Sự kết hợp giữa phương pháp sai phân và phương pháp newton raphson giải phương trình vi phân tuyến tính

BỘ GIÁO DỤC VÀ ĐÀO TẠO
TRƯỜNG ĐẠI HỌC SƯ PHẠM HÀ NỘI 2







ĐÀM THỊ THẢO





SỰ KẾT HỢP GIỮA PHƯƠNG PHAP SAI PHÂN
VÀ PHƯƠNG PHÁP NEWTON-RAPHSON GIẢI PHƯƠNG
TRÌNH VI PHÂN PHI TUYẾN






LUẬN VĂN THẠC SĨ TOÁN HỌC

HÀ NỘI, 2014

L (X, Y )
L (X, Y )
R
n
C
[a, b]
C
n
[a,b]

X
d X × X
R
(∀x, y ∈ X) d (x, y) ≥ 0, d (x, y) = 0 ⇔ x = y,
(∀x, y ∈ X) d (x, y) = d (y, x) ,

(∀x, y, z ∈ X) d (x, y) ≤ d (x, z) + d (z, y) ,
d X d (x, y)
x y X
M = (X, d)
x, y ∈ R
d (x, y) = |x − y|.
R
R
R
1
R
C
m
[a,b]
m (m ∈ N, m ≥ 1) [a, b]
x (t) , y (t) ∈ C
m
[a,b]
d (x, y) =
m

k=0
t∈[a,b]



x
(k)
(t) − y
(k)

(t)



.
d C
m
[a,b]
M = (X, d)
X
0
= ∅ X d X
M = (X
0
, d)
M = (X, d) {x
n
} ⊂
X M
(∀ε > 0) (∃n
0
∈ N
∗
) (∀m, n ≥ n
0
) , d (x
n
, x
m
) < ε

lim
n,m→+∞
d (x
n
, x
m
) = 0.
{x
n
} ⊂ X M
X T : X → X
d (T x, Ty) ≤ αd (x, y)
α < 1 x, y ∈ X.
x
∗
∈ X x
∗
= T x
∗
, x
0
∈ X {x
n
}
n∈N
x
k+1
= T x
k
, ∀k ∈ N x

∗
d (x
n
, x
∗
) ≤
α
n
1 − α
d (x
1
, x
0
) .
d (x
k+1
, x
k
) = d (T x
k
, T x
k−1
) ≤ αd (x
k
, x
k−1
) ≤ ≤ α
k
d (x
1

, x
0
) , ∀k ∈ N.
∀n ∈ N, ∀p ∈ N,
d (x
n+p
, x
n
) ≤ d (x
n+p
, x
n+p−1
) + + d (x
n+1
, x
n
)
≤

α
n+p−1
+ + α
n

d (x
1
, x
0
) ,
d (x

n+p
, x
n
) ≤
α
n
1 − α
d (x
1
, x
0
) .
{x
n
}
n∈N
X
x
∗
∈ X lim
n→∞
x
n
= x
∗
p → ∞
x
n+1
= T x
n

n → ∞ x
∗
= T x
∗
x
∗
T x
∗
= x
∗
.
¯x T ¯x = ¯x
d (x
∗
, ¯x) = d (T x
∗
, T ¯x) ≤ αd (x
∗
, ¯x) , α < 1.
x
∗
= ¯x, x
∗

X
P P = R P = C X → R
.
(∀x ∈ X) x ≥ 0, x = 0 ⇔ x = θ
(∀x ∈ X) (∀α ∈ P) αx = |α|. x
(∀x, y ∈ X) x + y ≤ x + y

x x
X
{x
n
} X x ∈ X
lim
n→∞
x
n
− x = 0, lim
n→∞
x
n
= x x
n
→ x (n → ∞)
{x
n
}
X lim
m,n→∞
x
n
− x
m
 = 0.
X
X
k R
k

x ∈ R
k
x = (x
1
, x
2
, , x
k
)
x
i
∈ R i = 1, 2, , k x =

k

i=1
|x
i
|
2
R
k
R
k
{x
n
}
R
k
lim

m,n→∞
x
n
− x
m
 = 0
(∀ε > 0) (∃M ∈ N
∗
) (∀m, n ≥ M) :
x
n
− x
m
 < ε ⇔
n

j=1
|x
n,j
− x
m,j
|
2
< ε
2
.
j ∈ N (∀ε > 0) (∃M
j
∈ N
∗

) (∀m, n ≥ M
j
) :
x
n,j
− x
m,j
 < ε.
j ∈ N {x
n,j
}
x
j
= lim
n→∞
x
n,j
, j =
1, k
(∀ε > 0) (∀j = 1, 2, , k) (∃M
j
∈ N
∗
) (∀n ≥ M
j
) : x
n,j
− x
j
 <

