スペクトル法を用いた数値計算 – 一次元線形移流方程式の場合 – 荻原 ...

– –
22070012
,

1
1 4
1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2 6
2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.3 . . . . . . . . . . . . . . . . . . . . . . 9
2.3 . . . . . . . . . . . . . . . . . . . . . . . 11
3 1 13
3.1 1 . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.1 1 . . . . . . . . . . . . . . . . 13
3.1.2 1 . . . . . . 14
3.1.3 1 . . . . . . . . . . 17
3.1.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 1 . . . . . . . . . . . . . . . . . . . . . . . . 19
( )

2
3.2.1 1 . . . . . . . . . . . . . . . 20
3.2.2 1 . . . . . 21
3.2.3 . . . . . . . . . . . . . . . . . . 22
4 (FFT) 25
4.1 FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.1.1 FFT . . . . . . . . . . . . . . . . . . . . . . . 26
4.1.2 FFT . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3 FFT . . . . . . . . . . . . . . . . . . . . . . . 36
4.3.1 IDFT FFT . . . . . . . . . . . . . . . 36
4.3.2 DFT FFT . . . . . . . . . . . . . . . 38
4.3.3 FFT . . . . . . . 39
5 40
5.1 - . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.1.1 1 1 . . . . . . . . . . . . 42
5.1.2 2 2 . . . . . . . . . . . . 43
5.1.3 3 3 . . . . . . . . . . . . 44
5.1.4 4 4 . . . . . . . . . . . . 46
5.1.5 - 5 . . . . . . . . . . . . 52
5.2 . . . . . . . . . . . . . . . . . . . . . . . . . 53
( )

3
5.2.1 - . . . . . . . . . . . . . . . . . 53
6 61
6.1 - . . . . . . . 62
6.1.1 1 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 62
6.1.2 2 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.3 3 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1.4 4 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7 85
8 86
9 123
10 124
( )

1 4
1
1.1
70 km
−60 ◦ ∼ 60 ◦ 100 m/s .
1.5 m/s 60 .
. 100 m/s
. , 1 ) , 100 m/s
, 2 ) 500 m/s
. , 13000 m/s, 9800 m/s .
, 60 .
.
.
.
.
.
. 2 , .
. , .
,
(2004) ,
1 .
,
.
1 ) 0 ◦
2 ) 0 ◦ ( )

1 5
1
.
1.2
.
1 . 2
.
, . 3
1 . ,
1 1
. 4 1
5 4
, FFT
6
. 7 1
.
8 .
( )

2 6
2
,
. 2 u = u(x, t)
,
∂u
∂t = ∂2 u
(0 < x < 1) (2.0.1)
∂x 2
: u(x, 0) = f(x),
: u(0, t) = u(1, t) = 0.
.
, (2.0.1)
. (2.0.1)
.
.
. .
2.1
j j = 0, 1, · · ·, J , [0, 1] x j
x j = j∆x
(j = 0, 1, · · ·, J),
∆x = 1 J
. , u(x, t) x , x
, . ,
u j (t) = u(x j , t) (j = 0, 1, · · ·, J)
( )

2 7
, .
, , 2
1 ) . 2
( ) ∂ 2 u
≈ u j+1 − 2u j + u j−1
(j = 1, 2, · · ·, J − 1) (2.1.2)
∂x 2 (∆x) 2
.
x=x j
, (2.0.1)
du j (t)
dt
= u j+1 − 2u j + u j−1
(∆x) 2 (j = 1, 2, · · ·, J − 1)
. u 0 = u j = 0 . , ,
u j (0) = f(x j ) (j = 1, 2, · · ·, J − 1)
, .
u j (t) 2 ) .
2.2
2.1 J − 1 ,
u j (t) = u(x j , t)
, .
ϕ j (x) (j = 1, 2, · · ·, J − 1) 3 )
J − 1
∑J−1
u(x, t) = α j (t)ϕ j (x) (2.2.3)
j=1
1 ) A .
2 ) ,
. 5 .
3 ) ϕ j 1 . ,
,
.
J−1
∑
0 = α j (t)ϕ j (x)
j=1
α j (t) = 0 (j = 1, 2, · · ·, J − 1)
( )

2 8
. ϕ j (x) , ϕ j (0) =
ϕ j (1) = 0 (j = 1, 2, · · ·, J − 1) . ,
α j (t) . 2
. .
2.2.1
J − 1 x j ,
. (2.0.1) ,
∑J−1
j=1
dα j (t) ∑J−1
ϕ j (x i ) =
dt
j=1
( ) ∂ 2 ϕ j (x)
α j (t)
(i = 1, 2, · · ·, J − 1)
∂x 2 x=x i
α j (t) .
. ,
, .
2.2.2
(2.0.1) R
R(x, t) ≡ ∂u
∂t − ∂2 u
∂x 2 (2.2.4)
. , J − 1 W m (x) (m = 1, 2, · · ·, J − 1)
. R 0 .
∫
W m (x)R(x, t)dx = 0 (m = 1, 2, · · ·, J − 1) (2.2.5)
(2.0.1) ,
∑J−1
j=1
dα j (t)
dt
∫ 1
0
∑J−1
∫ 1
∂ 2 ϕ j (x)
ϕ j (x)W m (x)dx = α j (t)
W
0 ∂x 2 m (x)dx (m = 1, 2, · · ·, J − 1)
j=1
(2.2.6)
. (2.2.6) α j (t) .
, . ,
0 , (weighted
residual method) .
( )

2 9
. W m ϕ j
.
. ,
2 .
2.2.3
J − 1 x j , ϕ j (x)
.
⎧
0 (x < x j−1 )
x−x j−1
⎪⎨
ϕ j (x) =
⎪⎩
∆x
(x j−1 ≤ x < x j )
x j+1 −x
∆x
(x j ≤ x < x j+1 )
0 (x j+1 ≤ x)
(j = 1, 2, · · ·, J − 1)
φ j(x)
1
x j−1 x j x j+1
2.2.1: .
ϕ j (x) (2.2.6)
. , (2.2.6) . (2.2.6) ,
∑J−1
j=1
∫ 1
∂ 2 ϕ j (x)
∑J−1
α j (t)
ϕ
0 ∂x 2 m (x)dx =
j=1
[ 1 ∂ϕj (x)
α j (t)
m(x)]
∂x ϕ ∑J−1
−
0
j=1
∫ 1
α j (t)
0
∂ϕ j (x)
∂x
(m = 1, 2, · · ·, J − 1)
∂ϕ m (x)
dx.
∂x
( )

2 10
ϕ j (0) = ϕ j (1) = 0 , ϕ m (0) = ϕ m (1) = 0. ,
∑J−1
j=1
∫ 1
α j (t)
0
∂ 2 ϕ j (x)
∂x 2
∑J−1
ϕ m (x)dx = −
j=1
(2.2.7) (2.2.9)
∑J−1
j=1
dα j (t)
dt
∫ 1
.
0
∑J−1
ϕ j (x)ϕ m (x)dx = −
j=1
∫ 1
α j (t)
0
∫ 1
α j (t)
0
∂ϕ j (x)
∂x
∂ϕ j (x)
∂x
∂ϕ m (x)
dx. (m = 1, 2, · · ·, J − 1)
∂x
(2.2.7)
∂ϕ m (x)
dx. (m = 1, 2, · · ·, J − 1)
∂x
ϕ j (x) ,
. (2.0.1) , sin (jπx)
. ,
∑J−1
u(x, t) = α j (t) sin (jπx) (2.2.8)
j=1
. (2.2.8) (2.2.6) ,
,
. ,
∑J−1
j=1
dα j (t)
dt
∫ 1
0
W m (x) = sin (mπx) (m = 1, 2, · · ·, J − 1)
∑J−1
sin (jπx) sin (mπx)dx = α j (t)
j=1
. , j = m ,
.
dα m (t)
dt
∫ 1
0
(jπ) 2 sin (jπx) sin (mπx)dx
= −(mπ) 2 α m (t) (m = 1, 2, · · ·, J − 1) (2.2.9)
α m , . (2.2.9)
u(x, 0) = f(x) W m (x) = sin (mπx) (m = 1, 2, · · ·, J − 1)
, 0 ≤ x ≤ 1 ,
α m (0) = 2
∫ 1
0
f(x) sin (mπx)dx (m = 1, 2, · · ·, J − 1)
( )

2 11
.
(2.2.9) ϕ j (x) ∂2
∂x 2 ,
.
. .
2.3
, ,
,
, 4 ) . ,
.
,
, 5 )
.
.
,
6 ) .
.
4 ) .
( )

2 12
.
5 ) (2.2.9) . L .
∂u(x, t)
∂t
= L[u(x, t)]
. u
u = ∑ j
α j (t)ϕ j (x)
. ϕ j (x) L , β j ,
L[ϕ j (x)] = β j ϕ j (x)
. (2.2.5)
∑
∫
dα j (t)
ϕ j (x)ϕ m (x)dx = ∑ dt
j
. j = m ,
j
∫
α j (t)
β j ϕ j (x)ϕ m (x)dx
dα m (t)
dt
= β m α m (t)
.
.
6 ) sin (mπx) , x
.
( )

3 1 13
3
1
, , 1
.
3.1 1
u(x, 0) = f(x), u(x, t) = u(x + 2π, t) , x, t
1 .
∂u
∂t + ∂u = 0. (3.1.1)
∂x
:
:
u(x, 0) = f(x),
u(x, t) = u(x + 2π, t).
.
3.1.1 1
, .
.
ξ ≡ x − t , x, t ξ t .
u t
u(x, t) = U(ξ, t).
∂u
∂t = ∂U ∂ξ
∂ξ ∂t + ∂U ∂t
∂t ∂t
(3.1.2)
= − ∂U
∂ξ + ∂U
∂t . (3.1.3)
( )

3 1 14
u x
(3.1.3) (3.1.5) (3.1.1) .
∂U(ξ, t)
∂t
∂u
∂x = ∂U ∂ξ
∂ξ ∂x
(3.1.4)
= ∂U
∂ξ . (3.1.5)
∂u
∂t + ∂u
∂x = −∂U ∂ξ + ∂U
∂t + ∂U
∂ξ
= ∂U
∂t
= 0
= 0 . F 1
∂U(ξ, t)
= 0
∂t
U(ξ, t) = F (ξ)
u(x, t) = F (x − t).
u(x, t) = f(x − t) (3.1.6)
. f(x) 1 x
. ,
, 1 .
.
3.1.2 1
, . u(x, t)
.
u(x, t) =
N∑
k=−N
û k (t)e ikx . (3.1.7)
, N , (cutting of the number of
waves) 1 ) . 2
1 ) f(x) ,
f(x) = 1 ∞∑
∞∑
2 a 0 + a n cos (nx) + b n sin (nx)
n=1
n=1
( )

3 1 15
, (2N + 1) × 2 , u(x, t)
,
û ∗ −k = û k (3.1.8)
2 ) , 2N + 1 3 ) .
û k (t) , (3.1.7) (3.1.1)
, R .
R =
N∑
k=−N
dû k (t)
e ikx +
dt
N∑
k=−N
ikû k (t)e ikx . (3.1.9)
. .
f(x) ≈ 1 N∑
N∑
2 a 0 + a n cos (nx) + b n sin (nx)
n=1
. N , N .
2 ) u(x, t)
u(x, t) =
. , u(x, t) u ∗ (x, t) ,
u(x, t) ∗ =
=
N∑
m=−N
N∑
l=−N
N∑
l=−N
n=1
û m e imx
û ∗ l (e ilx ) ∗
û ∗ l e −ilx
. u(x, t) u(x, t) = u ∗ (x, t) ,
N∑
m=−N
u(x, t) = u ∗ (x, t)
N∑
û m e imx = û ∗ l e −ilx .
l=−N
e −ikx [0, 2π] ,
∫ 2π
0
N∑
m=−N
.
3 ) ∗ .
û m e imx e −ikx dx =
∫ 2π
û m δ mk = û ∗ l δ l,−k
û ∗ −k = û k
N ∑
l=−N
û ∗ l e −ilx e −ikx dx
( )

3 1 16
e −ikx , [0, 2π] , 0
,
∫ 2π
0
e −ikx Rdx =
=
= 0
∫ 2π
0
N∑
k=−N
,
N∑
k=−N
dû k (t)
dt
∫
dû k (t)
2π
e ikx e −ikx dx +
dt
0
∫ 2π
0
e ikx e −ikx dx +
N∑
k=−N
N∑
k=−N
ikû k (t)
ikû k (t)e ikx e −ikx dx
∫ 2π
0
e ikx e −ikx dx.
dû k
dt + ikû k = 0 (3.1.10)
. û k (t) ,
,
û k (0) = 1 ∫ 2π
f(x)e −ikx dx (k = −N, · · ·, N) (3.1.11)
2π 0
. (3.1.11) ,
4 ) .
(3.1.12) (3.1.7) ,
.
û k (t) = û k (0)e −ikt (k = −N, · · ·, N). (3.1.12)
u(x, t) =
=
=
N∑
k=−N
N∑
k=−N
N∑
k=−N
û k (t)e ikx
û k (0)e −ikx e ikx
û k (0)e ik(x−t) (3.1.13)
x, t , (x − t) ,
1 x . .
4 ) 6
( )

3 1 17
3.1.3 1
, (3.1.1) .
, 2N + 1 ,
,
u j (t) = u(x j , t)
x j =
2π
2N + 1 j (j = 0, · · ·, 2N) (3.1.14)
. x ,
( ) ∂u
≃ u j+1 − u j−1
∂x
2∆x
x=x j
∆x =
2π
2N + 1
(3.1.15)
. (3.1.14) , (3.1.15) (3.1.1) ,
du j
dt + u j+1 − u j−1
2∆x
= 0 (j = 0, · · ·, 2N) (3.1.16)
. j = 0, 2N , ,
u 2N+1 = u 0
u 2N = u −1
, . (3.1.16) (A kj )
. (A kj )
A kj = e ikx j
(k = −N, · · ·, N; j = 0, · · ·, 2N) (3.1.17)
. 5 ) .
(b.1) , (3.1.14)
u j (t) =
=
N∑
k=−N
N∑
k=−N
û k (t)A kj
5 ) B .
û k (t)e ikx j
(j = 0, · · ·, 2N) (3.1.18)
( )

3 1 18
û k (t) . (3.1.18) (3.1.16) ,
du j
dt + u j+1 − u j−1
2∆x
=
=
=
N∑
k=−N
N∑
k=−N
N∑
k=−N
dû k (t)
e ikx j
+ 1
dt 2∆x
dû k (t)
e ikx j
+ 1
dt 2∆x
dû k (t)
e ikx j
+ 1
dt 2∆x
N∑
k=−N
N∑
k=−N
N∑
k=−N
û k (t) ( e ikx j+1
− e ikx j−1 )
û k (t) ( e ik(x j+∆x) − e ik(x j−∆x) )
û k (t)e ikx j ( e ik∆x − e −ik∆x) = 0
(3.1.19)
. eik∆x − e −ik∆x
2i
N∑
k=−N
dû k (t)
e ikx j
+ 1
dt 2∆x
N∑
k=−N
= sin (k∆x) , (3.1.19) ,
N∑
k=−N
dû k (t)
e ikx j
+ 1
dt i∆x
N∑
(
e ikx dûk (t)
j
dt
k=−N
û k (t)e ikx j ( e ik∆x − e −ik∆x) = 0
N∑
k=−N
û k (t)e ikx j
(sin (k∆x)) = 0
+ i
∆x sin (k∆x)û k(t)
)
= 0 (3.1.20)
. , (A kj ) = (e ikx j
) 6 ) (3.1.20) () 0
,
dû k (t)
+ i
dt ∆x sin (k∆x)û k(t) = 0,
( )
dû k (t) sin (k∆x)
= −i
û k (t) (k = −N, · · ·N)
dt
k∆x
. ,
( ( ) )
sin (k∆x)
û k (t) = û k (0) exp ik
t (k = −N, · · ·, N)
k∆x
. ,
( (
û k (t)e ikx j
sin (k∆x)
= û k (0) exp ik
k∆x
( (
= û k (0) exp ik x j −
) )
t
sin (k∆x)
t
k∆x
exp (ikx j )
))
(3.1.21)
.
6 ) , 0
.
( )

