Phương pháp sai phân hữu hạn và ứng dụng giải phương trình Poisson với điều kiện biên hỗn hợp

const
||A|| A
∀x x
∃x x
∈
R
d
d
max
min
Σ
Ω Ω
∆
x
x
∆
t
t
∂u
∂x
u x
∂u
∂y
u y
∂
2

u
∂x
2
u x
∂
2
u
∂y
2
u y
u

(x, y)



Γ
u(x, y) Γ
Ox x = a x = a + L = b
Ω Γ
ν Σ
Ω
(x, y) Ω
n n






a
11
x
1
+ a
12
x
2
+ + a
1n
x
n
= a
1n+1
a
21
x
1
+ a
22
x
2
+ + a
2n
x
n
= a
2n+1

a
n1
x
1
+ a
n2
x
2
+ + a
nn
x
n
= a
nn+1
a
ij
(i, j = 1, n)
a
in+1
(i = 1, n) x
i
(i = 1, n)
A =



a
11
a
12

a
1n
a
21
a
22
a
2n

a
n1
a
n2
a
nn



b =



a
1n+1
a
2n+1
a
nn+1




x =



x
1
x
2
x
n



Ax = b.
A
det(A) =







a
11
a
12
a
1n

a
21
a
22
a
2n

a
n1
a
n2
a
nn







= 0
Ax = −f,
A
A =




c
1

b
1
0 0 0
a
2
−c
2
b
2
0 0

0 0 0 −c
n−1
b
n−1
0 0 0 a
n
−c
n




.



−c
1
x

1
+ b
1
x
2
= −f
1
a
i
x
i−1
− c
i
x
i
+ b
i
x
i+1
= f
i
, (i = 2, . . . n − 1)
a
n
x
n−1
− c
n
x
n

= −f
n
.
c
1
= 0
x
1
=
b
1
c
1
x
2
+
f
1
c
1
.
i = 2
x
2
x
3
x
i
= α
i

x
i+1
+ β
i
α
i
β
i
x
i−1
= α
i−1
x
i
+ β
i−1
a
i
α
i−1
x
i
+ a
i
β
i−1
− c
i
x
i

+ b
i
x
i+1
= −f
i
.
x
i
(c
i
− a
i
α
i−1
) = b
i
x
i+1
+ a
i
β
i−1
+ f
i
.
c
1
− a
i

α
i−1
= 0
x
i
= −
b
i
c
i
− a
i
α
i−1
x
i+1
+
a
i
β
i−1
+ f
i
c
i
− a
i
α
i−1
.

α
i
=
b
i
c
i
− a
i
α
i−1
β
i
=
a
i
β
i−1
+ f
i
c
i
− a
i
α
i−1
, (i = 2, , n − 1).
α
1
=

b
1
c
1
, β
1
=
f
1
c
1
.
i n − 1
x
i−1
= α
n−1
x
n
+ β
n−1
.
α
n
(α
n−1
x
n
+ β
n−1

) − c
n
x
n
= −f
n
.
x
n
=
α
n
β
n−1
+ f
n
c
n
− a
n
α
n−1
, c
n
− a
n
α
n−1
= 0.


















α
1
=
b
1
c
1
, β
1
=
f
1
c
1

;
α
i
=
b
i
c
i
− a
i
α
i−1
, β
i
=
a
i
β
i−1
+ f
i
c
i
− a
i
α
i−1
, (i = 2, , n − 1);
x
n

=
α
n
β
n−1
+ f
n
c
n
− a
n
α
n−1
;
x
i
= α
i
x
i+1
+ β
i
, (i = n − 1, , 1).
α
1
, β
1
α
2
, β

2
α
n−1
, β
n−1
x
n
x
n−1
, , x
1
x
n
8n
a
i
, b
i
, c
i
= 0, (i = 1, , n) |c
1
| ≥ |b
1
| , |c
n
| ≥ |a
n
| , |c
i

| ≥ |a
i
| +
|b
i
| , (i = 2, , n − 1)
∆
i
= c
i
− a
i
α
i−1
= 0 ; |α
i−1
| ≤ 1, (i = 2, , n)
|c
1
| ≥ |b
1
| = 0 |α
1
| =
|b
1
|
|c
1
|

≤ 1.
|c
2
− a
2
α
1
| ≥ |c
2
| − |a
2
| |α
1
| ≥ |a
2
| + |b
2
| − |a
2
| |α
1
|
= |a
2
| (1 − |α
1
| + |b
2
|) ≥ |b
2

| > 0 ⇒ |c
2
− a
2
α
1
| = 0.
|α
2
| =
|b
2
|
|c
2
− a
2
α
1
|
≤ 1.
|α
2
| ≤ 1 |α
3
| ≤ 1, , |α
i−1
| ≤ 1, (i = 2, , n)
|c
i