ε
√
n
.
x = (x
j
)
j=1,k
{x
n
} x
M
0
= max {M
1
M
2
, , M
k
} (∀n ≥ M
0
) :
|x
n,j
− x
j
| <
ε

√

n
, j = 1, 2, , k ⇒ |x
n,j
− x
j
|
2
<
ε
2
n
⇒
k

j=1
|x
n,j
− x
j
|
2
< ε
2
⇒




k


j=1
|x
n,j
− x
j
|
2
< ε.
{x
n
} x
L (X, Y )
X ψ :
X × X → R
ψ (x, x) ≥ 0, ∀x ∈ X;
ψ (x, x) = 0 ⇒ x = θ;
ψ (x, y) = ψ (y, x) , ∀x, y ∈ X;
ψ (αx
1
+ βx
2
, y) = αψ (x
1
, y) + βψ (x
2
, y) ; ∀x
1
, x
2
, y ∈ X ∀α, β ∈ R

X ψ (x, y)
x, y (x, y) .
X
(.) . : X → R
x =

(x, x) X X
d : X × X → R
d (x, y) = x − y =

(x − y, x − y)
X (X, d)
d
X
(.) d (X, d)
X
(.)
X = R
k
x = (x
1
, x
2
, , x
k
) ∈ R
k
, y = (y
1
, y

2
, , y
k
) ∈ R
k
.
(x, y) =
k

i=1
x
i
y
i
. R
k
H = θ x, y, z,
H
H P
H (., .)
H x =

(x, x), x ∈ H
H
H
R
k
k x =
(x
j

) ∈ R
k
, y = (y
j
) ∈ R
k
(x, y) =
k

j=1
x
j
y
j
.
x =

(x, x) =




k

j=1
x
2
j
, x = (x
j

) ∈ R
k
x =

k

j=1
|x
j
|
2
R
k
R
k
L (X, Y )
X Y L (X, Y )
X Y
L (X, Y )
A, B ∈ L (X, Y ) A + B
(A + B) (x) = Ax + Bx, ∀x ∈ X.
α ∈ P P = R P = C
A ∈ L (X, Y ) αA
(αA) (x) = α (Ax) , ∀x ∈ X.
A + B ∈ L (X, Y ) α ∈ (X, Y )
L (X, Y )
P
A ∈ L (X, Y )
A = sup
x=1

Ax.
L (X, Y )
L (X, Y )
(A
n
) ⊂ L (X, Y ) A ∈
L (X, Y ) x ∈ X lim
n→∞
A
n
x − Ax = 0 Y
(A
n
) ⊂ L (X, Y ) A ∈ L (X, Y )
(A
n
) A Y
Y L (X, Y )
(A
n
) ⊂ L (X, Y )
(∀ε > 0) (∃n
0
∈ N
∗
) (∀n, m ≥ n
0
) A
n
− A

m
 < ε.
x ∈ X
A
n
x − A
m
x = (A
n
− A
m
) x ≤ A
n
− A
m
. x < ε x.
(A
n
x) ⊂ Y Y
Y
lim
n→∞
A
n
x = y ∈ Y.
y = Ax
A X
Y m → +∞
A
n

x − Ax ≤ ε x, ∀n ≥ n
0
, ∀x ∈ X,
(A
n
− A) x ≤ ε x, ∀n ≥ n
0
, ∀x ∈ X.
A
n
− A ≤ ε, ∀n ≥ n
0
.
A = A
n
1
− (A
n
1
− A) ∈ L (X, Y ) n
1
> n
0

n
− A → 0 n → ∞.
(A
n
) ⊂ L(X, Y ) A
L (X, Y ) L (X, Y ) 