3 1 19
3.1.4
sin (k∆x)
1 (3.1.21)
k∆x
. k ≪ 1
sin (k∆x)
, k∆x → 0 , → 1
∆x k∆x
. , |k| . k
N ,
2π
k∆x = N ·
(
2N + 1
= N + 1 ) 2π
2 2N + 1 − 1 2 · 2π
2N + 1
( ) 2N + 1 2π
=
2π 2N + 1 − π
2N + 1
π
= π −
2N + 1
. N ≫ 1 ,
π
sin (k∆x) sin (π −
= ) 2N+1
k∆x π −
π
= 1
2N
2N+1
π
) 2N+1
π
2N+1
≈ 1
2N
, ,
.
.
. ,
. .
, . ,
. , (3.1.1)
.
sin (
3.2 1
(3.1.1) 1 , u .
,
∂u
∂t + u∂u = 0. (3.2.22)
∂x
( )

3 1 20
:
:
u(x, 0) = f(x),
u(x, t) = u(x + 2π, t).
, (3.1.1) 1
.
3.2.1 1
, .
(3.2.22) , u(x, t) xt u(x, t) x
. s
,
. ,
,
(3.2.22) ,
∂u
∂t + u∂u ∂x
. U = F (s) ,
∂U
∂t
s = x − tF (s)
u(x, t) = U(s, t)
∂u
∂t = ∂U ∂s
∂s ∂t + ∂U
∂t
= −F (s) ∂U
∂s + ∂U
∂t .
u ∂u
∂x = U ∂U ∂s
∂s ∂x
= U ∂U
∂s .
= −F (s)∂U
∂s + ∂U
∂t + U ∂U
∂s
= ∂U + (U − F (s))∂U
∂t ∂s
= 0
+ (U − F (s))∂U
∂s = 0
∂U
∂t = 0
U(s, t) = F (s)
( )

3 1 21
. ,
u(x, t) = U(s, t)
= F (s).
u(x, 0) = f(x) ,
u(x, t) = F (s)
= f(s)
. ,
x = s + tf(s),
u(x, t) = f(s)
(3.2.23)
.
3.2.2 1
(3.2.22) . , (3.1.7) (3.2.22)
, R ,
(
N∑
N
) (
dû k (t) ∑
N
)
∑
R(x, t) =
e ikx + û k (t)e ikx ikû k (t)e ikx
dt
k=−N
k=−N
k=−N
. 2 2 l ,
(
N∑
N
) (
dû k (t) ∑
N
)
∑
R(x, t) =
e ikx + û k (t)e ikx ilû l (t)e ilx
dt
=
k=−N
N∑
k=−N
dû k (t)
e ikx +
dt
k=−N
N∑
N∑
k=−N l=−N
l=−N
ilû l (t)û k (t)e ikx e ilx
. , W m = e −imx ,
[0, 2π] 0 ,
N∑
k=−N
dû k (t)
dt
∫ 2π
0
e ikx e −imx dx +
N∑
N∑
k=−N l=−N
∫ 2π
ilû l (t)û k (t) e i(k+l)x e −imx = 0
0
. ,
dû m (t)
dt
+
N∑
N∑
k=−N l=−N
ilû l (t)û k (t)δ k+l,m = 0
( )

3 1 22
. k
. ,
dû m (t)
dt
dû m (t)
dt
dû m (t)
dt
dû m (t)
dt
+
+
+
+
N∑
N∑
k=−N l=−N
N∑
m−l=−N l=−N
N+m
∑
ilû l (t)û k (t)δ k+l,m = 0
N∑
N∑
l=−N+m l=−N
min(N,N+m)
∑
l=max(−N+m)
ilû l (t)û m−l (t) = 0
ilû l (t)û m−l (t) = 0
ilû l (t)û m−l (t) = 0 (m = −N, · · ·, N) (3.2.24)
. , (3.1.11) . ,
, .
(3.2.24) .
∑
ilû m−l û L , m 2N + 1 ,
l
l 2N + 1 O((2N + 1) 2 ) 7 ) . , (3.2.22)
, (3.1.15) ,
du j
dt + u j
∂u
∂t + u∂u ∂x = 0
u j+1 − u j−1
2∆x
= 0 (j = 0, · · ·, 2N)
, O(2N + 1) . ,
.
3.2.3
. .
7 ) , , .
( )

3 1 23
.
(3.2.24) u (3.1.7)
∫
1 2π
u ∂u
2π 0 ∂x e−imx dx (k = −N, · · ·, N) (3.2.25)
. (3.2.24) .
, (3.2.25) (3.2.24)
(3.2.25) . (3.2.25) u =
,
N∑
l=−N
û l e ilx
∫
1 2π
u ∂u
2π 0 ∂x e−imx dx = 1 ∫ 2π N∑ ∑ N
il û k e ikx û l e ilx e −imx dx
2π 0
k=−N l=−N
= 1 ∫ 2π N∑ N∑
il û k û l e ikx e ilx e −imx dx
2π 0
k=−N l=−N
N∑ N∑
( ∫ 1 2π
)
= il û k û l e ikx e ilx e −imx dx
2π
k=−N l=−N
0
(m = −N, · · ·, N)
8 ) . 3N .
J > k + l + m , J ≥ 3N + 1
.
,
. .
.
1. ()
u j =
( ) ∂u
=
∂x
j
N∑
k=−N
N∑
k=−N
û k e ikx j
,
ikû k e ikx j
(j = 0, · · ·, J − 1)
(3.2.26)
8 ) C
( )

3 1 24
2.
(
F j = u ∂u )
∂x
j
( ) ∂u
= u j
∂x
j
(j = 0, · · ·, J − 1)
3. ()
ˆF k = 1 ∑J−1
F j e −ikx j
(k = −N, · · ·, N) (3.2.27)
J
j=0
. 1
3 , (Inverse Discrete Fourier Transform ; IDFT
) , (Discrete Fourier Transform ; DFT ) .
. ( ) 1
∂u
J , 2N + 1 , u j , 2 . ,
∂x
j
1 ,
J × (2N + 1) × 2 = 2J × (2N + 1).
2 , J ,
3 , J , 2N + 1 ,
,
J.
J × (2N + 1).
2J × (2N + 1) + J + J × (2N + 1) = 3J × (2N + 1) + J
, O(6JN) .
J ≥ 3N + 1 , J J = 3N + 1 ,
O(18N 2 ) .
. m − l 2N + 1 , l
2N + 1 , O(4N 2 ) .
, . ,
IDFT DFT (Fast Fourier Transform ; FFT)
.
( )

4 (FFT) 25
4
(FFT)
DFT IDFT FFT(IFFT) .
x(ˆk) =
N−1
∑
k=0
kˆk
2πi
A(k)e N (ˆk = 0, · · ·, N − 1) (4.0.1)
. ,
x ∗ (ˆk) =
N−1
∑
k=0
A(k) ∗ kˆk
−2πi
e N (4.0.2)
, . , IDFT
.
4.1 FFT
(4.0.1) ,
⎛ ⎞ ⎛
⎞ ⎛ ⎞
(ˆk, k)
⎜ ⎟ ⎜ kˆk
⎝x(ˆk) ⎠ = 2πi
⎟ ⎜ ⎟
⎝ e N ⎠ ⎝A(k) ⎠
N × N
. N × N N ,
O(N 2 ) . , FFT ,
N × N ,
⎛ ⎞ ⎛ ⎞ ⎛
⎞ ⎛ ⎞ ⎛ ⎞
⎜ ⎟ ⎜
⎝x(ˆk) ⎠ = ⎝
⎟ ⎜
⎟
⎠ ⎝O(2N) ⎠ · · ·
N × N
⎜
⎝
⎟ ⎜ ⎟
⎠ ⎝A(k) ⎠
( )

4 (FFT) 26
. N × N O(log 2 N) . ,
O(2N log 2 N) . N
1 ) .
4.1.1 FFT
FFT . (4.0.1) , N = P S
) ,
0 ≤ ˆp ≤ P − 1,
0 ≤ ŝ ≤ S − 1
(P, S
(4.1.3)
ˆp, ŝ s k, ˆk
ˆk = S ˆp + ŝ
k = P s + ˆp
(4.1.4)
. 0 ≤ k ≤ N − 1 ,
0 ≤k ≤ N − 1
0 ≤ P s+ˆp ≤ P S − 1. (4.1.5)
, 0 ≤ ˆp ≤ P − 1 ,
0 ≤ˆp ≤ P − 1
P s ≤ P s+ˆp ≤ P (1 + s) − 1 (4.1.6)
(4.1.5) (4.1.6)
P s ≥ 0
s ≥ 0,
P (1 + s) − 1 ≤ P S − 1
1 + s ≤ S
1 ) N = 2 10 , FFT O(N 2 ) N 2 = 1048576
. , FFT O(2N log 2 N) , 2N log 2 N = 2 × 1024 × 10 = 20480
1 50 .
( )

4 (FFT) 27
. , 0 ≤ s ≤ S − 1 (4.0.1) . (4.0.1)
ˆk = S ˆp + ŝ, k = P s + ˆp ,
∑S−1
(P s+ˆp)(S ˆp+ŝ)
2πi
x(S ˆp + ŝ) = A(P s + ˆp)e P S
s=0
(
)
∑S−1
x(S ˆp + ŝ) = A(P s + ˆp)e 2πi ˆps+ ˆp2
P + ˆpŝ
P S + ŝs
S
s=0
ˆp2
∑S−1
2πi
x(S ˆp + ŝ) = e P e
2πi ˆpŝ
P S A(P s + ˆp)e 2πiˆps ŝs
2πi
e S
ˆp2
−2πi
x(S ˆp + ŝ)e P
ˆp2
−2πi
x(S ˆp + ŝ)e P
∑S−1
= e
2πi ˆpŝ
P S
s=0
s=0
= X S (S ˆp + ŝ) ,
ŝs
2πi
A(P s + ˆp)e S .
ˆpŝ
∑S−1
ŝs
2πi 2πi
X S (S ˆp + ŝ) = e P S A(P s + ˆp)e S (4.1.7)
s=0
(4.1.7) S = 1, P = N 0 ≤ ŝ ≤ 0
X 1 (ˆp) = e 0
0∑
A(Ns + ˆp)e 0
s=0
X 1 (ˆp) = A(ˆp).
, S = N, P = 1 0 ≤ ˆp ≤ 0
∑S−1
X N (ŝ) = e 0 ŝs
2πi
A(s)e S
s=0
= x(ŝ)
. , A(ˆp) = X 1 (ˆp) , x(ŝ) = X N (ŝ)
. ,
X S S .
P, Q, R, S N, S
N = P S,
S = QR
. , q, ˆq, r, ˆr
0 ≤ q, ˆq ≤ Q − 1,
0 ≤ r, ˆr ≤ R − 1
(4.1.8)
( )

4 (FFT) 28
.
0 ≤ ˆq ≤ Q − 1
0 ≤ Rˆq ≤ RQ − R
0 ≤ Rˆq + ˆr ≤ RQ − 1.
RQ = S ,
0 ≤ Rˆq + ˆr ≤ S − 1.
0 ≤ ŝ ≤ S − 1 ,
ŝ = Rˆq + ˆr (4.1.9)
. ,
0 ≤ r ≤ R − 1
0 ≤ Qr ≤ QR − Q
0 ≤ Qr + q ≤ QR − 1
0 ≤ Qr ≤ S − 1.
0 ≤ s ≤ S − 1 ,
s = Qr + q (4.1.10)
. (4.1.9) (4.1.10) (4.1.7) ,
ˆpŝ
∑S−1
ŝs
2πi 2πi
X S (S ˆp + ŝ) = e P S A(P s + ˆp)e S
s=0
ˆp(Rˆq+ˆr)
2πi
= e
P QR
ˆpˆq
2πi
= e
ˆpˆq
2πi
= e
∑ ∑
Q−1 R−1
q=0 r=0
Q−1 R−1
ˆpˆr
P Q e
2πi P QR
(Rˆq+ˆr)(Qr+q)
2πi
A(P (Qr + q) + ˆp)e
QR
∑ ∑
q=0 r=0
∑ ∑
Q−1 R−1
ˆpˆr
P Q e
2πi P QR
Q−1
ˆpˆq ∑
2πi
= e
P Q
q=0
q=0 r=0
A(P Qr + P q + ˆp)e 2πi (ˆqr+ ˆrr
R + ˆqq Q + qˆr
QR)
ˆrr
2πi
A(P Qr + P q + ˆp)e R e
2πi ˆqq Q e
2πi qˆr
QR
∑
R−1
e 2πi ˆpˆr
P QR e
2πi ˆqq
Q e
2πi qˆr
QR
Q−1
ˆpˆq ∑
2πi
= e
P Q
e 2πi ˆqq Q e
2πi P (qˆr)+ˆpˆr
P QR
q=0
Q−1
ˆpˆq ∑
2πi
= e
P Q
q=0
e 2πi ˆqq Q
{
(P q+ˆp)ˆr
2πi
e
P QR
∑R−1
r=0
r=0
∑R−1
r=0
ˆrr
2πi
A(P Qr + P q + ˆp)e R
ˆrr
2πi
A(P Qr + P q + ˆp)e R
ˆrr
2πi
A(P Qr + P q + ˆp)e R
}
(4.1.11)
( )

4 (FFT) 29
, Q = 1 ,(4.1.8) 0 ≤ q, ˆq ≤ 0 ,
ˆp = (P q + ˆp),
ŝ = ˆr,
s = Qr = r
. , S = R , (4.1.7)
(P q+ˆp)ˆr ∑R−1
2πi ˆrr
X R (R(P q + ˆp) + ˆr) = e
P QR
2πi
A(P Qr + P q + ˆp)e R
. , (4.1.11) {} X R ,
r=0
Q−1
ˆpˆq ∑ ˆqq
2πi 2πi
X S (S ˆp + ŝ) = e
P Q
e
Q XR (R(P q + ˆp) + ˆr)
q=0
Q−1
ˆpˆq ∑ ˆqq
2πi 2πi
= e
P Q
e
Q XR (P Rq + Rˆp + ˆr).
q=0
, (4.1.9) (4.1.10) ,
. ,
X S (S ˆp + ŝ) = X QR (QRˆp + Rˆq + ˆr)
Q−1
ˆpˆq ∑
2πi
X QR (QRˆp + Rˆq + ˆr) = e
P Q
e 2πi ˆqq Q XR (P Rq + Rˆp + ˆr)
q=0
(0 ≤ ˆp ≤ P − 1, 0 ≤ ˆq ≤ Q − 1, 0 ≤ ˆr ≤ R − 1) (4.1.12)
X S S R QR .
4.1.2 FFT
(4.1.12) . ˆq Q , q Q , ˆp P ,
, ˆr R . ,
Q × Q × P × R = Q 2 P R
= P QR × Q
= NQ
( )