− a
i
α
i−1
| ≥ |c
i
| − |a
i
| |α
i−1
| ≥ |a
i
| + |b
i
| − |a
i
| |α
i−1
|
= |a
i
| (1 − |α
i−1
|) + |b
i
|  |b
i
| > 0, ∀
i
.

c
i
− a
i
α
i−1
= 0, (i = 2, , n).
Ax = b
a
ii
x
i
+

j=i
a x
j
= b
i
, i = 1, 2, , n
A ∈ M
n
, b =

b
1

b
n


, x =

x
1

x
n

x
(0)
a
ii
x
(k+1)
i
+

j=i
a x
(k)
j
= b
i
, i = 1, 2, , n; k = 0, 1, 2, . . .
∀i, a
ii
= 0
x
(k+1)
i

= −

j=i
a
a
ii
x
(k)
j
+
b
i
a
ii
, i = 1, 2, , n; k = 0, 1, 2, . . .
0 < q < 1
n

j=i
,j=1
|a
ij
| ≤ q |a
ii
| , ∀i = 1, 2, , n
Ax = b
x
(0)




x
(k)
− x



∞
≤
q
k
1 − q



x
(0)
− x
(1)



∞
, k = 0, 1, 2,



x
(k)
− x




∞
≤
q
1 − q



x
(k)
− x
(k−1)



∞
, k = 0, 1, 2,
x

1 3 1
5 1 −1
2 1 6

→

5 1 −1
1 3 1
2 1 6


R
n
||.||
1
x
1
=
n

i=1
|x
i
|
S
1
= max
1≤j≤n
n

i=1
|S
ij
| = max
1≤j≤n

i=j
|a
ij
|

|a
ij
|
.
0 < q
1
< 1
∀j = 1, , n,
n

j=i
i=j
|a
ij
| ≤ q
1
|a
ii
|
.
∞
.
1
L(cm)
S(cm
2
) ρ(g/cm
3
)
C(cal/g.

o
C) V (cm
3
)
u(
o
C) H(cal)
ρ .
k(cm
2
/s)
c = kρC
[cal/(s.cm.
o
C)]
k c
Ox x = a
x = a + L = b
Ox x = a x = a + L = b
u(x, t) x t
x
q(cal/(cm
2
.s)) x
S
u x
∂u
∂x
q = −kρC
∂u

∂x
.
−
S∆x x x+ ∆x ∆l
−
q(x, t)S∆t
q(x + ∆x, t)S∆t
S∆xρC∆u, ∆u
∆t
q(x, t)S∆t − q(x + ∆x, t)S∆t = S∆xρC∆u.
S∆x∆t
q(x, t) − q(x + ∆x, t)
∆x
= ρC
∆u
∆t
.
∆x → 0, ∆t → 0
−
∂q
∂x
= ρC
∂u
∂t
.
k
∂
2
u
∂x

2
=
∂u
∂t
, a < x < b, t > 0, k = c t > 0.
k = const
∂
∂x
(k
∂u
∂x
) =
∂u
∂t
, a < x < b, t > 0.
k x, t, u k = k(x, t, u)
∂
∂x
[k(x, t, u)
∂u
∂x
] =
∂u
∂t
, a < x < b, t > 0.
f(x, t)
∂
∂x
[k(x, t, u)
∂u

∂x
] + f(x, t, u) =
∂u
∂t
, a < x < b, t > 0.
k f u
∂u
∂t
=
∂
∂x
[k(x, t, u)
∂u
∂x
] − q(x, t)u + f(x, t), a < x < b, t > 0.
∂u
∂t
=
∂
∂x
[k(x, t)
∂u
∂x
] + r(x, t)
∂u
∂x
− q(x, t)u + f(x, t), a < x < b, t > 0,
r(x, t)
∂u
∂x

Ω
Γ Oxy
∂u
∂t
= k

∂
2
u
∂x
2
+
∂
2
u
∂y
2

, (x, y) ∈ Ω, t > 0, k = c t
Ω
Γ
∂u
∂t
=
∂
∂x

k
1
(x, y, t, u)