X = Y L (X, X)
X
A, B X AB X
(AB) x = A (Bx) , ∀x ∈ X.
AB
(AB) x = A (Bx) ≤ A. Bx ≤ A. Bx,
AB
AB ≤ A. B.
L (X, X)
L (X, X)
AB ≤ A. B;
I I = 1
L (X, X) L (X, X)
A
0
= I, A
n
= A
n−1
A (n = 1, 2, ) .
R
n
R
n
R
n
d (x, y) =

n


j=1
(x
j
− y
j
)
2
.
x, y ∈ R
n
, x = (x
1
, x
2
, , x
n
); y = (y
1
, y
2
, , y
n
)
d (x, y) =




n


j=1
(x
j
− y
j
)
2
.
2n
a
j
, b
j
(j = 1, 2, , n)





n

j=1
a
j
b
j






≤




n

j=1
a
2
j
.




n

j=1
b
2
j
0 ≤
n

i=1

n


j=1
(a
i
b
j
− a
j
b
i
)
2

=
n

i=1
n

j=1
a
2
i
b
2
j
− 2
n

i=1
n


j=1
a
i
b
j
a
j
b
i
+
n

i=1
n

j=1
a
2
j
b
2
i
= 2

n

j=1
a
2

j

.

n

j=1
b
2
j

− 2

n

j=1
a
j
b
j

2
x, y, z ∈ R
n
, x = (x
1
, x
2
, , x
n

) , y = (y
1
, y
2
, , y
n
) , z =
(z
1
, z
2
, , z
n
)
d
2
(x, y) =
n

j=1
(x
j
− y
j
)
2
=
n

j=1

[(x
j
− z
j
) + (z
j
− y
j
)]
2
=
n

j=1
(x
j
− z
j
)
2
+ 2
n

j=1
(x
j
− z
j
) (z
j

− y
j
) +
n

j=1
(z
j
− y
j
)
2
≤ d
2
(x, z) + 2d (x, z) .d (z, y) + d
2
(z, y)
= [d (x, z) + d (z, y)]
2
⇒ d (x, y) ≤ d (x, z) + d (z, y) .
R
n
R
n
R
n
R
n
x
1

=
n

i=1
|x
i
|, x
2
=




n

i=1
x
2
i
, x
∞
= m
i=1,n
|x
i
|.
R
n
R
n

∀x, y ∈ R
n
, x = (x
1
, x
2
, , x
n
) , y = (y
1
, y
2
, , y
n
)
(x, y) =
n

j=1
x
j
y
j
.
x =

(x, x) =





n

j=1
x
2
j
, x = (x
1
, x
2
, , x
n
) ∈ R
n
.
R
n
C
[a, b]
C
[a,b]
= {x (t) ∀t ∈ [a, b]}, −∞ < a < b < +∞.
C
[a,b]
∀x, y ∈ C
[a,b]
, d (x, y) = m
a≤t≤b
|x (t) − y (t)|.

C
[a, b]
x = m
a≤t≤b
|x (t)|.
C
[a,b]
C
[a,b]
C
[a,b]
C
[a,b]
B = {S
n
, n ∈ N
∗
}
S
n
=

x ∈ C
[a,b]
, x <
1
n

.
C

[a,b]
.
x = m
a≤t≤b
|x (t)|.
C
n
[a,b]
C
n
[a,b]
x (t) [a, b]
n
x = m
a≤t≤b

|x (t)|, |x

(t)|, ,



x
(n)
(t)




.

f : R → R h
0
∆
0
f (x) = f (x) y = f (x) .
∆
1
f (x) = f (x + h) − f (x) y = f (x) .
∆
2
f (x) = ∆

∆
1
f (x)

= ∆f (x + h) − ∆f (x)
= f (x + 2h) − 2f (x + h) + f (x)
y = f (x) .
∆
n
f (x) = ∆

∆
n−1
f (x)

(∀n ∈ N
∗
) n

y = f (x) .
k x
n
∆
k
x
n
= ∆

∆
k−1
x
n

= ∆
k−1
x
n+1
− ∆
k−1
x
n
=
k

i=0
(−1)
i
C
i

k
x
n+k−i
.
k = 1
∆x
n
= x
n+1
− x
n
= C
0
1
x
n+1
− x
n
= C
0
1
x
n+1
− C
1
1
x
n
.
k