4 (FFT) 30
. N = Q 1 · Q 2 · · · Q l , X 1 → X Q1 →
X Q1 Q 2
→ · · · → X Q1 Q 2···Q l
= X N
NQ 1 , NQ 2 , · · ·, NQ l C N
. N = 2 l
C N = NQ 1 + NQ 2 + · · · + NQ l
= N(Q 1 + Q 2 + · · · + Q l )
Q 1 · Q 2 · · · Q l = 2 l .
Q N Q 1 = Q 2 = · · · = Q l = 2 . ,
. , N = 2 l ,
C N
Q 1 + Q 2 + · · · + Q l = 2 + 2 + · · · + 2
N = 2 l
= 2 × l
l = log 2 N
.
C N = N(Q 1 + Q 2 + · · · + Q l )
= N(2 × l)
= 2N log 2 N
4.2 FFT
A(k) x(ˆk)
, x(ˆk) ,
. . N
.
4.2.1
x(j) (j = 0, · · ·, N − 1) DFT
A(k) = 1 N−1
∑
(
x(j) exp −2πi jk )
N
N
j=0
(k = 0, 1, · · ·, N − 1) (4.2.13)
( )

4 (FFT) 31
. N N FFT
2
. (4.2.13) ,
A ∗ (k) = 1 N
= 1 N
N−1
∑
j=0
N−1
∑
j=0
(
x(j) ∗ exp ∗ −2πi jk )
N
(
x(j) ∗ exp 2πi jk )
N
x(j)
A ∗ (k) = 1 N
N−1
∑
j=0
(
x(j) exp 2πi jk )
.
N
, A(N − k) ,
A(N − k) = 1 N
= 1 N
N−1
∑
j=0
N−1
∑
j=0
= A ∗ (k)
(
x(j) exp 2πi
(
x(j) exp 2πi jk )
N
)
j(N − k)
N
. , A(k)
(k = 0, · · ·, N )
2 − 1 A(k) A ∗ (k)
. A(k) , A(0) , A(k)
(k = 1, · · ·, N )
2 − 1 ( ) N
, A , N . , N x(j)
2
N .
x(j) ,
A(k) = 1 N
= 1 N
N
2 −1
∑
j=0
j=0
(
x(2j) exp −2πi 2jk )
+ 1 N N
N
∑2 −1 (
x(2j) exp −2πi jk N
2
)
+ 1 N
N
2 −1
∑
j=0
j=0
(
x(2j + 1) exp −2πi
N
∑2 −1
(
x(2j + 1) exp −2πi jk N
2
)
(2j + 1)k
N
)
(
exp −2πi k )
.
N
(4.2.14)
( )

4 (FFT) 32
,
N
2 −1 (
∑
B 0 (k) ≡ x(2j) exp 2πi jk N
2
j=0
N
2 −1
(
∑
B 1 (k) ≡ x(2j + 1) exp
j=0
2πi jk N
2
)
,
)
(4.2.14)
A(k) = 1 {
(
B ∗
N
0(k) + B1(k) ∗ exp −2πi k )}
N
(4.2.15)
. , x(j) B 0 , B j .
y(j) = x(2j) + ix(2j + 1)
(j = 0, 1, · · ·, N )
2 − 1
N 2 . N y(j)
2
C(k) , C(k) B 0 , B 1
N
2 −1 ( )
∑
C(k) = y(j) exp 2πi jk N
j=0
2
N
2 −1 ( ) N
∑
= x(2j) exp 2πi jk 2 −1
( )
∑
+ i x(2j + 1) exp 2πi jk N
N
j=0
2 j=0
2
= B 0 (k) + iB 1 (k)
(k = 1, · · ·, N )
2 − 1 . (4.2.16)
,
N
( ) N
2 −1 ( )
C
2 − k ∑
= y(j) exp −2πi jk N
j=0
2
N
2 −1 ( ) N
∑
= x(2j) exp −2πi jk 2 −1
( )
∑
+ i x(2j + 1) exp −2πi jk N
N
j=0
2 j=0
2
= B0(k) ∗ + iB1(k)
(k ∗ = 1, · · ·, N )
2 − 1 (4.2.17)
. (4.2.16) (4.2.17) B 0 , B 1 ,
B0 ∗ = 1 { ( ) }
N
C
2 2 − k + C ∗ (k) ,
B1 ∗ = 1 { ( ) }
N
C
2i 2 − k − C ∗ (k)
( )

4 (FFT) 33
. k = 1, · · ·, N − 1 k
2
A(k) = 1 [{ ( ) } N
C
2N 2 − k + C ∗ (k) + 1 (
i exp −2πi k ) { ( ) }]
N
C
N 2 − k − C ∗ (k)
(4.2.18)
. , B 0 (0), B 1 (0) , (4.2.16) ,
B 0 (0) = Re(C(0))
B 1 (0) = Im(C(0))
( ) N
, A(0) A
2
A(0) = 1 N
N
2 −1
∑
x(2j) + 1 N
j=0
N
2 −1
∑
x(2j + 1)
j=0
= 1 N {B 0(0) + B 1 (0)}
= 1 N
{Re(C(0)) + Im(C(0))} , (4.2.19)
( ) N
A = 1 2 N
N
2 −1
∑
x(2j) − 1 N
j=0
N
2 −1
∑
x(2j + 1)
j=0
= 1 N {B 0(0) − B 1 (0)}
= 1 N
{Re(C(0)) − Im(C(0))} (4.2.20)
C(0) . (4.2.18) , (4.2.19) , (4.2.20)
, DFT A , y(j) = x(2j) + ix(2j + 1) N 2
FFT .
4.2.2
.
x(j) =
N−1
∑
k=0
(
A(k) exp 2πi jk )
N
(j = 0, 1, · · ·, N − 1). (4.2.21)
( )

4 (FFT) 34
x(j)
x ∗ (j) =
N−1
∑
k=0
(
A ∗ (k) exp −2πi jk )
.
N
, x(j) ,x(j) = x ∗ (j) ,
N−1
∑
m=0
(
A(k) exp 2πi jm N
N−1
∑
m=0
N−1
∑
m=0
)
exp
(
A(m) exp 2πi
x(j) = x ∗ (j)
) N−1
∑
= A ∗ (l) exp
(
A(m) exp 2πi jm N
l=0
(
) N−1
j(N − k) ∑
−2πi =
N
l=0
)
j(m − N + k)
N
N−1
∑
=
l=0
(
−2πi jl
N
(
A ∗ (l) exp −2πi jl
N
(
A ∗ (l) exp −2πi
)
) (
exp −2πi
j(l − k)
N
)
)
j(N − k)
N
. , m = N − k, l = k
A(N − k) = A ∗ (k)
, . N 2
.
FFT
(4.2.21) k N 2 ,
N
( ) N ∑2 −1
x(j) = A(0) + (−1) j A +
2
k=1
N
( ) N ∑2 −1
= A(0) + (−1) j A +
2
. 4 ,
N
2 −1
∑
k=1
k=1
(
A ∗ (k) exp −2πi jk )
=
N
=
N
(
A(k) exp 2πi jk )
∑2 −1
+
N
k=1
N
(
A(k) exp 2πi jk )
∑2 −1
+
N
k=1
(
A(N − k) exp 2πi
(
A ∗ (k) exp −2πi jk )
N
N
∑2 −1 ( ) (
N
A ∗ 2 − k′ exp −2πi j ( N
− k′)
)
2
N
k ′ =1
N
2 −1
∑
( ) N
A ∗ 2 − k′ (−1) j exp
k ′ =1
) (2πi jk′
N
(4.2.22)
)
j(N − k)
N
( )

4 (FFT) 35
, x(j) ,
N
( ) N ∑2 −1
x(2j) = A(0) + A + A(k) exp
2
k=1
N
( ) N ∑2 −1
x(2j + 1) = A(0) + −A +
2
k=1
k=1
(
A(k) exp
N
∑2 −1 ( ) (
N
− A ∗ 2 − k exp 2πi jk N
2
)
2πi jk N
2
(
) N
∑2 −1 ( ) (
N
+ A ∗ 2 − k exp
)
2πi jk N
2
(
exp 2πi k N
k=1
exp
)
(
2πi k N
)
2πi jk N
2
)
,
. x(2j) + ix(2j + 1) ,
( ) N
x(2j) + ix(2j + 1) = A(0) + A
2
{ ( )} N
+ i A(0) + A
2
N
2 −1 ( )
∑
[{
+ exp 2πi jk ( )} N
A(k) + A ∗ N
2 − k k=1
2
(
+i exp 2πi k ) ( )}] N
{A(k) − A ∗
N
2 − k
(4.2.23)
. D(k)
{ ( )} N
D(k) ≡ A(k) + A
2 − k
(
+ i exp 2πi k ) ( )} N
{A(k) − A ∗
N
2 − k (k = 0, 1, · · ·, N )
2 − 1
( ) N
. A
2
(4.2.24)
( ) ( )
N N
A = A ∗
2 2
, (4.2.23)
N
2 −1 (
∑
x(2j) + ix(2j + 1) = D(k) exp 2πi jk N
2
k=0
)
(4.2.25)
. (4.2.25) , A N D FFT
2
, x(j) .
( )

4 (FFT) 36
4.3 FFT
FFT .
4.3.1 IDFT FFT
(3.2.26) . , J ≥ 3N + 1
N∑
u j = û k e ikx j
,
k=−N
(4.3.26)
x j = 2π J j (j = 0, · · ·, J − 1)
. FFT . u j û −k = û ∗ k
. , û , û 0 , û k (k = 1, · · ·, N)
2N + 1 .
a 0 = û 0 ,
a k = Re(û k ),
b k = Im(û k ) (k = 1, · · ·, N)
, (4.3.26) ,
u j = û 0 +
= a 0 +
= a 0 +
= a 0 + 2
N∑
[û k exp (ikx j ) + û ∗ k exp (−ikx j )]
k=1
N∑
[(a k + ib k ) exp (ikx j ) + (a k − ib k ) exp (−ikx j )]
k=1
N∑
[(a k + ib k )(cos (kx j ) + i sin (kx j )) + (a k − ib k )(cos (kx j ) − i sin (kx j )]
k=1
N∑
[a k cos (kx j ) − b k sin (kx j )]
k=1
. , J J FFT
( )

4 (FFT) 37
(4.2.22) ,
∑J−1
(
x(j) = A(k) exp 2πi jk )
J
k=0
( ) J ∑
= A(0) + (−1) j A +
2
J
( ) J
= A(0) + (−1) j A +
2
( ) J
= A(0) + (−1) j A
2
J
2 −1
k=1
∑
2 −1
k=1
J
2 −1
J
(
A(k) exp 2πi jk )
∑2 −1
+
J
k=1
(
A ∗ (k) exp −2πi jk )
J
[ (
A(k) exp 2πi jk )
(
+ A ∗ (k) exp −2πi jk )]
J
J
∑
+ 2 [Re(A(k)) cos (kx j ) − Im(A(k)) sin (kx j )]
,
k=1
A(0) = a 0 ,
(Re(A(k)), Im(A(k))) = (a k , b k )
(
A(k) = 0 k = N + 1, · · ·, J )
2
(k = 1, · · ·, N),
FFT x(j) , u j .
D(k) ,
(4.2.24) 0 ≤ k ≤ N − 1
2
N − 1
2
≤ k ≤ N + 1
D(k) =
A(k) = a k + ib k
D(k) = A(k) + i exp
(
2πi k )
(A(k)).
J
{ ( )} (
J
A(k) + A
2 − k + i exp 2πi k ) ( )} J
{A(k) − A ∗
J
2 − k .
N + 1 ≤ k ≤ J 2 − 1
( ) (
J
D(k) = A
2 − k + i exp 2πi k ( ) J
)A ∗
J 2 − k
FFT .
( )

4 (FFT) 38
∂u ,
∂x
,
( ) ∂u
= 2
∂x
j
N∑
[−(ka k ) sin (kx j ) − (kb k ) cos (kx j )]
k=1
A(0) = 0,
(Re(A(k)), Im(A(k))) = (−kb k , ka k )
(
A(k) = 0 k = N + 1, · · ·, J )
2
(k = 1, · · ·, N),
,D(k) .
4.3.2 DFT FFT
(3.2.27)
ˆF k = 1 ∑J−1
F j exp (−ikx j ),
J
j=0
( ) ∂u
F j = u j (k = −N, · · ·, N)
∂x
j
. , ˆF 0 , ˆF k (k = 1, · · ·, N)
. , FFT , (4.2.13)
A(k) = 1 ∑J−1
x(j) exp (−ikx j )
J
( ) J
, x(j) , A(0) , A , A(k)
2
j=0
(k = 1, · · ·, J 2 − 1 )
. , F j x(j) FFT
A(k) ,
ˆF k = A(k) (k = 0, · · ·, N)
ˆF k . k = N + 1, · · ·, J 2
.
( )

4 (FFT) 39
4.3.3 FFT
2 2N + 1 3N + 1
FFT . 3 2
, 3 2 . ( )

5 40
5
x(t), f(x, t) ,
, ,
dx
dt
= f(x, t)
: x(t 0 ) = x 0
.
x(t), f(x, t) .
t t 0 , t 1 , , t k
x k .
,
t n+1 = t n + h
x n = x(t n )
. , 1 -.
H .
5.1 -
f(x, t) (x, t) , ,
-
.
( )

5 41
p - (i = 1, 2, · · ·, p) ,
k i = hf
(
x n +
c i =
)
p∑
α ij k j , t n + c i h , (5.1.1a)
j=1
p∑
α ij ,
j=1
x n+1 = x n +
(5.1.1b)
p∑
ω i k i (5.1.2)
. α ij x .
α ij ω i ,
. .
i=1
.
dx
dt
= f(x, t)
. p ω
x n+1 − x n =
p∑
ω i hf i (x n+1 , t n+1 ) (5.1.3)
i=1
. f i (x n+1 , t n+1 )
k i = hf i (x n+1 , t n+1 )
. (5.1.3) t n+1
α ij
f i (x n+1 , t n+1 )
f i (x n+1 , t n+1 ) = f
= f
(
(
t n+1 = t n +
x n +
x n +
p∑
α ij h.
j=1
p∑
α ij hf j (x n+1 , t n+1 ), t n +
j=1
p∑
α ij hk j , t n +
j=1
)
p∑
α ij h
j=1
)
p∑
α ij h
j=1
( )

5 42
,
c ij =
p∑
j=1
α ij
(
)
p∑
k i = hf x n + α ij k j , t n + c i h
j=1
(5.1.3)
x n+1 = x n +
p∑
ω i k i
i=1
. , (5.1.1a) , (5.1.1b) , (5.1.2) .
(5.1.1a) . -
. , -
. (5.1.1a) k i
α ij = 0 (i ≤ j)
1 ) . ,
2 ) .
5.1.1 1 1
p = 1 . (5.1.1b) (5.1.2) ,
k 1 = hf(x n , t n ),
1 ) i = (1, 2, 3) .
x n+1 = x n + ω 1 k 1
= x n + ω 1 hf(x n , t n )
k 1 = hf(x n , t n ),
k 2 = hf(x n + α 21 k 1 , t n + c 2 h),
k 3 = hf(x n + α 31 k 1 + α 32 k 2 , t n + c 3 h),
k i .
2 ) . ,
. G .
( )

