Алексеев Г.В. Классические методы математической физики. Часть 2
Подождите немного. Документ загружается.
Q
T
u
∂u
∂t
= a
2
∂
2
u
∂x
2
+ f(x, t),
u|
x=0
= 0, u|
x=l
= 0 (0, T ]
u|
t=0
= 0 (0, l).
f ∈ C
1
(Q
T
) f (0, t) = f(l, t) = 0 ∀t ∈ [0, T ]
u
u(x, t) =
∞
X
k=1
T
k
(t) sin
kπx
l
T
k
(t) f
f(x, t) =
∞
X
k=1
f
k
(t) sin
kπx
l
,
f
k
(t) =
2
l
l
Z
0
f(x, t) sin
kπx
l
dx.
∞
X
k=1
"
T
′
k
(t) +
kπa
l
2
T
k
(t) − f
k
(t)
#
sin
kπx
l
= 0.
T
k
T
′
k
(t) + ω
2
k
T
k
(t) = f
k
(t), k = 1, 2, ...,
ω
k
= kπa/l u
u(x, 0) =
∞
X
k=1
T
k
(0) sin
kπx
l
= 0,
T
k
T
k
(0) = 0, k = 1, 2, .. .
T
k
k
T
k
(t) =
Z
t
0
e
−ω
2
k
(t−τ)
f
k
(τ)dτ.
T
k
(t)
u(x, t) =
∞
X
k=1
Z
t
0
e
−ω
2
k
(t−τ)
f
k
(τ)dτ
sin
kπx
l
.
§1
t x
Q
T
g
i
∈ C
1
[0, T ], i = 1, 2
w(x, t) = g
1
(t) + [g
2
(t) − g
1
(t)]
x
l
.
u
u(x, t) = v( x, t) + w(x, t),
v
v Q
T
∂v
∂t
= a
2
∂
2
v
∂x
2
+
f(x, t),
f(x, t) ≡ f(x, t) −
∂w
∂t
+ a
2
∂
2
w
∂x
2
,
v|
x=0
= u(0, t) − w(0, t) = 0, v|
x=l
= u(l, t) − w(l, t) = 0, t ∈ (0, T ]
v|
t=0
= ϕ(x) ≡ ϕ(x) − w(x, 0), x ∈ (0 , l).
v
∂u
∂x
x=0
= 0,
∂u
∂x
x=l
= 0
u
λ
k
X
k
X
′′
+ λX = 0 (0, l), X
′
(0) = X
′
(l) = 0.
λ
k
X
k
§
R
R
Q
T
= (−∞, ∞ ) × ( 0, T ] T < ∞ Q
T
= (−∞, ∞) × (0, ∞)
T = ∞ R
u
∂u
∂t
= a
2
∂
2
u
∂x
2
Q
T
0 < T ≤ ∞
u|
t=0
= ϕ(x), −∞ < x < ∞.
ϕ R
T ≤ ∞
(T < ∞) (T = ∞)
Q
T
= (−∞, ∞)×[0, T ] T < ∞ Q
T
= (−∞, ∞)×[0, ∞)
T = ∞
u ∈ C
2,1
(Q
T
) ∩ C( Q
T
) Q
T
(x, t) ∈ Q
T
x ∈ (−∞, ∞)
u
1
u
2
|u
1
| ≤ M, |u
2
| ≤ M Q
T
, M = < ∞.
u = u
1
−u
2
u|
t=0
= 0 |u| ≤ 2M Q
T
T < ∞ Q
LT
= {(x, t) : |x| < L, 0 < t ≤ T }
v(x, t) =
4M
L
2
x
2
2
+ a
2
t
.
Σ
LT
Q
LT
v
Σ
LT
v(x, 0) ≥ |u(x, 0)| = 0, x ∈ [−L, L], v(±L, t) ≥ 2M ≥ |u(±L, t)|, t ∈ [0, T ] ,
−v(x, t) ≤ u(x, t) ≤ v(x, t) Σ
LT
v ±u Q
LT
v(x, t) − u(x, t) ≥ 0 v(x, t) + u(x, t) ≥ 0 Q
LT
−v(x, t) ≤ u(x, t) ≤
v(x, t) ∀(x, t) ∈
Q
LT
|u(x, t)| ≤ v(x, t) =
4M
L
2
x
2
2
+ a
2
t
∀(x, t) ∈
Q
LT
.
