t) − g
′′
1
(t)]
x
l
.
v
∂
2
v
∂t
2
= a
2
∂
2
v
∂x
2
+
¯
f(x, t), v|
x=0
= 0, v|
x=l
= 0, v|
t=0
=
ϕ
0
(x),
∂v
∂t
t=o
=
ϕ
1
(x).
v
R
2
R
3
R
2
R
3
Ω S
S
S
S
S
ρ(x)
∂
2
u
∂t
2
=
∂
∂x
p(x)
∂u
∂x
− q(x)u ≡ p(x)
∂
2
u
∂x
2
+ p
′
(x)
∂u
∂x
− q(x)u
Q
T
.
ρ, p q [0, l] Q
T
= (0, l)×(0, T ] 0 <
T < ∞
αu(0, t) − β
∂u(0, t)
∂x
= 0, γu(l, t) + δ
∂u(l, t)
∂x
= 0, t ∈ (0, T ]
u |
t=0
= ϕ
0
(x),
∂u
∂t
|
t=0
= ϕ
1
(x), x ∈ (0, l).
α β γ δ ϕ
0
ϕ
1
(0, l)
ρ, p, p
′
, q [0, l] p(x) ≥ p
0
= const > 0
ρ(x) ≥ ρ
0
= const > 0 q(x) ≥ 0 ∀x ∈ [0, l]
α, β, γ, δ ≥ 0 α + β 6= 0 γ + δ 6= 0
u(x, t) = X(x)T (t).
ρ(x)X(x)T
′′
(t) = T (t)[p(x)X
′
(x)]
′
− q(x)X(x)T (t),
[p(x)X
′
(x)]
′
− q(x)X(x)
ρ(x)X(x)
=
T
′′
(t)
T (t)
.
x
t
λ
T X
T
′′
(t) + λT (t) = 0,
[p(x)X
′
(x)]
′
+ [λρ(x) − q(x)]X(x) = 0.
X
αX(0) − βX
′
(0) = 0, γX(l) + δX
′
(l) = 0.
λ
λ
λ
1
< λ
2
< λ
3
< ... < λ
k
< ... , lim
k→∞
λ
k
= +∞.
λ
λ
λ
k
X
k
l
Z
0
ρ(x)X
2
k
(x)dx = 1.
ρ
ρ (0, l)
l
Z
0
ρ(x)X
k
(x)X
m
(x)dx = 0 (k 6= m).
λ
k
λ
m
X
k
X
m
[p(x)X
′
k
(x)]
′
+ [λ
k
ρ(x) − q(x)]X
k
(x) = 0,
[p(x)X
′
m
(x)]
′
+ [λ
m
ρ(x) − q(x)]X
m
(x) = 0.
X
m
X
k
X
m
(x)[p(x)X
′
k
(x)]
′
− X
k
(x)[p(x)X
′
m
(x)]
′
+ (λ
k
− λ
m
)ρ(x)X
k
(x)X
m
(x) = 0,