5 43
. x n+1 x(t n+1 ) h 1
ω 1 hf(x n , t n ) = hf(x n , t n )
,
ω 1 = 1
.
x n+1 = x n + hf(x n , t n )
. . 1 1 .
5.1.2 2 2
p = 2 . (5.1.1b) i = 1 ,
i = 2 ,
k 1 = hf(x n , t n ).
k 2 = hf(x n + α 21 k 1 , t n + c 2 h).
(α 21 k 1 , c 2 h) f 1
(
( ) ( )
∂f(xn , t n ) ∂f(xn , t n )
k 2 = h f(x n , t n ) + α 21 k 1 + c 2 h
∂x
∂t
= h ( f + α 21 k 1 f x + α 21 hf t + O(h 2 ) ) .
( ) ( )
∂f(xn , t n ) ∂f(xn , t n )
f(x n , t n ) = f,
= f x ,
= f t .
∂x
∂t
(5.1.2) ,
x n+1 = x n + ω 1 k 1 + ω 2 k 2
= x n + ω 1 hf + ω 2 h ( f + α 21 hff x + α 21 hf t + O(h 2 ) )
= x n + h(ω 1 + ω 2 )f + h 2 ω 2 α 21 (ff x + f t ) + O(h 3 )
. , F (f.1)
x ′′ = ff x + f t
)
+ O(h 2 )
. , 2 ,
ω 1 + ω 2 = 1,
ω 2 α 21 = 1 2
. 1 . ω 1 , ω 2 , α 21
. 2 2 .
( )

5 44
ω 1 = 0, ω 2 = 1,
. ,
c 2 = α 21 = 1 2
k 1 = hf(x n , t n ),
k 2 = hf
(x n + 1 2 k 1, t + 1 )
2 h ,
x n+1 = x n + k 2
. .
5.1.3 3 3
p = 3 . (5.1.1b) i = 1 ,
k 1 = hf(x n , t n ).
i = 2 ,
k 2 = hf(x n + α 21 k 1 , t n + c 2 h).
(α 21 k 1 , c 2 h) f 2
(
k 2 = h f + α 21 k 1 f x + c 2 hf t + 1 2 c2 2h 2 f 2 f x 2 + 1 )
2 c2 2h 2 f t 2 + O(h 3 )
i = 3 ,
= hf + c 2 h 2 ff x + c 2 h 2 f t + 1 2 c2 2h 3 f 2 f x 2 + 1 2 c2 2h 3 f t 2 + O(h 4 ).
( ) ( )
∂ 2 f(x n , t n ) ∂ 2 f(x n , t n )
= f
x 2 x 2,
= f
t 2 t 2 .
k 3 = hf(x n + α 31 k 1 + α 32 k 2 , t n + c 3 h).
( )

5 45
k 2 (α 31 k 1 + α 32 k 2 , c 3 h) f 3
k 3 = h
(f + (α 31 k 1 + α 32 k 2 )f x + (α )
31k 1 + α 32 k 2 ) 2
f x 2 + c 3 hf t + c2 3
2
2 f t 2 + O(h3 )
= hf + h 2 α 31 ff x + α 32 h 2 ff x + c 3 h 2 f t
+α 32 α 21 h ( ) 3 ffx 2 1
+ f t f x +
2 c2 3h 3 f t 2 + h3 f 2 f x 2 ( )
α
2
2 31 + α32 2 1
+ 2α 31 α 32 +
2 h3 c 2 3f t 2 + O(h 4 )
= hf + h 2 c 3 (ff x + f t ) + c2 )
3
2 h3 f 2 f x 2 + f t 2 + α32 c 2 h 3 (ff x + f t f x ) + O(h 4 ).
(5.1.2) ,
x n+1 = x n + ω 1 k 1 + ω 2 k 2 + ω 3 k 3
)
= x n + ω 1 hf + ω 2
(hf + c 2 h 2 ff x + c 2 h 2 f t + 1 2 c2 2h 3 f 2 f x 2 + 1 2 c2 2h 3 f t 2
(
+ω 3 hf + h 2 c 3 (ff x + f t ) + c2 ( )
)
3
2 h3 f 2 f x 2 + f t 2 + α32 c 2 h 3 (ff x + f t f x )
= x n + h(ω 1 + ω 2 + ω 3 )f + h 2 (ω 2 c 2 + ω 3 c 3 ) (ff x + f t )
+ h3 (c 2 2ω 2 + c 2 3ω 3 ) ( ) ( )
f 2 f x 2 + f t 2 + h 3 ω 3 α 32 c 2 ff
2
2
x + f t f x + O(h 4 ).
. F (f.2)
+ O(h 4 )
x ′′′ = f t 2 + f t f x + ff 2 x + f 2 f x 2
, x n+1
1
6 (f 2 f x 2 + f t 2) + 1 6 (f tf x + ff 2 x)
. , 3 ,
ω 1 + ω 2 + ω 3 = 1,
ω 2 c 2 + ω 3 c 3 = 1 2 ,
c 2 3ω 3 + c 2 2ω 2 = 1 3 ,
ω 3 α 32 c 2 = 1 6
. 6 4 . 2
. 1 .
( )

5 46
3
ω 1 = 1 4 , ω 2 = 0, ω 3 = 3 4 ,
c 2 = 1 3 , c 3 = 2 3 ,
α 32 = 2 3 ,
α 21 = c 2 = 1 3 ,
α 31 = c 3 − α 32 = 0
. ,
k 1 = hf(x n , t n ),
k 2 = hf
(x n + 1 3 k 1, t n + 1 )
3 h ,
k 3 = hf
(x n + 2 3 k 2, t n + 2 )
3 h ,
x n+1 = x n + 1 4 (k 1 + 3k 3 ) .
3 3 3 .
5.1.4 4 4
p = 4 . (5.1.1b) i = 1 ,
k 1 = hf(x n , t n ).
i = 2 ,
k 2 = hf(x n + α 21 k 1 , t n + c 2 h).
(α 21 k 1 , c 2 h) f 3
(
k 2 = h f + α 21 k 1 f x + c 2 hf t + 1 2 c2 2h 2 f 2 f x 2 + 1 2 c2 2h 2 f t 2 + 1 6 c3 2h 3 f 3 f x 3 + 1 )
6 c3 2h 3 f t 3 + O(h 4 )
= hf + c 2 h 2 ff x + c 2 h 2 f t + 1 2 c2 2h 3 f 2 f x 2 + 1 2 c2 2h 3 f t 2 + 1 6 c3 2h 4 f 3 f x 3 + 1 6 c3 2h 4 f t 3 + O(h 5 ).
( )

i = 3 ,
5 47
( ) ( )
∂ 2 f(x n , t n ) ∂ 2 f(x n , t n )
= f
x 3 x 2,
= f
t 2 t 3 .
k 3 = hf(x n + α 31 k 1 + α 32 k 2 , t n + c 3 h).
k 2 (α 31 k 1 + α 32 k 2 , c 3 h) f 4
k 3 = h
(f + (α 31 k 1 + α 32 k 2 )f x + (α 31k 1 + α 32 k 2 ) 2
f x 2 + (α 31k 1 + α 32 k 2 ) 3
f x 3
2
6
)
+c 3 hf t + c2 3
2 h2 f t 2 + c3 3
6 h3 f t 3 + O(h 4 )
= hf + h 2 α 31 ff x + α 32 h 2 ff x + c 3 h 2 f t
+α 32 α 21 h ( ) 3 ffx 2 h 3 f 2 f x 2 ( )
+ f t f x + α
2
2 31 + α32 2 1
+ 2α 31 α 32 +
2 h3 c 2 3f t 2
( )
1
+h 4 f 2 f x 2f x
2 α 32c 2 2 + α 31 α 32 c 2 + α32c 2 2 + α 32
( )
2 h4 c 2 2f x f t 2 + ff x 2f t α
2
31 c 2 + α 31 α 32 c 2
+ h4 f 3
6
( )
α
3
31 + 3α32α 2 31 + 3α 31 α32 2 + α32
3 fx 3 + 1 6 h4 c 3 3f t 3 + O(h 5 )
= hf + h 2 c 3 (ff x + f t ) + c2 ( )
3
2 h3 f 2 f x 2 + f t 2 + α32 c 2 h 3 (ff x + f t f x )
+ c3 ( )
3
6 h4 f 3 α 32 c 2
f x 3 + f t 3 +
2
2 h4 f x f t 2 + α 32 c 3 c 2 h 4 ff x 2f t + c 2α 32 (c 2 + 2c 3 )
2
h 4 f 2 f x 2f x + O(h 5 ).
( )

5 48
i = 4 ,
k 4 = hf(x n + α 41 k 1 + α 42 k 2 + α 43 k 3 , t n + c 4 h).
(α 41 k 1 + α 42 k 2 + α 43 k 3 , c 4 h) f 4
k 4 = h
(f + (α 41 k 1 + α 42 k 2 + α 43 k 3 )f x + (α 41k 1 + α 42 k 2 + α 43 k 3 ) 2
f x 2
2
+ (α )
41k 1 + α 42 k 2 + α 43 k 3 ) 3
f x 3 + c 4 hf t + c2 4
6
2 f t 2 + c3 4
6 f t 3 + O(h4 )
= hf + h 2 α 41 ff x + α 42 h 2 ff x + α 43 h 2 ff x + c 4 h 2 f t
+α 42 α 21 h ( ) 3 ffx 2 + f t f x + α43 c 3 h ( )
3 ffx 2 + f t f x
+ h3 f 2 f x 2
2
+ α 42c 2 2h 4 f 2 f x 2f x
2
(
α
2
41 + α 2 42 + α 2 43 + 2α 41 α 42 + 2α 42 α 43 + 2α 43 α 41
)
+
1
2 h3 c 2 3f t 2
+ + α 43c 2 3h 4 (f 2 f x 2f x + f x f t 2
2
+ (α2 42c 2 + α43c 2 3 + α 41 α 42 c 2 + α 41 α 43 c 3 + α 42 α 43 c 3 + α 42 α 43 c 2 ) (f 2 f x 2f x + ff x 2f t )
+ α 42
2 h4 c 2 2f x f t 2
2
+ h4 f 3 (
α
3
6 41 + α42 3 + α43
3 )
+3α41α 2 42 + 3α 41 α42 2 + 3α41α 2 43 + 3α 41 α43 2 + 3α42α 2 43 + 3α 42 α43 2 + 6α 41 α 42 α 43 fx 3
+ 1 6 h4 c 3 4f t 3 + O(h 5 )
= hf + h 2 c 4 (ff x + f t ) + c2 ( )
4
2 h3 f 2 f x 2 + f t 2 + (α42 c 2 + α 43 c 3 )h 3 (ff x + f t f x )
+ c3 ( )
4
6 h4 f 3 f x 3 + f t 3 + α43 α 32 c 2 h 4 (ffx 3 + fxf 2 t ) + (α 42c 2 2 + α 43 c 2 3)
h 4 f x f t 2
2
+(α 42 c 2 + α 43 c 3 )c 4 h 4 ff x 2f t + (α 42c 2 2 + α 43 c 2 3 + 2α 43 c 3 c 4 + 2α 42 c 2 c 4 )
h 4 f 2 f x 2f x + O(h 5 ).
2
( )

5 49
(5.1.2) ,
x n+1 = x n + ω 1 k 1 + ω 2 k 2 + ω 3 k 3 + ω 4 k 4
= x n + ω 1 hf
)
+ω 2
(hf + c 2 h 2 ff x + c 2 h 2 f t + 1 2 c2 2h 3 f 2 f x 2 + 1 2 c2 2h 3 f t 2 + 1 6 c3 2h 4 f 3 f x 3 + 1 6 c3 2h 4 f t 3
(
+ω 3 hf + h 2 c 3 (ff x + f t ) + c2 ( )
3
2 h3 f 2 f x 2 + f t 2 + α32 c 2 h 3 (ff x + f t f x )
+ c3 ( )
3
6 h4 f 3 α 32 c 2
f x 3 + f t 3 +
2
+ω 4
(
hf + h 2 c 4 (ff x + f t ) + c2 4
2 h3 ( f 2 f x 2 + f t 2
+ c3 4
6 h4 ( f 3 f x 3 + f t 3
2 h4 f x f t 2 + α 32 c 3 c 2 h 4 ff x 2f t + α )
32c 2 (c 2 + 2c 3 )
h 4 f 2 f x 2f x
2
)
+ (α42 c 2 + α 43 c 3 )h 3 (ff x + f t f x )
)
+ α43 α 32 c 2 h 4 (ffx 3 + fxf 2 t ) + (α 42c 2 2 + α 43 c 2 3)
h 4 f x f t 2
2
+(α 42 c 2 + α 43 c 3 )c 4 h 4 ff x 2f t + (α )
42c 2 2 + α 43 c 2 3 + 2α 43 c 3 c 4 + 2α 42 c 2 c 4 )
h 4 f 2 f x 2f x + O(h 5 )
2
= x n + h(ω 1 + ω 2 + ω 3 + ω 4 )f + h 2 (ω 2 c 2 + ω 3 c 3 + ω 4 c 4 ) (ff x + f t )
+ h3 (c 2 2ω 2 + c 2 3ω 3 + c 2 4ω 4 ) (
f 2 f x 2 + f t 2)
+ h 3 (α 32 c 2 ω 3 + (α 42 c 2 + α 43 c 3 )ω 4 ) ( )
ffx 2 + f t f x
2
+ h4 (c 3 2ω 2 + c 3 3ω 3 + c 3 4ω 4 )
(f 2 f x 2 + f t 2) + h 4 α 32 α 43 c 2 ω 4 (ffx 3 + f 2
6
xf t )
+ h4 (α 32 c 2 2ω 3 + (α 42 c 2 2 + α 43 c 2 3)ω 4 )
f x f t 2 + h 4 (α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 )ff x 2f t
2
+ h4 (α 32 c 2 2 + α 42 c 2 2 + α 43 c 2 3 + 2α 32 c 2 c 3 + 2α 42 c 2 c 4 + 2α 43 c 3 c 4 )
f 2 f x 2f x + O(h 5 ).
2
. F (f.3)
x ′′′′ = f 3 f x 3 + f t 3 + f 2 xf t + ff 3 x + +f x f t 2 + 3ff x 2f t + 4f 2 f x 2f x .
. x n+1 4
1
24 (f 3 f x 3 + f t 3) + 1
24 (f xf 2 t + ffx) 3 + 1
24 f xf t 2 + 1 8 ff x 2f t + 1 6 f 2 f x 2f x
. .
(f 3 f x 3 + f t 3)
c 3 2ω 2 + c 3 3ω 3 + c 3 4ω 4
6
= 1 24
c 3 2ω 2 + c 3 3ω 3 + c 3 4ω 4 = 1 4 .
( )

5 50
f 2 xf t + ff 3 x α 32 α 43 c 2 ω 4 = 1 24 .
f x f t 2
α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4
2
= 1
24
α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 = 1
12 .
ff x 2f t
α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 = 1 8 .
f 2 f x 2f x
α 32 c 2 2 + α 42 c 2 2 + α 43 c 2 3 + 2α 32 c 2 c 3 + 2α 42 c 2 c 4 + 2α 43 c 3 c 4 2 = 1 6
α 32 c 2 2 + α 42 c 2 2 + α 43 c 2 3 + 2α 32 c 2 c 3 + 2α 42 c 2 c 4 + 2α 43 c 3 c 4 = 1 3 .
, α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 = 1 12
α 32 c 2 2 + α 42 c 2 2 + α 43 c 2 3 + 2α 32 c 2 c 3 + 2α 42 c 2 c 4 + 2α 43 c 3 c 4 = 1 3
2α 32 c 2 c 3 + 2α 42 c 2 c 4 + 2α 43 c 3 c 4 + 1 12 = 1 3
α 32 c 2 c 3 + (α 42 c 2 + α 43 c 3 )c 4 = 1 8
ff x 2f t . , 4
( )