(x, t) L u(x, t) ≡
0 ∀(x, t) ∈ Q
LT
T = ∞ Q
LT
Q
LT
0
T
0
< ∞
x t Q
T
u
u(x, t) = X(x)T (t).
T
′
(t)
a
2
T (t)
=
X
′′
(x)
X(x)
= −λ
2
,
λ
2
T X
T
′
(t) + a
2
λ
2
T (t) = 0,
X
′′
(x) + λ
2
X(x) = 0.
X(x) = α λx + β λx,
α β λ
T (t) = e
−a
2
λ
2
t
.
X λ
λ
u
λ
(x, t) = e
−a
2
λ
2
t
[α(λ) λx + β(λ) λx]
Q
T
α(λ)
β(λ)
u(x, t) =
Z
∞
−∞
u
λ
(x, t)dλ =
Z
∞
−∞
e
−a
2
λ
2
t
[α(λ)
λx + β(λ) λx]dλ
Q
T
t x
α(λ) β(λ)
t = 0
ϕ(x) =
Z
∞
−∞
[α(λ)
λx + β(λ) λx]dλ.
ϕ λx λx
α(λ) β(λ)
ϕ
α(λ) =
1
2π
Z
∞
−∞
ϕ(ξ)
λξdξ, β(λ) =
1
2π
Z
∞
−∞
ϕ(ξ)
λξdξ.
ϕ(x) =
1
2π
Z
∞
−∞
dλ
Z
∞
−∞
ϕ(ξ)cosλ(ξ − x)dξ,
ϕ
ϕ
(−∞, ∞)
(−∞, ∞)
R
∞
−∞
|ϕ(x)|dx
λ
u(x, t) =
1
2π
Z
∞
−∞
dλ
Z
∞
−∞
ϕ(ξ)e
−a
2
λ
2
t
λ(ξ − x)dξ =
=
1
π
Z
∞
0
dλ
Z
∞
−∞
ϕ(ξ)e
−a
2
λ
2
t
λ(ξ − x)dξ.
u(x, t) =
1
π
Z
∞
−∞
ϕ(ξ)dξ
Z
∞
0
e
−a
2
λ
2
t
λ(ξ −x)dλ.
x t λ ξ z µ
aλ
√
t = z, λ(ξ − x) = µz ⇒ µ =
ξ − x
a
√
t
, dλ =
dz
a
√
t
, t > 0.
Z
∞
0
e
−a
2
λ
2
t
λ(ξ − x)dλ =
1
a
√
t
Z
∞
0
e
−z
2
µzdz ≡
1
a
√
t
J(µ).
J
µ ∈ (−∞, ∞) J µ
J
′
(µ) = −
R
∞
0
e
−z
2
z
µzdz
Z
∞
0
u
′
vdz = −
Z
∞
0
uv
′
dz + uv|
∞
0
u(z) = (1/2)e
−z
2
v(z) = sin µz u
′
= −ze
−z
2
v
′
= µ cos µz
J
′
(µ) = −
µ
2
Z
∞
0
e
−z
2
µzdz = −
µ
2
J(µ).
J
J
′
(µ) = −(µ/2)J(µ)
J(µ) = Cexp −(µ
2
/4) C
µ = 0
C = J(0) =
Z
∞
0
e
−z
2
dz =
√
π
2
.
Z
∞
−∞
e
−z
2
dz =
√
π.
J(µ) =
√
π
2
e
−
µ
2
4
.
Z
∞
0
e
−a
2
λ
2
t
λ(ξ − x)dλ =
√
π
2a
√
t
e
−
(ξ−x)
2
4a
2
t
, t > 0.
u(x, t) =
Z
∞
−∞
ϕ(ξ)
1
2a
√
πt
e
−
(ξ−x)
2
4a
2
t
dξ.
