Лекции по курсу Математическая физика
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[α, β]
φ,
β
Z
α
¡
−(p · y
0
)
0
+ q · y
¢
· φ =
β
Z
α
f · φ. (4.5.1)
β
Z
α
¡
−(p · y
0
)
0
+ q · y
¢
· φ = −(py
0
)φ
¯
¯
β
α
+
β
Z
α
¡
p · y
0
· φ
0
+ q · y · φ
¢
=
= p(α)µ
α
y(α)φ(α)+p(β)µ
β
y(β)φ(β)+
β
Z
α
¡
p·y
0
·φ
0
+q·y·φ
¢
=
β
Z
α
f· φ. (4.5.2)
{φ
k
} k =
1, . . . , N
φ = φ
n
n = 1, . . . , N
{φ
k
} N
N y
N
φ
k
(x) = A
k
cos(γ
k
x) + B
k
sin(γ
k
x)
γ
k
, k ∈ N .
γ
k
A
k
B
k
α = 0, β = 1
N [0, 1]
x
n
= nh; n = 0, . . . N + 1; h =
1
N + 1
. (4.5.3)
Π
0
(x) =
½
1 −
x
h
0 ≤ x ≤ h
0 x > h;
Π
n
(x) =
0 x < (n − 1) · h
x
h
− (n − 1) (n − 1) · h ≤ x ≤ n · h
(n + 1) −
x
h
n · h ≤ x ≤ (n + 1) · h
0 x > (n + 1) · h
, n = 1, . . . , N;
Π
N+1
(x) =
½
0 x < N · h
x
h
− N N · h ≤ x ≤ (N + 1) · h.
b b b b b b b
A
A
A
A
A
A
A
A
A
A
A
A¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
¢
0 h (n − 1)h nh (n + 1)h Nh (N + 1)h = 1
ey(x) =
N+1
X
k=0
c
k
· Π
k
(x). (4.5.4)
Π
k
(x
n
) = δ
nk
,
c
k
= ey(x
k
)
φ = Π
n
,
N+1
X
k=0
a
nk
· c
k
= b
n
, n = 0, . . . , N + 1, (4.5.5)
a
nk
=
1
Z
0
¡
p · Π
0
n
· Π
0
k
+ q · Π
n
· Π
k
¢
, n, k = 1, . . . , N; (4.5.6)
a
0k
= a
k0
= p(0)µ
0
· δ
0k
+
1
Z
0
¡
p · Π
0
0
· Π
0
k
+ q · Π
0
· Π
k
¢
; (4.5.7)
a
N+1,k
= a
k,N+1
= p(1)µ
1
·δ
N+1,k
+
1
Z
0
¡
p·Π
0
N+1
·Π
0
k
+q ·Π
N+1
·Π
k
¢
; (4.5.8)
b
n
=
1
Z
0
f · Π
n
, n = 0, . . . , N + 1. (4.5.9)
(N + 2) ×(N + 2) A
N+1
X
k,n=0
a
nk
c
k
c
n
= hLey, eyi. (4.5.10)
q
c 6= θ A
N
{φ
k
}
|n − k| > 1
a
nk
= 0 A
p [0, 1]
q, f [0, 1], q ≥ 0. y
ey
x ∈ [0, 1]
|ey(x) − y(x)| ≤ C · h, (4.5.11)
C
k = 0, 1, . . . , N + 1 |c
k
− y(x
k
)| ≤ C · h.
p, q, f
C
h
ε
Π
k
.
µ
0
= µ
1
= 0 q
φ
ey(x) =
N
X
k=1
c
k
· Π
k
(x). (4.5.4
0
)
N +2, N;
Oz
Oz
∆z
Oz
(x, y)
u uu
u
u
∆x
∆y
∆t
∆Q
1
= γ · ∆t · ∆x · ∆y · ∆z.
λ
∆Q
2
= λ · ∆t · ∆y · ∆z ·
T (x + ∆x, y) − T (x, y)
∆x
.
∆Q
3
= λ · ∆t · ∆y · ∆z ·
T (x − ∆x, y) − T (x, y)
∆x
.
∆Q
4
= λ · ∆t · ∆x · ∆z ·
T (x, y + ∆y) − T (x, y)
∆y
.
∆Q
5
= λ · ∆t · ∆x · ∆z ·
T (x, y − ∆y) − T (x, y)
∆y
.
5
X
j=0
∆Q
j
= 0.
∆t∆x∆y∆z (∆ x = ∆ y = 0)
T
D
x
(λ · D
x
T ) + D
y
(λ · D
y
T ) + γ = 0. (5.1.1)
Lu ≡ −div(λ · ∇u) = f, (5.2.1)
Ω ⊂ R
n
n = 2
n = 3
λ Ω,
λ(x) ≥ λ
0
> 0, x ∈ Ω;
f Ω
λ = const,
− 4 u = f
±
λ
Ω,
u
¯
¯
¯
∂Ω
= φ
1
λ · D
n
u
¯
¯
¯
∂Ω
= φ
2
(λ · D
n
u + α · u)
¯
¯
¯
∂Ω
= φ
3
Ω
∂Ω
Ω ⊂ R
n
n = 2 , 3
∂Ω. D
L
Ω
∂Ω
Lu = f, u ∈ D
L
. (5.3.1)
L
Lϕ = ν · ϕ, ϕ ∈ D
L
. (5.3.2)
Ω
hϕ, ψi =
Z
Ω
ϕ · ψ.
λ ≡ 1,
− 4 ϕ = ν · ϕ. (5.3.3)
Ω =]0, a
1
[×]0, a
2
[×]0, a
3
[⊂ R
3
∂Ω
ϕ
jkm
= sin
³
jπx
1
a
1
´
· sin
³
kπx
2
a
2
´
· sin
³
mπx
3
a
3
´
, j, k, m ∈ N .
ϕ
jkm
ν
jkm
= π
2
·
³
j
2
a
2
1
+
k
2
a
2
2
+
m
2
a
2
3
´
. (5.3.4)
ϕ
jkm
= cos
³
jπx
1
a
1
´
· cos
³
kπx
2
a
2
´
· cos
³
mπx
3
a
3
´
, j, k, m ∈ N ∪ {0},
Ω = {x ∈ R
2
¯
¯
¯
x
2
1
+ x
2
2
< R
2
}
(ρ, ϑ)
−D
ρ
(ρ · D
ρ
ϕ) −
1
ρ
· D
2
ϑϑ
ϕ = ν · ρ · ϕ.
ϕ
¯
¯
ρ=R
= 0
ϕ
km
= J
|k|
³
µ
(k)
m
ρ
R
´
· exp(i kϑ), k ∈ Z, m ∈ N .
J
|k|
|k|, µ
(k)
m
m ϕ
km
ν
km
=
¡
µ
(k)
m
/R
¢
2
n = 3
u =
X
j,k,m
f
jkm
ν
jkm
· ϕ
jkm
, (5.3.5)
f
jkm
=
hf, ϕ
jkm
i
kϕ
jkm
k
2
=
³
Z
Ω
f · ϕ
jkm
´
Á
³
Z
Ω
|ϕ
jkm
|
2
´
.
ϕ
jkm
Ω
φ,
∂Ω