5 51
,
ω 1 + ω 2 + ω 3 + ω 4 = 1,
c 2 ω 2 + c 3 ω 3 + c 4 ω 4 = 1 2 ,
c 2 2ω 2 + c 2 3ω 3 + c 2 4ω 4 = 1 3 ,
α 32 c 2 ω 3 + (α 42 c 2 + α 43 c 3 )ω 4 = 1 6 ,
c 3 2ω 2 + c 3 3ω 3 + c 3 4ω 4 = 1 4 ,
(5.1.4)
α 32 α 43 c 2 ω 4 = 1 24 ,
α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 = 1 8 ,
α 32 c 2 2ω 3 + (α 42 c 2 2 + α 43 c 2 3)ω 4 = 1
12
. 10 8 . , 2
. .
.
ω 3 .
ω 1 = 1 6 , ω 2 = 2 3 − ω 3, ω 4 = 1 6
c 2 = 1 2 , c 3 = 1 2 , c 4 = 1,
α 21 = c 2 = 1 2 ,
α 32 = 1
6ω 3
,
α 31 = c 3 − α 32 = 1 2 − 1
6ω 3
,
α 42 = 1 − 3ω 3 , α 43 = 3ω 3 ,
α 41 = c 4 − α 42 − α 43 = 0.
(5.1.5)
( )

5 52
, ω 3 = 1 , (5.1.5)
3
ω 1 = 1 6 , ω 2 = 1 3 , ω 3 = 1 3 , ω 4 = 1 6
c 2 = 1 2 , c 3 = 1 2 , c 4 = 1,
α 21 = 1 2 ,
α 31 = 0, α 32 = 1 2 ,
α 42 = 0, α 43 = 1, α 41 = 0
,
k 1 = hf(x n , t n ),
k 2 = hf
(x n + 1 2 k 1, t n + 1 )
2 h ,
k 3 = hf
(x n + 1 2 k 2, t n + 1 )
2 h ,
(5.1.6)
k 4 = hf (x n + k 3 , t n + h) ,
x n+1 = x n + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 )
. , 4 4 .
5.1.5 - 5
p ≥ 5 , p p . p 3 ) q
, p q 5.1.1 .
5 .
, 4 . ,
4 ) . 4 4 .
3 ) p
4 ) f .
( )

5 53
p 1 2 3 4 5 6 7 8 9 10
q 1 2 3 4 4 5 6 6 7 7
5.1.1: p q (, 2004:
)
5.2
. x(0) = x 0
dx
dt
= f(x, t) = λx (5.2.7)
, λ . (5.2.7) t → ∞
, 5 ) .
x(t) = x 0 e λt .
5.2.1 -
-
h . ,
. , .
1 1
(5.2.7) 1 1 .
k 1 = hf(x, t)
= λhx
˜x(t + h) = x + k 1
= (1 + λh)x
5 ) ,
, 1 , λ
(5.2.7) . ,
.
( )

5 54
. ˜x . ,
x(t + h) = x 0 e λ(t+h)
= x 0 e λt e λh
= x(t)e λh
= (1 + λh + · · ·) x(t)
. x(t + h) ˜x(t + h) 1
. , 1 h 1 , ,
1 1
. λh = z ,
˜x(t + h)
x
= 1 + z
. , 1 |1 + z| .
,
|1 + z| ≤ 1
6 ) . a b z = a + bi ,
,
|1 + z| = |1 + a + bi|
√
= (1 + a) 2 + b 2
= √ a 2 + 2a + b 2 + 1 (5.2.8)
a 2 + 2a + +b 2 + 1 ≤ 1 (5.2.9)
. (a, b) .
2 2
2 2 1 1 . 2 z = λh
2 . , 2 2
, 1 ,
6 ) ,
˜x(t + h)
x
˜x(t + h)
x
= 1 + z + 1 2 z2
.
( )

5 55
5.2.1: 1 1 -.
,z = a + bi ,
∣ 1 + z + 1 ∣ ∣∣∣
2! z2 ≤ 1
(a, b) . ,
∣ 1 + z + 1 ∣ ∣∣∣ 2 z2 =
∣ 1 + a + bi + 1 ∣ ∣∣∣
2 (a + bi)2 =
∣ 1 + 1 2 (a2 + 2a − b 2 ) + b(a + 1)i
∣
=
√ (
1 + 1 ) 2
2 (a2 + 2a − b 2 ) + b 2 (a + 1) 2
=
√
1
4 (a4 + 2a 2 b 2 + b 4 ) + a 3 + a 2 + 2a 2 + 2a + 1 (5.2.10)
,
1
4 (a4 + 2a 2 b 2 + b 4 ) + a 3 + ab 2 + 2a 2 + 2a + 1 ≤ 1 (5.2.11)
. (a, b) .
( )

5 56
5.2.2: 2 2 -.
3 3
3 3 2 2 . 3 z = λh
3 . , 3 3
, 1 ,
˜x(t + h)
x
= 1 + z + 1 2! z2 + 1 3! z3
, z = a + bi ,
∣ 1 + z + 1 2! z2 + 1 ∣ ∣∣∣
3! z3 ≤ 1
( )

5 57
(a, b) . ,
∣ 1 + z + 1 2! z2 + 1 ∣ ∣∣∣ 3! z3 =
∣ 1 + a + bi + 1 2 (a2 + 2abi + b 2 ) + 1 6 (a3 + 3a 2 bi − 3a 2 − b 3 i)
∣
=
∣ 1 + 1 6 (a3 + 3a 2 + 6a − 3b 2 − 3ab 2 ) + 1 6 b(3a2 + 6a − b 2 )i
∣
=
=
√ (
1 + 1 ) 2 ( 2
1
6 (a3 + 3a 2 + 6a − 3b 2 − 3ab 2 ) +
6 b(3a2 + 6a − b )) 2
[ ( 1
1 +
36 a6 + 1 6 a5 + 7
12 a4 + 4 3 a3 + 2a 2 + 2a + 1
12 a4 b 2
+ 1 3 a3 b 2 + 1 2 a2 b 2 + 1
12 a2 b 4 + 1 6 ab4 + 1 36 b6 − 1 )] 1
2
12 b4
(5.2.12)
,
( 1
1 +
36 a6 + 1 6 a5 + 7 12 a4 + 4 3 a3 + 2a 2 + 2a + 1
12 a4 b 2
+ 1 3 a3 b 2 + 1 2 a2 b 2 + 1
12 a2 b 4 + 1 6 ab4 + 1
36 b6 − 1 )
12 b4
.
≤ 1 (5.2.13)
4 4
4 4 . 4 z = λh 4
. , 4 4 .
,
˜x(t + h)
x
= 1 + z + 1 2! z2 + 1 3! z3 + 1 4! z4
, z = a + bi ,
∣ 1 + z + 1 2! z2 + 1 3! z3 + 1 ∣ ∣∣∣
4! z4 ≤ 1 (5.2.14)
(a, b) . ,
( )

5 58
5.2.3: 3 3 -.
∣ 1 + z + 1 2! z2 + 1 3! z3 + 1 ∣ ∣∣∣
4! z4 =
=
=
=
∣ 1 + a + bi + 1 2 (a2 + 2abi + b 2 ) + 1 6 (a3 + 3a 2 bi − 3a 2 − b 3 i)
+ 1
24 (a4 + 4a 3 bi − 6a 2 b 2 − 4ab 3 i + b 4 )+
∣
∣ 1 + 1
24 (a4 + 4a 3 + 12a 2 + 24a − 12b 2 + b 4 − 6a 2 b 2 − 12ab 2 )
+ 1
24 b(12a2 + 24a − 4b 2 + 4a 3 b − 4ab 3 )i
∣
[ (
1 + 1
24 (a4 + 4a 3 + 12a 2 + 24a − 12b 2 + b 4 − 6a 2 b 2 − 12ab 2 )
( ) ] 1
2 2
1
+
24 b(12a2 + 24a − 4b 2 + 4a 3 b − 4ab 3 )
[ ( 1
1 +
576 a8 + 1
72 a7 + 5
72 a6 + 1 4 a5 + 2 3 a4 + 4 3 a3 + 2a 2 + 2a
+ 1
144 a6 b 2 + 1
24 a5 b 2 + 1 8 a4 b 2 + 1
96 a4 b 4 + 1 6 a3 b 2 + 1
24 a3 b 4
+ 1
24 a2 b 4 + 1
144 a2 b 6 − 1
12 ab4 + 1
72 ab6 + 1
576 b8 − 1 )] 1
2
72 b6 (5.2.15)
) 2
( )

5 59
,
( 1
1 +
576 a8 + 1 72 a7 + 5
72 a6 + 1 4 a5 + 2 3 a4 + 4 3 a3 + 2a 2 + 2a
.
+ 1
144 a6 b 2 + 1 24 a5 b 2 + 1 8 a4 b 2 + 1
96 a4 b 4 + 1 6 a3 b 2 + 1
24 a3 b 4
+ 1
24 a2 b 4 + 1
144 a2 b 6 − 1
12 ab4 + 1
72 ab6 + 1
576 b8 − 1
72 b6 )
≤ 1 (5.2.16)
( )

5 60
5.2.4: 4 4 -.
( )

6 61
6
f(x) x = π
) N
( 1 − cos x
f(x) =
2
(
= sin x ) 2N
(6.0.1)
2
, (3.1.1) 1 ) . (6.0.1)
(3.1.1) f(x − t) t = 0 6.0.1 . (3.1.1)
6.0.1: (6.0.1) ,t = 0 f(x − t) .
1 ) I .
( )

6 62
(3.1.10)
. ,
dû k
dt
= f(t)
= −ikû k (6.0.2)
.
. 5 4 -
6.1 -
, -
.
6.1.1 1 1
1 1 (5.2.9) . (6.0.2) a = 0, b = −kh
. , 1 1
b 2 = (−kh) 2
≤ 0.
, . (5.2.8)
√
1 + b2 = √ 1 + (−kh) 2
= √ 1 + (kh) 2
. . (6.0.2) - 1 1
,
k 1 = hf(û k )
= −ikhû k
ũ k = û k + k 1
. ũ k t n+1 û k .
( )

6 63
6.1.2: (6.0.2) N = 100, h = 2π t = 0.12π
100
.
N = 100,h = 2π t = 0.12π 6.1.2 . 6.1.2
100
. k = N ,
.
√
1 + (kh)2 = √ 1 + (2π) 2
∼ 6.3623
( )

6 64
, kh
. 6.1.3 N = 75, h = 2π t = 0.1π
500
. . ,
. k = N ,
√
( )
√ 2 3π
1 + (kh)2 = 1 +
10
.
∼ 1.3741
6.1.3: (6.0.2) N = 75, h = 2π t = 0.1π
500
.
1 1 1
1 kh ≫ 1 -
. , kh
. , kh
1 .
( )

6 65
6.1.2 2 2
2 2 (5.2.11) . (6.0.2) a = 0, b = −kh
2 2
1
4 b2 = 1 4 (−kh)2
≤ 0.
, . (5.2.10)
√ √
1 1
4 b4 + 1 =
4 (−kh)4 + 1
=
√
1
4 (kh)4 + 1
. 2 2 .(6.0.2) -
2 2 ,
.
k 1 = hf(û k )
= −ikhû k
(
k 2 = hf û k + 1 )
2 k 1
(
= −ikh û k + 1 )
2 k 1
ũ k = û k + k 2
, 2 2 1 1 .
N = 100,h = 2π t = 0.12π 6.1.4 . 6.1.4
100
1 1 . k = N
√ √
1 1
4 (kh)4 + 1 =
4 (2π)4 + 1
∼ 19.7645
. 1 1
.
2 2 kh ≫ 1 1 1 kh
1 1 . 6.1.5 N = 75, h = 2π
500
t = 0.1π .
( )

6 66
6.1.4: (6.0.2) N = 100, h = 2π t = 0.12π
100
.
6.1.5 1 1 6.1.3
. k = N
√
1
4 (kh)4 + 1 =
√
1
4
∼ 1.0942
( ) 2 3π
+ 1
10
. 1 1
. , 1 t .
2 2 . 6.1.5
t = 0.6π . 6.1.6 .
.
( )

6 67
6.1.5: (6.0.2) N = 75, h = 2π t = 0.1π
500
.
t = π 6.1.7 . .
1 2 2
1 ,
. ,
, . , 2 2
1 .
( )

6 68
6.1.6: (6.0.2) N = 75, h = 2π t = 0.6π
500
.
( )

6 69
6.1.7: (6.0.2) N = 75, h = 2π t = π
500
.
( )

6 70
6.1.3 3 3
3 3 (5.2.13) . (6.0.2) a = 0, b = −kh
1
36 b6 − 1
12 b4 = 1 36 (−kh)6 − 1
12 (−kh)4
= 1 36 (−kh)4 ((−kh) 2 − 3) ≤ 0.
− √ 3 ≤ kh ≤ √ 3 . (5.2.12)
√
1 + 1
36 b6 − 1 √
12 b4 = 1 + 1 36 (−kh)6 − 1
12 (−kh)4
√
= 1 + 1 36 (kh)6 − 1
12 (kh)4
.
3 3 3 . (6.0.2) -
3 3 3 ,
.
k 1 = hf(û k )
= −ikhû k
(
k 2 = hf û k + 1 )
3 k 1
(
= −ikh û k + 1 )
3 k 1
(
k 3 = hf û k + 2 )
3 k 2
(
= −ikh û k + 2 )
3 k 2
ũ k = û k + 1 4 (k 1 + 3k 3 )
3 3
2 ) . 3 3 , 2 2
. , . 1 2
, N = 100,h = 2π . t = 0.12π 6.1.8
100
.
( )

6 71
6.1.8: (6.0.2) 3 N = 100, h = 2π
100 t =
0.12π .
6.1.8 1 1 2 2 .
k = N
√
1 + 1 36 (kh)6 − 1
√
12 (kh)4 = 1 + 1 36 (2π)6 − 1
12 (2π)4
∼ 39.7525
. 2 2
.
N = 75, h = 2π
2π
. 6.1.9 N = 75, h =
500 500
t = 0.1π . 2 2 . 3
3 .
2 ) 5
( )

6 72
6.1.9: (6.0.2) 3 N = 75, h = 2π t = 0.1π
500
.
k = N
√
1 + 1 36 (kh)6 − 1
12 (kh)4 =
√
1 + 1 36
∼ 0.9943
( 2π
5
) 6
− 1
12
( 2π
5
) 4
. 1 .
t = 0.6π . 6.1.10 . 2 2
3 3 .
( )

6 73
6.1.10: (6.0.2) 3 N = 75, h = 2π t = 0.6π
500
.
( )

6 74
6.1.8
6.1.11 . . ,
.
6.1.11: (6.0.2) 3 N = 27, h = 2π
100 t =
0.12π .
k = N
√
1 + 1 36 (kh)6 − 1
12 (kh)4 =
√
1 + 1
36
∼ 0.9859
( ) 6 27π
− 1
50 12
( ) 4 27π
50
. t = 2π 6.1.12 .
, .
( )

6 75
6.1.12: (6.0.2) 3 N = 27, h = 2π t = 2π
100
.
( )