F (ξ, x, t) ≡
1
2a
√
πt
e
−
(ξ−x)
2
4a
2
t
,
x t
(x, t) t > 0
ϕ
ϕ ∈ C(−∞, ∞) |ϕ(x)| ≤ M < ∞ ∀x ∈ (−∞, ∞)
(x, t) ∈ Q
T
x t
Π = Π(L, t
0
, T
0
) = {(x, t) : −L ≤ x ≤ L, t
0
≤
t ≤ T
0
} t
0
> 0 T
0
≤ T
x t
I(x, t) =
1
t
k
Z
∞
−∞
ϕ(ξ)(ξ −x)
m
e
−
(ξ−x)
2
4a
2
t
dξ,
k m
z =
ξ − x
2a
√
t
(t > 0), ξ = x + 2a
√
tz, dξ = 2a
√
tdz
I(x, t) = (2a)
m+1
t
m+1
2
−k
Z
∞
−∞
ϕ(x + 2az
√
t)z
m
e
−z
2
dz.
M|z|
m
e
−z
2
(−∞, ∞)
t ≥ t
0
> 0 t ≥ t
0
t = 0
u
x t
x t
t > 0
(x, t) t > 0 u
t > 0
lim
t→0
u(x, t) = ϕ( x) ∀x ∈ (−∞, ∞).
u(x, t) =
1
√
π
Z
∞
−∞
ϕ(x + 2az
√
t)e
−z
2
dz.
u |x| < ∞ t > 0
|ϕ(x)| ≤ M ∀x ∈ (−∞, ∞)
|u(x, t)| ≤
1
√
π
Z
∞
−∞
|ϕ(x + 2az
√
t)|e
−z
2
dz ≤
M
√
π
Z
∞
−∞
e
−z
2
dz = M,
1
√
π
Z
∞
−∞
e
−z
2
dz = 1.
ϕ(x)
u(x, t) − ϕ(x) =
1
√
π
Z
∞
−∞
h
ϕ(x + 2az
√
t) − ϕ(x)
i
e
−z
2
dz.
|u(x, t) − ϕ(x)| ≤
1
√
π
Z
∞
−∞
|ϕ(x + 2az
√
t) − ϕ(x)|e
−z
2
dz.
|ϕ(x + 2az
√
t) − ϕ(x)| ≤ 2M ∀x, z ∈ (−∞, ∞), t ∈ [0, T ] .
ε > 0
N
2M
√
π
Z
−N
−∞
e
−z
2
dz ≤
ε
3
,
2M
√
π
Z
∞
N
e
−z
2
dz ≤
ε
3
.
(−∞, ∞) (−∞, −N)
[−N, N] (N, ∞)
|u(x, t) − ϕ(x)| ≤
2ε
3
+
1
√
π
Z
N
−N
|ϕ(x + 2az
√
t) − ϕ(x)|e
−z
2
dz.
ϕ (−∞, ∞)
t > 0 |x| ≤ L |z| ≤ N |ϕ(x + 2az
√
t) − ϕ(x)| ≤ ε/3
|u(x, t) − ϕ(x)| ≤
2ε
3
+
ε
3
1
√
π
Z
N
−N
e
−z
2
dz ≤
2ε
3
+
ε
3
1
√
π
Z
∞
−∞
e
−z
2
dz = ε.
|u(x, t) − ϕ(x)| < ε ∀x ∈ (−∞, ∞)
t ε
∂u
∂t
= a
2
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
R
3
× (0, ∞), u|
t=0
= ϕ(x, y, z) R
3
u(x, y, z, t) =
∞
Z
−∞
∞
Z
−∞
∞
Z
−∞
ϕ(ξ, η, ζ)
1
(2a
√
πt)
3
e
−
(ξ−x)
2
+(η−y)
2
+(ζ−z)
2
4a
2
t
dξdηdζ.
∂u
∂t
= a
2
∂
2
u
∂x
2
+ f(x, t)
Q
T
u(x, 0) = 0, x ∈ (−∞, ∞)
lim
x→±∞
u(x, t) = 0, lim
x→±∞
∂u(x, t)
∂x
= 0, t > 0.
ϕ
R
ϕ
ˆϕ(λ) =
1
√
2π
Z
∞
−∞
ϕ(ξ)e
iλξ
dξ,
−∞ < λ < ∞ ˆϕ
ϕ
ϕ
ˆϕ(λ)
ϕ(x) =
1
√
2π
Z
∞
−∞
ˆϕ(λ)e
−iλx
dλ, x ∈ (−∞, ∞).