∆
k
x
n
=
k

i=0
(−1)
i
C
i
k
x
n+k−i
.
k + 1
∆
k+1
x
n
= ∆
k
x
n+1
− ∆
k
x
n
=

k

i=0
(−1)
i
C
i
k
x
n+k+1−i
−
k

i=0
(−1)
i
C
i
k
x
n+k−i
.
i = i

−1 i

i
k

i=0

(−1)
i
C
i
k
x
n+k−i
=
k+1

i

=1
(−1)
i

−1
C
i

−1
k
x
n+k+1−i

= −
k+1

i=1
(−1)

i
C
i−1
k
x
n+k+1−i
.
∆
k+1
x
n
=
k

i=0
(−1)
i
C
i
k
x
n+k+1−i
+
k+1

i=1
(−1)
i
C
i−1

k
x
n+k+1−i
=
k

i=1
(−1)
i
C
i
k
x
n+k+1−i
+ x
n+k+1
+
k

i=1
(−1)
i
C
i−1
k
x
n+k+1−i
+ (−1)
k+1
x

n
=
k

i=1
(−1)
i

C
i
k
+ C
i−1
k

x
n+k+1−i
+ x
n+k+1
+ (−1)
k+1
x
n
=
k

i=1
(−1)
i
C

i
k+1
x
n+k+1−i
+ x
n+k+1
+(−1)
k+1
x
n
=
k

i=1
(−1)
i
C
i
k+1
x
n+k+1−i
.

∆
k
(ax
n
+ by
n
) = a∆

k
x
n
+ b∆
k
y
n
.
∆
k
(ax
n
+ by
n
) =
k

i=0
(−1)
i
C
i
k
(ax
n+k−1
+ by
n+k−1
)
=
k


i=0
(−1)
i
C
i
k
(ax
n+k−1
) +
k

i=0
(−1)
i
C
i
k
(by
n+k−1
)
= a
k

i=0
(−1)
i
C
i
k

x
n+k−1
+ b
k

i=0
(−1)
i
C
i
k
y
n+k−1
= a∆
k
x
n
+ b∆
k
y
n
.

k m
m − k k < m
k = m
k > m.
P
m
(n) = n

m
∆n
m
= (n + 1)
m
− n
m
= C
0
m
+ C
1
m
n + + C
n
m
n
m
− n
m
= C
n
m
+ C
1
m
n + + C
m−1
m
n

m−1
= P
m−1
(n) .
k = s < m
k = s + 1 < m
∆
s+1
n
m
= ∆ (∆
s
n
m
) = ∆
s
(n + 1)
m
−∆
s
n
m
= ∆P
m−s
(n) = P
m−s−1
(n) .
k = m
∆
m

n
m
= P
m−m
(n) = P
0
(n) = C = .
k > m
∆
k
n
m
= ∆
k−m
∆
m
n
m
= ∆
k−m
C = ∆
k−m−1
∆C = 0.

N

n=a
∆
k
x

n
= ∆
k−1
x
N+1
− ∆
k−1
x
n
, k ∈ Z
+
.
N

n=a
∆
k
x
n
=
N

n=a
∆

∆
k−1
x
n


= ∆
k−1
x
a+1
− ∆
k−1
x
a
+ ∆
k−1
x
a+2
− ∆
k−1
x
a−1
+ + ∆
k−1
x
N+1
− ∆
k−1
x
N
= ∆
k−1
x
N+1
− ∆
k−1

x
a
.

k = 1
N

n=a
∆x
n
= x
N+1
− x
a
.
X, Y f : X → Y
x
0
X f : X → Y
x
0
A (x
0
) : X → Y
A (x
0
) ∈ L (X, Y )
f (x
0
+ h) − f (x

0
) = A (x
0
) (h) + α (x
0
, h) h ∈ X,
lim
h→0
α (x
0
, h)
h
= 0,
A (x
0
) f
x
0
f

(x
0
)
A (x
0
) (h) f x
0
df (x
0
, h) df (x

0
, h) = f

(x
0
) (h)
f

(x
0
) f

(x
0
) (h)
f

(x
0
) h
f : U → Y U
X f x
0
∈ U
f
f : U → Y g : U → Y
X, Y U
X f, g x
0
∈ U

(f + g)

(x
0
) = f

(x
0
) + g

(x
0
) ,
(k.f )

(x
0
) = k.f

(x
0
) , k ∈ R