6 76
. N = 100, h = 2π
340
. t = 0.1π 6.1.13 .
6.1.13: (6.0.2) 3 N = 100, h = 2π
340 t =
0.1π .
k = N
√
1 + 1 36 (kh)6 − 1
12 (kh)4 =
√
1 + 1
36
∼ 1.0651
( ) 6 10π
− 1
17 12
( ) 4 10π
17
. 1 . , 1
.
t = 2π 6.1.14 . 6.1.14 .
( )

6 77
6.1.14: (6.0.2) 3 N = 100, h = 2π t = 2π
340
.
( )

6 78
1 3 3
1 − √ 3 ≤ kh ≤ √ 3
. , .
6.1.4 4 4
4 4 (5.2.16) . (6.0.2) a = 0, b = −kh
1
576 b8 − 1
72 b6 = 1
576 (−kh)8 − 1
72 (−kh)6
= 1
576 (−kh)6 ((−kh) 2 − 8) ≤ 0.
−2 √ 2 ≤ kh ≤ 2 √ 2 . (5.2.15)
√
1 + 1
576 b8 − 1 √
72 b6 = 1 + 1
576 (−kh)8 − 1
72 (−kh)6
√
= 1 + 1
576 (kh)8 − 1
72 (kh)6
. 4 4 -. (6.0.2)
- 4 4 -
,
k 1 = hf(û k )
= −ikhû k
(
k 2 = hf û k + 1 )
2 k 1
(
= −ikh û k + 1 )
2 k 1
(
k 3 = hf û k + 1 )
2 k 2
(
= −ikh û k + 1 )
2 k 2
k 4 = hf (û k + k 3 )
= −ikh (û k + k 3 )
ũ k = û k + 1 6 (k 1 + 2k 2 + 2k 3 + k 4 )
( )

6 79
.
3 3 3 3 . , N = 100,h = 2π
100
. t = 0.12π 6.1.15 .
6.1.15: (6.0.2) - N = 100, h = 2π
100
t = 0.12π .
6.1.15 .
k = N
√
1 + 1
576 (kh)8 − 1
√
72 (kh)6 = 1 + 1
576 (2π)8 − 1 72 (2π)6
∼ 57.9962
. 3 3 .
3 3 N = 100, h = 2π . , 3 3
340
t = 0.1π . 6.1.16 .
( )

6 80
6.1.16: (6.0.2) - N = 100, h = 2π
340
t = 0.1π .
k = N
√
1 + 1
576 (kh)8 − 1
72 (kh)6 =
√
1 + 1
576
∼ 0.8264
. 3 3 .
( ) 8 10π
− 1 17 72
( ) 6 10π
17
t = 2π . 6.1.17 . 3 3
6.1.17 .
3 3 6.1.15
6.1.18 . 6.1.15
6.1.18 .
( )

6 81
6.1.17: (6.0.2) - N = 100, h = 2π
340
t = 2π .
6.1.18: (6.0.2) - N = 45, h = 2π
100
t = 0.12π .
( )

6 82
k = N
√
1 + 1
576 (kh)8 − 1
72 (kh)6 =
√
1 + 1
576
∼ 0.9975
( 9π
10
) 8
− 1 72
( ) 6 9π
10
. t = 2π . 6.1.19
.
6.1.19: (6.0.2) - N = 45, h = 2π
100
t = 2π .
6.1.19 .
1 4 4
1 −2 √ 2 ≤ kh ≤ 2 √ 2
. , 1 -
.
( )

6 83
1
4 4 - 1
. N = 100, h = 2π t = 0,t = π,t = 2π
500
. , .
, 6.1.20, 6.1.21, 6.1.22 .
, .
.
6.1.20: t = 0 , :N = 100, h = 2π
500 -
, :.
( )

6 84
6.1.21: t = π , :N = 100, h = 2π
500 -
, :.
6.1.22: t = 2π , :N = 100, h = 2π
500 -
, :.
( )

7 85
7
.
.
, .
1 ,
. 1
. 1
.
.
1
6.1.20, 6.1.21, 6.1.22 . ,
.
. 1 , -
. 1 2
. 3
N, h
0 ≤ Nh ≤ √ 3
. ,
. 4
0 ≤ Nh ≤ 2 √ 2
, . 5
. . , 1
- 4 4
.
( )

8 86
8
A:
( ) d 2 u
x = x j = j∆x u(x)
.
dx 2 x=x j
2 .
( d 2 u
dx 2 )
x=x j
≈ u j+2 − 2u j+1 + u j
(∆x) 2 , (a.f1)
( d 2 u
dx 2 )
x=x j
≈ u j − 2u j−1 + u j−2
(∆x) 2
(a.f2)
. , , . .
u((j + 1)∆x) x = x j ,
( ) du
u((j + 1)∆x) = u(j∆x) +
(∆x) +
dx
x=j∆x
u((j + 2)∆x) ∆x = 0
( )
( du
d 2 u
u((j + 2)∆x) = u(j∆x) + 2
(∆x) + 2
dx
x=j∆x
( 1 d 2 u
(∆x)
2 dx
)x=j∆x
2 +
2
(∆x)
dx
)x=j∆x
2 +
2
( 1 d 3 u
(∆x)
6 dx
)x=j∆x
3 + · · ·.
3
(a.f3)
( 4 d 3 u
(∆x)
3 dx
)x=j∆x
3 + · · ·.
3
(a.f4)
( )

8 87
(a.f3) 2 , (a.f4) ,
( ( d 2 u
d
u((j + 2)∆x) − 2u((j + 1)∆x) = −u(j∆x) +
(∆x)
dx
)x=j∆x
2 3 u
+
(∆x) 2 dx
)x=j∆x
3
3
+ · · ·
( d 2 u
u((j + 2)∆x) − 2u((j + 1)∆x) + u(j∆x) =
(∆x)
dx
)x=j∆x
2 + O((∆x) 3 )
2
(
u((j + 2)∆x) − 2u((j + 1)∆x) + u(j∆x) d 2 u
=
+ O((∆x)) (a.f5)
(∆x) 2 dx
)x=j∆x
2
. (a.f5) , (a.f1) .
. R ,
ϵ =
≈
u((j + 2)∆x) − 2u((j + 1)∆x) + u(j∆x)
(∆x)
( )
( 2
)
d 2 u
d 2 u
+ O((∆x)) −
dx 2 dx 2
= O(∆x)
. , ∆x.
x=x j
− u j+2 − 2u j+1 + u j
(∆x) 2
u((j − 1)∆x) x x j ,
( ) du
u((j − 1)∆x) = u(j∆x) −
(∆x) +
dx
x=j∆x
u((j − 2)∆x) ∆x = 0
( )
( du
d 2 u
u((j − 2)∆x) = u(j∆x) − 2
(∆x) + 2
dx
x=j∆x
(a.f6) 2 , (a.f7) ,
u((j − 2)∆x) − 2u((j − 1)∆x) = −u(j∆x) +
( d 2 u
u(j∆x) − 2u((j − 1)∆x) + u((j − 2)∆x) =
( 1 d 2 u
(∆x)
2 dx
)x=j∆x
2 −
2
(∆x)
dx
)x=j∆x
2 −
2
( 1 d 3 u
(∆x)
6 dx
)x=j∆x
3 + · · ·.
3
(a.f6)
( 4 d 3 u
(∆x)
3 dx
)x=j∆x
3 + · · ·.
3
(a.f7)
( ) ( d 2 u
d
(∆x) 2 3 u
−
(∆x)
dx 2 dx
)x=j∆x
3 + · · ·
3
(∆x)
dx
)x=j∆x
2 + O((∆x) 3 )
2
(
u(j∆x) − 2u((j − 1)∆x) + u((j − 2)∆x) d 2 u
=
+ O((∆x)) (a.f8)
(∆x) 2 dx
)x=j∆x
2
( )

8 88
. R ,
u(j∆x) − 2u((j − 1)∆x) + u((j − 2)∆x)
R =
(∆x)
( 2
( )
d 2 u
d 2 u
≈
+ O((∆x)) −
dx 2
dx 2 )x=j∆x
= O(∆x)
. , ∆x .
x=x j
− u j − 2u j−1 + u j−2
(∆x) 2
u((j + 1)∆x) x = x j
( )
( ∂u
1 ∂ 2 u
u((j + 1)∆x) = u(j∆x) +
(∆x) +
∂x
x=j∆x
2
( ( 1 ∂ 3 u
1
+
(∆x)
6 ∂x
)x=j∆x
3 ∂ 4 u
+
3 24
u((j − 1)∆x) x x j ,
u((j − 1)∆x) = u(j∆x) −
−
∂x 2 )x=j∆x
∂x 4 )x=j∆x
( )
( ∂u
1 ∂ 2 u
(∆x) +
∂x
x=j∆x
2
( 1 ∂ 3 u
(∆x)
6 ∂x
)x=j∆x
3 +
3
∂x 2 )x=j∆x
(∆x) 2
(∆x) 4 + · · ·.
(∆x) 2
( 1 ∂ 4 u
(∆x)
24 ∂x
)x=j∆x
4 + · · ·.
4
(a.f9)
(a.f10)
(a.f9) (a.f10) ,
( ∂ 2 u
u((j + 1)∆x) + u((j − 1)∆x) = 2u(j∆x) +
+
( 1 ∂ 4 u
12
∂x 4 )x=j∆x
(∆x)
∂x
)x=j∆x
2
2
(∆x) 4 + · · ·
( ∂ 2 u
u((j + 1)∆x) − 2u(j∆x) + u((j − 1)∆x) =
(∆x)
∂x
)x=j∆x
2
2 ( 1 ∂ 4 u
+
(∆x)
12 ∂x
)x=j∆x
4 + O((∆x) 6 )
4
(
u((j + 1)∆x) − 2u(j∆x) + u((j − 1)∆x) ∂ 2 u
=
+ O((∆x)
(∆x) 2 ∂x
)x=j∆x
2 ) (a.f11)
2
( )

8 89
R
u((j + 1)∆x) − 2u(j∆x) + u((j − 1)∆x)
R =
(∆x)
( 2
( )
∂ 2 u
∂
≈
+ O((∆x) 2 2 u
) −
∂x 2
∂x 2 )x=j∆x
= O((∆x) 2 )
x=x j
− u j+1 − 2u j + u j−1
(∆x) 2
, 2 2 .
1 .
B:e ikx j
2 A kj
A kj = e ikx j
(k = −N, · · ·, N; j = 0, · · ·, 2N) (b.1)
. (A kj ) (j, k) A −1
jk
2N∑
j=0
, ,
A kj A −1
jl
= δ kl (k = −N, · · ·, N; j = −N, · · ·, N) (b.2)
. δ kl . A −1
jl
A −1 1
jl
=
2N + 1 e−ijx l
1
=
2N + 1 e−ij∆xl
1
=
2N + 1 e−il∆xj
1
=
2N + 1 e−ilx j
( )

8 90
,
2N∑
j=0
A kj A −1
jl
=
=
1
2N + 1
1
2N + 1
2N∑
e ikx j
e −ilx j
j=0
2N∑
e i(k−l)j∆x
j=0
2N∑
1
)
=
(e 2πi(k−l) j
2N+1
2N + 1
j=0
{
1 (k − l ≡ 0 mod(2N + 1))
= e 2πi(k−l) −1
(k − l ≢ 0 mod(2N + 1))
e
2πi(k−l)
2N+1 −1
(b.3)
1 ) 2 ) . k ≠ l (k − l) . , e −2πi(k−l) = 1 .
, (b.3) δ kl (k = −N, · · ·, N; l = −N, · · ·, N) , (b.2)
. k, l −N ∼ N , (A kj ) (A −1
kj ) , (A kj) ,
.
C:
. ,
S = 1
2π
∫ 2π
0
g(x)dx
(c.1)
. . J
x j = 2πj (j = 0, 1, · · ·, J − 1) g(x J ) , S ,
J
Ŝ = 1 J
∑J−1
g(x j )
j=0
(c.2)
1 ) a ≡ b mod n , a − b n .
2 ) a n = r n−1
.
S = rn − 1
r − 1
( )

8 91
. , g(x) e ikx
,
(k = −N, · · ·, N) (c.1)
S = 1 ∫ 2π
2π
= 1
2π
{
=
0
∫ 2π
0
g(x)dx
e ikx dx
1 (k = 0)
0 (k ≠ 0)
(c.3)
, , (c.2) ,
Ŝ = 1 J
= 1 J
= 1 J
= 1 J
∑J−1
g(x j )
j=0
∑
J−1
e ikx j
j=0
∑
J−1
e ik 2π J j
j=0
∑J−1
j=0
( ) j
e ik 2π J
, r = e ik 2π J
, ,
Ŝ = 1 J
= 1 J
{
=
∑J−1
( ) j
e ik 2π J
j=0
∑
J−1
r j
j=0
1
J
1 (r = 1)
r J −1
(r ≠ 1)
r−1
. r J = e ik 2π J ·J = e i2πk = 1 ,
{
1 (r = 1)
Ŝ =
1 r J −1
(r ≠ 1)
J r−1
=
{
1 (r = 1)
0 (r ≠ 1) . (c.4)
S Ŝ (c.3) (c.4) ,
{
= 1 (k = 0)
r =
≠ 1 (k ≠ 0)
(c.5)
( )

8 92
. k = 0 ,
r = e ik 2π J
= e 0
= 1
. , k ≠ 0 , −N ≤ k ≤ N ,
k J > k J
3 ) . J J ≥ N + 1 . , (−N, · · ·, N)
e ikx ,
.
1
2π
∫ 2π
0
e ikx dx = 1 ∑J−1
e ikx j
(N + 1 ≤ J)
J
j=0
D: 1
(3.2.24) .
. . F (u)
∂F (u)
u (3.2.22)
∂u
,
∂F (u) ∂u (u) ∂u
+ udF
∂u ∂t du ∂x = 0
(d.1)
, u G(u)
∫ u
dF (η)
G(u) ≡ η dη (d.2)
dη
. (d.2) ,
∫ u
∂F (η)
G(u) = η
∂η
dη
= [ηF (η)] u 0 − ∫ u
F (η)dη
= uF (u) −
∫ u
F (η)dη
3 ) J ≤ k , k = ±J . k ≠ 0 r = 1
(c.5) .
( )

8 93
. G(u) x ,
∂G(u)
∂x
= ∂ ( uF (u) − ∫ u F (η)dη )
∂x
= ∂u
(u)
F (u) + u∂F − ∂
∂x ∂x ∂x
= ∂u
∂x
= ∂u
= u
∂x
(u)
F (u) + u∂F
∂u
(u)
F (u) + u∂F
∂F (u)
∂u
∂u
∂x
∂u
. (d.1) , (d.3) ,
∫ u
F (η)dη
∂u
∂x − ∂
∂u
∂u
∂x − F (u)∂u ∂x
∫ u
F (η)dη ∂u
∂x
(d.3)
∂F (u) ∂u (u) ∂u
+ u∂F
∂u ∂t ∂u ∂x = 0
∂F (u) ∂F (u) ∂u
+ u
∂t ∂u ∂x = 0
∂F (u)
+ ∂G(u) = 0 (d.4)
∂t ∂x
. (d.4) [0, 2π] u ,
∫ 2π
0
∫ 2π
∂
∂t
∂
∂t
∂
∂t
∂F (u)
dx +
∂t
0
∫ 2π
0
∫ 2π
0
∫ 2π
0
∂G(u)
∂x dx = 0
F (u)dx + [G(u)] u(x+2π,t)
u(x,t)
= 0
F (u)dx +
F (u)dx = 0
∫ u(x+2π,t)
u(x,t)
∂F (η)
η
∂η dη = 0
(d.5)
, C F
, (d.5) ,
C F = 1 ∫ 2π
F (u)dx
2π 0
∫
∂ 2π
F (u)dx = 0
∂t 0
2π ∂C F
∂t
= 0
∂C F
∂t
= 0
(d.6)
( )