∂u
∂x

+
∂
∂y

k
2
(x, y, t, u)
∂u
∂y

+ f(x, y, t, u), (x, y) ∈ Ω, t > 0
k
1
, k
2
, f u
∂u
∂t
=
∂
∂x

k
1
(x, y, t)
∂u
∂x


+
∂
∂y

k
2
(x, y, t)
∂u
∂y

− q(x, y, t)u
+ f(x, y, t), (x, y) ∈ Ω, t > 0.
ν
Σ Oxyz
∂u
∂t
= k

∂
2
u
∂x
2
+
∂
2
u
∂y
2

+
∂
2
u
∂z
2

, k = c ∈ ν, t > 0
ν Σ
∂u
∂t
=
∂
∂x

k
1
(x, y, z, t, u)
∂u
∂x

+
∂
∂y

k
2
(x, y, z, t, u)
∂u
∂y


+
∂
∂z

k
3
(x, y, z, t, u)
∂u
∂z

+ f(x, y, z, t, u), (x, y, z) ∈ ν, t > 0.
k
1
, k
2
, k
3
, f u
∂u
∂t
=
∂
∂x

k
1
(x, y, z, t)
∂u
∂x


+
∂
∂y

k
2
(x, y, z, t)
∂u
∂y

+
∂
∂z

k
3
(x, y, z, t)
∂u
∂z

− q(x, y, z, t) + f(x, y, z, t), (x, y, z) ∈ ν, t > 0.
∂u
∂t
= 0
d
2
u
dx
2

= 0, a < x < b
d
dx

k(x, u)
du
dx

= f(x, u), a < x < b
d
dx

k(x)
du
dx

− q(x)u = f(x), a < x < b.
∂
2
u
∂x
2
+
∂
2
u
∂y
2
= 0, (x, y) ∈ Ω
∂

∂x

k
1
(x, y, u)
∂u
∂x

+
∂
∂y

k
2
(x, y, u)
∂u
∂y

= f(x, y, u), (x, y) ∈ Ω
∂
∂x

k
1
(x, y)
∂u
∂x

+
∂

∂y

k
2
(x, y)
∂u
∂y

− q(x, y)u = f(x, y), (x, y) ∈ Ω.
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
= 0, (x, y, z) ∈ ν
f(x, y, z, u) =
∂
∂x


k
1
(x, y, z, u)
∂u
∂x

+
∂
∂y

k
2
(x, y, z, u)
∂u
∂y

+
∂
∂z

k
3
(x, y, z, u)
∂u
∂z

, (x, y, z) ∈ ν
f(x, y, z) =
∂

∂x

k
1
(x, y, z)
∂u
∂x

+
∂
∂y

k
2
(x, y, z)
∂u
∂y

+
∂
∂z

k
3
(x, y, z)
∂u
∂z

− q(x, y, z)u, (x, y, z) ∈ ν.
∂

2
u
∂x
2
+
∂
2
u
∂y
2
= f(x, y), (x, y) ∈ Ω
∂
2
u
∂x
2
+
∂
2
u
∂y
2
+
∂
2
u
∂z
2
= f(x, y, z), (x, y, z) ∈ ν
a b a < b u = u(x) a < x < b

Lu = − (ku

)

+ qu = f (x) , a < x < b
u(a) = α, u(b) = β
k = k(x), q = q(x), f(x)
0 < c
0
≤ k (x) ≤ c
1
, c
0
, c
1
= const, q (x) ≥ 0,
α, β
u [a, b]
[a, b] N
h = (b − a)/N x
i
= a + ih, i = 0, 1, , N
x
i
h
Ω
h
= {x
i
, 1 ≤ i ≤ N − 1}

Γ
h
= {x
0
, x
N
}
Ω
h
= Ω
h
∪ Γ
h
[a, b]
Ω
h
v x
i
v
i
u(x) x ∈ [a, b] u
x
i
u
i
= u(x
i
)
v
v

xi
=
v
i+1
− v
i
h
, v
xi
=
v
i
− v
i−1
h
.
v
xx
v
xxi
=
v
xi+1
− v
xi
h
=

v
i+1

− v
i
h
−
v
i
− v
i−1
h

h
=
v
i+1
− 2v
i
+ v
i−1
h
2
.
a
(av
x
)
xi
=
a
i+1
v

xi+1
− a
i
v
xi
h
=
a
i+1
v
i+1
− (a
i+1
+ a
i
) v
i
+ a
i
v
i−1
h
2
.
u(x
i
)
x
i
∈ Ω

h
v
i
v
i
L
h
v ≡ − (av
x
)
xi
+ q
i
v
i
= f
i
,
v
0
= α, v
N
= β,
a
i
= k (x
i
− h/2) , q
i
= q (x

i
) , f
i
= f (x
i
) .
a
i
v
i−1
−

a
i
+ a
i+1
+ h
2
q
i

v
i
+a
i+1
v
i+1
= −h
2
f

i
, i = 1, 2, , N−1
v
0
= α, v
N
= β.
A
i
y
i−1
− C
i
y
i
+ B
i
y
i+1
= −F
i
, i = 1, 2, , N − 1
y
0
= m
1
y
1
+ n
1