8 94
, C F t .
, u , F (u) = u C F ,
F (u) = 1 2 u2 . M,
E ,
. , .
M (3.1.7) ,
,
M = 1
2π
M = 1 ∫ 2π
udx,
2π 0
(d.7)
E = 1 ∫ 2π
1
2π 0 2 u2 dx (d.8)
= 1
2π
∫ 2π
0
∫ 2π
0
M = 1 ∫ 2π
2π 0
udx
N∑
k=−N
N∑
k=−N
û k e ikx dx.
û k e ikx dx
. , E (3.1.7) ,
,
E = 1
2π
= 1
2π
= û 0 (d.9)
∫ 2π
0
∫ 2π
0
1
2 u2 dx
1
2
N∑
N∑
k=−N l=−N
û k û l e ikx e ilx .
E = 1 ∫ 2π
1
2π 0 2
=
=
N∑
k=−N l=−N
N∑
k=−N
N∑
N∑
1
2ûkû
−k
N∑
k=−N l=−N
1
2ûkû
l δ k+l,0
û k û l e ikx e ilx
(d.10)
( )

8 95
u ,
E =
=
=
N∑
k=−N
N∑
k=−N
N∑
k=−N
1
2ûkû
−k
1
2ûkû ∗ k
1
2 |û k| 2
. M , E , (3.2.24) .
.
M
(d.9) (3.2.24) ,
dM
dt
= dû 0
dt
= −
N∑
l=−N
ilû l û −l .
, 2 , 2 , l = −l ′ ,
dM
dt
N∑
= − ilû l û −l
= − 1 2
= 0
l=−N
( N
∑
)
N∑
ilû l û −l + (−il ′ )û −l ′û l ′
l=−N
l ′ −N
, M . , (3.2.24)
, u (3.1.7) ,
1
2π
∫ 2π
1 ,
1
2π
0
∫ 2π
0
∂u
∂t e−ikx dx + 1
2π
∂u
∂t e−ikx dx + 1
2π
dû k
dt + 1
2π
∫ 2π
0
∫ 2π
0
∫ 2π
0
u ∂u
∂x e−ikx dx = 0.
u ∂u
∂x e−ikx dx = 0
u ∂u
∂x e−ikx dx = 0.
(d.11)
( )

8 96
(d.11) k = 0 ,
dû k
dt + 1 ∫ 2π
2π
dû 0
dt + 1
2π
dû 0
dt
= − 1
2π
0
∫ 2π
0
∫ 2π
0
u ∂u
∂x e−ikx dx = 0
u ∂u
∂x dx = 0
u ∂u
∂x dx
,
dM
dt
= dû 0
dt
= − 1 ∫ 2π
2π
= − 1
2π
0
∫ 2π
0
u ∂u
∂x dx
1 ∂u 2
2 ∂x dx
= u 2 (2π, t) − u 2 (0, t)
= 0
, M .
E
M E (3.2.24) 2 .
(d.10) (3.2.24) ,
dE
dt =
= 1 2
N
∑
k=−N
1 d (û k û −k )
2 dt
N∑
( dûk
dt û−k + dû −k
dt
k=−N
= − 1 2 i N
∑
k=−N
⎧⎛
⎨
⎝
⎩
min(N,N+k)
∑
û k
)
l=max(−N,−N+k)
⎞ ⎛
lû k−l û l
⎠ û −k + ⎝
min(N,N−k)
∑
l=max(−N,−N−k)
lû −k−l û l
⎞
⎠ û k
⎫
⎬
⎭ .
(d.12)
( )

8 97
, 1 . 1 ,
N∑
min(N,N+k)
∑
k=−N l=max(−N,−N+k)
lû k−l û l û −k =
N∑
N∑
k=−N l=−N
lû k−l û l û −k
(−N ≤ k − l ≤ N)
(d.13)
. l = −k ′ , k = −l ′ ,
N∑
N∑
k=−N l=−N
lû k−l û l û −k (−N ≤ k − l ≤ N) =
. , k ′ = k, l ′ = l ,
N∑
N∑
l ′ =−N k ′ =−N
N∑
N∑
l ′ =−N k ′ =−N
−k ′ û −l ′ +k ′û −k ′û l ′ (−N ≤ −l′ + k ′ ≤ N) =
. , (d.13) k − l = l ′ ,
N∑
N∑
k=−N l=−N
lû k−l û l û −k (−N ≤ k − l ≤ N) =
, l ′ = l ,
N∑
N∑
k=−N k−l ′ =−N
−k ′ û −l ′ +k ′û −k ′û l ′
(−N ≤ −l ′ + k ′ ≤ N)
N∑
N∑
k=−N k−l ′ =−N
N∑
k=−N l=−N
−kû k−l û −k û l
(−N ≤ k − l ≤ N) (d.14)
N∑
(k − l ′ )û l ′û k−l ′û −k
(−N ≤ k − (k − l ′ ) ≤ N)
(k − l ′ )û l ′û k−l ′û −k (−N ≤ k − (k − l ′ ) ≤ N) =
. M
N∑
(d.15) 2 3 ,
N∑
min(N,N−k)
∑
k=−N l=max(−N,−N−k)
lû k−l û l û −k = 1 3
N∑
min(N,N+k)
∑
k=−N l=max(−N,−N+k)
min(N,N+k)
∑
k=−N l=max(−N,−N+k)
+(k − l)û l û −k û k−l }
N∑
k=−N l=−N
N∑
(k − l)û l û k−l û −k
(−N ≤ k − l ≤ N)
(d.15)
lû k−l û l û −k (d.14)
{lû k−l û l û −k + (−k)û k−l û −k û l
= 0 (d.16)
( )

8 98
. (d.12) 2 . 2 ,
N∑
min(N,N+k)
∑
k=−N l=max(−N,−N+k)
lû −k−l û l û k =
N∑
N∑
k=−N l=−N
lû −k−l û l û k (−N ≤ k − l ≤ N)
(d.17)
. (d.17) l = −k
′ , k = −l ′ ,
N∑
N∑
k=−N l=−N
lû −k−l û l û k (−N ≤ k − l ≤ N) =
. , k ′ = −k, l ′ = −l ,
N∑
N∑
l ′ =−N k ′ =−N
N∑
N∑
l ′ =−N k ′ =−N
−k ′ û k ′ +l ′û −k ′û −l ′ (−N ≤ l′ + k ′ ≤ N) =
. , (d.17) −k − l = l ′ ,
N∑
N∑
k=−N l=−N
lû −k−l û l û k (−N ≤ k − l ≤ N) =
, l ′ = l ,
N∑
k=−N k−l ′ =−N
.
,
N∑
N∑
N∑
N∑
N∑
−k ′ û k ′ +l ′û −k ′û −l ′ (−N ≤ l′ + k ′ ≤ N)
N∑
k=−N l=−N
N∑
k=−N k−l ′ =−N
kû −k−l û k û l (−N ≤ k − l ≤ N)
(k + l ′ )û l ′û −k−l ′û k
(−N ≤ −k + (k + l ′ ) ≤ N)
(−k − l ′ )û l ′û −k−l ′û k (−N ≤ −k + (k + l ′ ) ≤ N) =
min(N,N−k)
∑
k=−N l=max(−N,−N−k)
min(N,N+k)
∑
k=−N l=max(−N,−N+k)
lû k−l û l û −k = 1 3
N∑
k=−N l=−N
(d.18)
N∑
(−k − l)û l û −k−l û k
(−N ≤ −k − l ≤ N)
(d.19)
lû k−l û l û −k (d.18) (d.19) , 3
N∑
min(N,N−k)
∑
k=−N l=max(−N,−N−k)
+(−k − l)û l û −k−l û k }
{lû k−l û l û −k + kû −k−l û k û l
= 0 (d.20)
( )

8 99
. (d.12) (d.15) (d.20) ,
⎧⎛
⎞ ⎛
N
dE
dt = −1 2 i
∑ ⎨ min(N,N+k)
∑
⎝
lû k−l û l
⎠ û −k + ⎝
⎩
= 0
k=−N
l=max(−N,−N+k)
min(N,N−k)
∑
l=max(−N,−N−k)
lû −k−l û l
⎞
⎠ û k
⎫
⎬
⎭
, . E . E
(d.11) ,
,
,
dE
dt = 1 2
= 1 2
dE
dt = 1 2
N∑
k=−N
N∑
k=−N
N∑
k=−N
( ∫ 2π
= 1 1
2 2π
= 1 ( 1
2 2π
= − 1
2π
= − 1
2π
= − 1
6π
( dûk
dt û−k + dû )
−k
û k
dt
( ∫
1 2π
û −k (−u) ∂u
1
2π ∂x e−ikx dx + û k
2π
(
û −k
1
2π
0
k=−N
∫ 2π
0
∫ 2π
0
∫ 2π
N∑
0
N∑
k=−N
∫ 2π
0
û k e ikx =
= u
N∑
k=−N
û −k e −ikx
(−u) ∂u
∂x e−ikx dx + û k
1
2π
(−u) ∂u
∂x û−ke −ikx dx + 1
2π
(−u) ∂u
∂x udx + 1
2π
u 2 ∂u
∂x dx
1 ∂u 3
0 3 ∂x dx
(
u 3 (2π, t) − u 3 (0, t) ) = 0
, E .
∫ 2π
0
∫ 2π
0
∫ 2π
0
∫ 2π
0
(−u) ∂u
∂x udx )
(−u) ∂u )
∂x eikx dx .
(−u) ∂u )
∂x eikx dx
N∑
k=−N
(−u) ∂u
∂x ûke ikx dx
E ,
. , M E 3
, .
)
( )

8 100
E:
(step)
x n+1 , x n , x n−1 ,
, x n−m+1 (m , m . m = 1
1 (-), m ≥ 2 (-
) .
(stage)
x n x n+1 f(x, t) .
4 ) .
(order)
.
. . t n+1 = t n + h ,
,
x(t n+1 ) = x(t n ) + hx ′ (t n ) + h2
2 x′′ (t n ) + · · ·
,
x n+1 = x n + hx ′ n + h2
2 x′′ n + · · ·
, q , q
. q . , q 1
h q+1 . t 0 , t x
, t x x t , 1 ,
x 0 ,
x t ≈ x t−1 + hx ′ t−1
≈ x t−2 + hx ′ t−2 + hx ′ t−1.
x t ≈ x 0 + hx ′ 0 + hx ′ 1 + · · · + hx ′ t−2 + hx ′ t−1
4 ) f(x, t)
( )

8 101
t (0 ∼ t − 1). t = t t ,
t t = t t−1 + h
= t t−2 + 2h.
t 0 ,
t t = t 0 + th
t t − t 0
h
= t
. , t − t 0
h
h , t , h q+1 × h −1 = h q
5 ) .
/ (explicit/implicit)
,
, .
.
x n+1 = F (x n , x n−1 , · · ·)
. x x
.
.
x n+1 = F (x n+1 , x n , x n−1 , · · ·)
. x n+1 ,
( ) q
5 1 ) h , t .
2
( )

8 102
F:x
x q
dx
dt
= ∂ ∂t + dx ∂
dt ∂x
= ∂ ∂t + f ∂
∂x
,
( ∂
x (q)
n =
∂t + dx ) q−1
∂
f
dt ∂x
. x n (n = 2, 3, 4) .
x 2
n = 2
x ′′ = dx′
dt
= df
dt
=
( ∂
∂t + dx
dt
∂
∂x
)
f
(f.1)
.
( )

8 103
x 3
n = 3
x ′′′ = dx′′
dt
= d ( ∂
+ ) dx ∂
∂t dt ∂x f
dt
( ∂
=
∂t + f ∂ ) 2
f
∂x
( ∂
=
∂t + f ∂ ) ( ∂
∂x ∂t + f ∂ )
f
∂x
( ( ) ∂ ∂
=
+ f ∂ (
f ∂ )
+ ∂ ∂t ∂t ∂x ∂x ∂t
( ∂
2
=
∂t + f ∂2
2 ∂x∂t + ∂f
∂t
(
f ∂ )
+ f ∂
∂x ∂x
∂
∂
∂x + f ∂2
∂t∂x + f ∂f
∂x
(
f ∂
∂x
∂ )
∂x 2
))
f
∂x + f 2 f
= f t 2 + f t f x + ffx 2 + f 2 f x 2. (f.2)
x t f ∂2 f
∂x∂t = f ∂2 f
∂t∂x = 0 .
x 4
n = 4
( ∂
x ′′′′ =
∂t + f ∂ ) 3
f
∂x
( ∂
=
∂t + f ∂ ) ( ∂
∂x ∂t + f ∂ ) 2
f
∂x
( ∂
∂t + f ∂ ) ( ∂
2
∂x ∂t + f ∂
2 ∂x∂t + ∂f ∂
∂t ∂x + f ∂
∂t∂x + f ∂f ∂
∂x ∂x + f 2 ∂ )
f
∂x
( 2 ( ) ( 3 ( ∂
= + f ∂ ) ) 2 ∂
+ ∂ ( )
∂
f t + f ∂ ( )
∂
f t
∂t ∂x ∂t ∂t ∂x ∂x ∂x
+ ∂ ( )
∂
ff x + f ∂ ( ) (
∂
ff x + ∂ ( ) ) (
2 ∂
f 2 + f ∂ ( ) )) 2 ∂
f 2 f
∂t ∂x ∂x ∂x ∂t ∂x ∂x ∂x
= ∂3 f
∂t + f ∂f
3 t 2 ∂x + ff ∂ 2 f
t
∂x + f ∂f
xf 2 t
∂x + ∂f
ff2 x
∂x + f 2 ∂f
f x
∂x + f 2 ∂ 2 f
f x
∂x 2
∂ 2 f
+2ff t
∂x + 2f 2 ∂ 2 f
f 2 x
∂x + f 3 ∂3 f
2 ∂x 3
= f 3 f x 3 + f t 3 + fxf 2 t + ffx 3 + +f x f t 2 + 3ff x 2f t + 4f 2 f x 2f x . (f.3)
( )

8 104
x t
∂f
∂x∂t =
∂f
∂t∂x = 0 .
G:-
-
.
2 2
- 2 2
ω 1 + ω 2 = 1,
ω 2 α 21 = 1 2
.
ω 1 = 1 2 , ω 2 = 1 2 ,
c 2 = α 21 = 1
. ,
k 1 = hf(x n , t n ),
k 2 = hf (x n + k 1 , t + h) ,
x n+1 = x n + 1 2 (k 1 + k 2 )
. , .
3 3
- 3 3
ω 1 + ω 2 + ω 3 = 1,
ω 2 c 2 + ω 3 c 3 = 1 2 ,
c 2 3ω 3 + c 2 2ω 2 = 1 3 ,
ω 3 α 32 c 2 = 1 6
( )