, y
N
= m
2
y
N−1
+ n
2
,
A
i
> 0, B
i
> 0, D
i
= C
i
− A
i
− B
i
≥ 0,
0 ≤ m
1
≤ 1, 0 ≤ m
2
≤ 1, m
1
+ m
2

< 2.
A
i
= a
i
, B
i
= a
i+1
, C
i
= a
i
+ a
i+1
+ h
2
q
i
, F
i
= h
2
f
i
,
m
1
= 0, m
2

= 0, n
1
= α, n
2
= β.
y
i
= α
i+1
y
i+1
+ β
i+1
.
α
i
β
i
y
i
α
i
, β
i
i i − 1
y
i−1
= α
i
y

i
+ β
i
y
i−1
(C
i
− A
i
α
i
) y
i
= B
i
y
i+1
+ A
i
B
i
+ F
i
.
C
i
− A
i
α
i

= 0.
y
i
=
B
i
C
i
− A
i
α
y
i+1
+
A
i
B
i
+ F
i
C
i
− A
i
α
i
.
α
i+1
=

B
i
C
i
− A
i
α
, β
i+1
=
A
i
B
i
+ F
i
C
i
− A
i
α
i
.
i = 0
y
0
= α
1
y
1

+ β
1
.
α
1
= m
1
, β
1
= n
1
.
0 ≤ m
1
≤ 1
0 ≤ m
1
< 1 0 ≤ α
1
< 1
C
1
− A
1
α
1
≥ A
1
+ B
1

− A
1
α
1
= B
1
+ (1 − α
1
) A
1
> B
1
⇒ 0 ≤ α
2
=
B
1
C
1
− A
1
α
1
< 1.
0 ≤ α
2
< 1 0 ≤ α
3
< 1
0 ≤ m

1
< 1 ⇒ 0 ≤ α
i
< 1, i = 1, 2, , N.
m = 1 α
1
= 1
C
1
− A
1
α
1
≥ A
1
+ B
1
− A
1
α
1
= B
1
+ (1 − α
1
) A
1
= B
1
⇒ 0 ≤ α

2
=
B
1
C
1
− A
1
α
1
≤ 1.
0 ≤ α
2
≤ 1 0 ≤ α
3
≤ 1
0 ≤ m
1
≤ 1 ⇒ 0 ≤ 0 ≤ α
i
≤ 1, i = 1, 2, , N.
C
i
− A
i
α
i
≥ A
i
+ B

i
− A
i
α
i
= A
i
(1 − α
i
) + B
i
≥ B
i
> 0, ∀i.
α
i
, β
i
i = N − 1
y
N−1
= α
N
y
N
+ β
N
.
(1 − m
2

α
N
) y
N
= n
2
+ m
2
β
N
.
0 ≤ m
1
< 1 m
2
≤ 1 α
N
< 1
1 − m
2
α
N
> 0
m
1
= 1 m
2
< 1 α
N
≤ 1

1 − m
2
α
N
> 0 1 − m
2
α
N
> 0
y
N
=
n
2
+ m
2
β
N
1 − m
2
α
N
.
y
i
i = N − 1 i = 0
α
1
= m
1

, β
1
= n
1
α
i+1
=
B
i
C
i
− A
i
α
i
, β
i+1
=
A
i
B
i
+ F
i
C
i
− A
i
α
i

, i = 2, . . . , N
y
N
=
n
2
+ m
2
β
N
1 − m
2
α
N
y
i
= α
i+1
y
i+1
+ B
i+1
, i = N − 1, N − 2, . . . , 2, 1, 0.
y
i+1
= ξ
i+1
y
i
+ η

i+1
.
ξ
N
= m
2
, η
N
= n
2
ξ
i
=
A
i
C
i
− ξ
i+1
B
i
, η
i
=
B
i
η
i+1
+ F
i

C
i
− ξ
i+1
B
i
, i = 1, 2, , N − 1
y
0
=
n
1
+ m
1
η
1
1 − m
1
ξ
1
y
i+1
= ξ
i+1
y
i
+ η
i+1
, i = 0, 1, 2, N − 1.