8 105
.
ω 1 = 1 6 , ω 2 = 2 3 , ω 3 = 1 6 ,
c 2 = 1 2 , c 3 = 1,
α 32 = 2,
α 21 = c 2 = 1 2 ,
α 31 = c 3 − α 32 = −1
. ,
k 1 = hf(x n , t n ),
k 2 = hf
(x n + 1 2 k 1, t n + 1 )
2 h ,
k 3 = hf (x n − k 1 + 2k 2 , t n + h) ,
x n+1 = x n + 1 6 (k 1 + 4k 2 + k 3 ) .
3 3 3 .
4 4
- 3 3
ω 1 + ω 2 + ω 3 + ω 4 = 1,
c 2 ω 2 + c 3 ω 3 + c 4 ω 4 = 1 2 ,
c 2 2ω 2 + c 2 3ω 3 + c 2 4ω 4 = 1 3 ,
α 32 c 2 ω 3 + (α 42 c 2 + α 43 c 3 )ω 4 = 1 6 ,
c 3 2ω 2 + c 3 3ω 3 + c 3 4ω 4 = 1 4 ,
α 32 α 43 c 2 ω 4 = 1 24 ,
.
α 32 c 2 c 3 ω 3 + (α 42 c 2 + α 43 c 3 )c 4 ω 4 = 1 8 ,
α 32 c 2 2ω 3 + (α 42 c 2 2 + α 43 c 2 3)ω 4 = 1
12
( )

8 106
3 8
ω 1 = 1 8 , ω 2 = 3 8 , ω 3 = 3 8 , ω 4 = 1 8
c 2 = 1 3 , c 3 = 2 3 , c 4 = 1,
α 21 = 1 3 ,
α 31 = − 1 3 , α 32 = 1,
α 41 = 1, α 42 = −1, α 43 = 1,
. ,
k 1 = hf(x n , t n ),
k 2 = hf
(x n + 1 3 k 1, t n + 1 )
3 h ,
k 3 = hf
(x n − 1 3 k 1 + k 2 , t n + 2 )
3 h ,
k 4 = hf (x n + k 1 − k 2 + k 3 , t n + h) ,
x n+1 = x n + 1 8 (k 1 + 3k 2 + 3k 3 + k 4 )
. 3 8 .
( )

8 107
x, f N , (5.1.6)
O(4N) 6 ) .4
k 1 = hf(x n , t n ),
˜x 1 = x n + α 21 k 1 ,
k 2 = hf (˜x 1 , t n + c 2 h) ,
˜x 2 = ˜x 1 + (α 31 − α 21 )k 1 + α 32 k 2 ,
k 3 = hf (˜x 2 , t n + c 3 h) ,
˜x 3 = ˜x 2 + (α 41 − α 31 )k 1 + (α 42 − α 32 )k 2 + α 43 k 3 ,
k 4 = hf (˜x 3 , t n + c 4 h) ,
(g.1)
x n+1 = ˜x 3 + (ω 1 − α 41 )k 1 + (ω 2 − α 42 )k 2 + (ω 3 − α 43 )k 3 + ω 4 k 4
. , 2 (ω 1 − α 41 , ω 2 − α 42 ) (α 41 − α 31 , α 42 − α 32 )
,
(ω 1 − α 41 , ω 2 − α 42 ) = β(α 41 − α 31 , α 42 − α 32 ) (g.2)
, (g.1) ,
r 1 = hf(x n , t n ),
˜x 1 = x n + α 21 r 1 ,
k 2 = hf (˜x 1 , t n + c 2 h) ,
˜x 2 = ˜x 1 + (α 31 − α 21 )r 1 + α 32 k 2 ,
r 2 = (α 41 − α 31 )r 1 + (α 42 − α 32 )k 2 ,
k 3 = hf (˜x 2 , t n + c 3 h) ,
˜x 3 = ˜x 2 + r 2 + α 43 k 3 ,
r 3 = βr 2 + (ω 3 − α 43 )k 3 ,
k 4 = hf (˜x 3 , t n + c 4 h) ,
(g.3)
x n+1 = ˜x 3 + r 3 + ω 4 k 4
6 ) x n N , k i (i = 1, · · ·, 4) N x n + k 1
2
f N , k i x n+1 = x n + k 1
6 x n+1
N O(4N) . k x n+1 k
. x n k 1 , k 1 k . k 1 x n+1
, k 1 k 2 , k 2 k 1 .
.
( )

8 108
. , r i , ˜x i (i = 1, 2, 3), k j (j = 2, 3, 4) ,
.
O(3N) . (5.1.5) (g.2)
ω 3 , 2
, ω 3 =
( √
1 ±
3
1
2
)
18ω 2 3 − 12ω 3 + 1 = 0
7 ) . ,
ω 2 = 1 3
(
1 ∓
√
1
2)
,
α 31 = − 1 2 ± √
1
2 ,
√
1
α 32 = 1 ∓
2 ,
α 41 = 0,
1
α 42 = ∓√
2 ,
√
1
α 32 = 1 ±
2
7 ) (g.2)
(ω 1 − α 41 ) = β(α 41 − α 31 ), (1)
(5.1.5) ,ω 1 = 1 6 , α 41 = 0, α 31 = 3ω 3 − 1
6ω 3
, (1) ,
(5.1.5) ,ω 2 = 2 3 − ω 3, α 42 = 1 − 3ω 3 ,
.
(ω 2 − α 42 ) = β(α 42 − α 32 ). (2)
1
6 = −β 3ω 3 − 1
6ω 3
β = ω 3
1 − 3ω 3
. (3)
1
6ω 3
, (3) (2) ,
− 1 3 + 2ω 3 = (1 − 3ω 3 )β −
β
6ω 3
− 1 3 + 2ω 1
3 = ω 3 +
6(3ω 3 − 1)
2(3ω 3 − 1) 2 = 1
18ω 2 3 + 12ω 2 + 1 = 0
( )

8 109
, (g.2) β
√ ( √
2 1
β = ∓ 1 ±
3 2)
. , (g.3)
r 1 = hf(x n , t n ),
˜x 1 = x n + r 1
2 ,
(
k 2 = hf ˜x 1 , t n + h )
,
2
( √
1
2)
˜x 2 = ˜x 1 +
−1 ±
r 1 +
( √ )
1 1
r 2 =
2 ± r 1 − k 2 ,
2
(
k 3 = hf ˜x 2 , t n + h )
,
2
( √
1
2)
˜x 3 = ˜x 2 + r 2 +
1 ±
(
k 3 ,
1 ∓
√
1
2)
k 2 ,
√ ( √ (
2 1
r 3 = ∓ 1 ± r 2 +
3 2)
2 √
1
−1 ∓ k 3 ,
3 2)
k 4 = hf (˜x 3 , t n + h) ,
x n+1 = ˜x 3 + r 3 + k 4
6
.
r 1 = q 1 ,
r 2 = −
(
1 ±
√
1
2)
q 2 , r 3 = − q 3
3
( )

8 110
,
q 1 = hf(x n , t n ),
˜x 1 = x n + q 1
2 ,
(
k 2 = hf ˜x 1 , t n + h )
,
2
( √
1
2)
˜x 2 = ˜x 1 +
q 2 = q 1 +
(
1 ∓
1 ∓
√
1
2)
(
k 3 = hf ˜x 2 , t n + h )
,
2
( √
1
2)
˜x 3 = ˜x 2 +
q 3 = q 2 +
(
1 ±
1 ±
√
1
2)
k 4 = hf (˜x 3 , t n + h) ,
(k 2 − q 1 ) ,
(2k 2 − 3q 1 ),
(k 3 − q 2 ),
(2k 3 − 3q 2 ),
(g.3)
x n+1 = ˜x 3 + k 4 − 2q 3
6
. (g.3) 8 )
--. O(3N)
. , 4
,
. ,
.
H:
. x(t) t = t n , x n =
x(t n )
x(t n + h) = x n + hf(x n , t n ) + 1 2 h2 ˙ f(x n , t n ) + · · ·
8 )
( )

8 111
. f ˙ = df ( ∂f
dt = ∂t + dx )
∂f
.
dt ∂x
f, ˙ ¨f, · · · (x n , t n ) f ,
f .
2
t n−1 = t n − h x O(h 2 ) x n−1 , f
f n−1 = f(x n−1 , t n−1 ) , 2
f n−1 = f(x n−1 , t n−1 )
= f(x(t n − h), t n − h)
= f(x n − hf n + O(h 2 ), t n − h)
∂f n
= f n − hf n − h ∂f n
+ O(h 2 )
∂x n ∂t n
= f n − hf ˙ n + O(h 2 )
hf ˙ n = f n − f n−1 + O(h 2 )
. f n = f(x n , t n ) . x(t)
,
x(t n + h) = x(t n ) + hf n + 1 2 h2 ˙ f n + O(h 3 )
= x(t n ) + hf n + h 2 h ˙ f n + O(h 3 )
= x(t n ) + hf n + h 2 (f n − f n−1 + O(h 2 )) + O(h 3 )
( 3
= x(t n ) + h
2 f n − 1 )
2 f n−1 + O(h 3 )
. ,
x n+1 = x n + h
( 3
2 f n − 1 2 f n−1
)
.
2 .
3
2 3 .
( )

8 112
f n−1 2 ,
f n−1 = f(x n−1 , t n−1 )
= f(x(t n − h), t n − h)
)
= f
(x n − hf n + h2
f
2 ˙ n + 0(h 3 ), t n − h
(
∂f n
= f n − hf n − h ∂f n
+ h2
f
∂x n ∂t n 2 ˙ ∂f −hf n + h2 ˙
) 2
f n
2 n
n +
∂x n 2
= f n − hf ˙
( )
n + h2
ff ˙ x + f 2 f x 2 + f t 2 + O(h 3 )
2
= f n − h ˙ f n + h2
2
= f n − h ˙ f n + h2
2 ¨f n + O(h 3 )
( )
ff
2
x + f x f t + f 2 f x 2 + f t 2 + O(h 3 )
∂ 2 f n
∂x 2 n
+ h2
2
∂ 2 f n
∂t 2 n
+ O(h 3 )
h ˙ f n = f n − f n−1 + h2
2 ¨f n + O(h 3 ). (h.1)
f(x n , t n ) = f, ∂f(x n, t n )
= f x , ∂f(x n, t n )
= f t , ∂2 f n
= f
∂x n ∂t n ∂x 2 x 2,
n
∂ 2 f n
= f t 2 . f n−2 (h.1) h
∂t 2 n
2h ,
f n−2 = f(x n−2 , t n−2 )
= f(x(t n − 2h), t n − 2h)
)
= f
(x n − hf n + 4h2 f
2 ˙ n + 0(h 3 ), t n − 2h
= f n − 2h ˙ f n + 4h2
2 ¨f n + O(h 3 )
h 2 ¨fn = − f n
2 + f n−2
+ hf 2
˙ n + O(h 3 )
. (h.2) (h.1) ,
(h.2)
(h.1) (h.3) ,
h 2 ¨fn = − f n
2 + f n−2
2
+ h ˙ f n + O(h 3 )
= f n − 2f n−1 + f n−2 + O(h 3 ). (h.3)
h ˙ f n = f n − f n−1 + h2
2 ¨f n + O(h 3 )
= 3 2 f n − 2f n−1 + 1 2 f n−2 + O(h 3 ). (h.4)
( )

8 113
x(t) 3 ,
. ,
x(t n + h) = x(t n ) + hf n + 1 2 h2 ˙ f n + 1 6 h3 ¨fn + O(h 4 )
= x(t n ) + hf n + h 2 h f ˙ n + h 6 h2 ¨fn + O(h 4 )
= x(t n ) + hf n + h ( 3
2 2 f n − 2f n−1 + 1 )
2 f n−2 + O(h 3 )
+ h 6
(
fn − 2f n−1 + f n−2 + O(h 3 ) ) + O(h 4 )
= x(t n ) + h 12 (23f n − 16f n−1 + 5f n−2 ) + O(h 4 )
x n+1 = x n + h 12 (23f n − 16f n−1 + 5f n−2 ) .
3 .
( )

8 114
4
4 .
f n−1 3 ,
f n−1 = f(x n−1 , t n−1 )
= f(x(t n − h), t n − h)
)
= f
(x n − hf n + h2
f
2 ˙ n + h3
6 ¨f n + 0(h 4 ), t n − h
∂f n
= f n − hf n − h ∂f n
+ h2
f
∂x n ∂t n 2 ˙ ∂f n
n + h3
∂x n 6 ¨f ∂f n
n
∂x n
(
−hf n + h2 f ˙
2 n − h3 ¨f
) 2
6 n ∂ 2 f n
+
+ h2 ∂ 2 f n
2
∂x 2 n 2 ∂t 2 n
(
−hf n + h2 f ˙
2 n − h3 ¨f
) 3
6 n ∂ 3 f n
+
+ h2 ∂ 3 f n
+ O(h 4 )
6
∂x 3 n 2 ∂t 3 n
= f n − hf ˙ n + h2
2 ¨f
(
n − h3 ¨ffx + 3fff 6
˙
)
x 2 + f 3 f x 3 + f t 3 + O(h 4 )
= f n − hf ˙ n + h2
2 ¨f n − h3 ( )
f 3 f x 3 + f t 3 + f 2
6
xf t + ffx 3 + f x f t 2 + 3ff x 2f t + 4f 2 f x 2f x + O(h 4 )
= f n − hf ˙ n + h2
2 ¨f n − h3
6
hf ˙ n = f n − f n−1 + h2
2 ¨f n − h3
6
∂3 f n
∂x 3 n
= f x 3, ∂3 f n
...
f n + O(h 4 )
...
f n + O(h 4 ).
∂t 3 n
(h.5) h 2h ,
= f t 3 . f n−2
(h.5)
f n−2 = f(x n−2 , t n−2 )
= f(x(t n − 2h), t n − 2h)
= f n − 2hf ˙ n + 4h2
2 ¨f n − 8h3
6
h 2 ¨fn = 1 2 f n − hf ˙ n − 2h3
3
. (h.6) (h.5) ,
...
f n + O(h 4 )
...
f n + O(h 4 )
h 2 ¨fn = 1 2 f n − hf ˙ n − 2h3 ...
f
3 n + O(h 4 )
= f n − 2f n−1 + f n−2 + h 3... f n + O(h 4 )
(h.6)
(h.7)
( )

8 115
(h.5) (h.7) ,
hf ˙ n = f n − f n−1 + h2
2 ¨f n − h3 ...
f
6 n + O(h 4 )
= 3 2 f n − 2f n−1 + 1 2 f n−2 + h3
3
...
f n + O(h 4 ).
f n−3 (h.5) h 3h ,
(h.8)
f n−3 = f(x n−3 , t n−3 )
= f(x(t n − 3h), t n − 3h)
= f n − 3hf ˙ n + 9h2
2 ¨f n − 27h3 ...
f
6 n + O(h 4 )
h 3 ¨fn = 2 9 f n − 2 9 f n−3 − 2 3 h f ˙ n + ¨f n + O(h 4 ).
(h.9)
(h.9) (h.7) (h.8) ,
(h.10) (h.8) ,
(h.10) (h.7) ,
h 3 ¨fn = 2 9 f n − 2 9 f n−3 − 2 3 h ˙ f n + ¨f n + O(h 4 )
= f n − 3f n−1 + 3f n−2 − f n−3 + O(h 4 ). (h.10)
hf ˙ n = f n − f n−1 + h2
2 ¨f n − h3 ...
f
6 n + O(h 4 )
= 4 3 f n − 3f n−1 + 3 2 f n−2 + 1 3 f n−3 + O(h 4 ). (h.11)
h 2 ¨fn = 1 2 f n − hf ˙ n − 2h3 ...
f
3 n + O(h 4 )
= 2f n − 5f n−1 + 4f n−2 + f n−3 + O(h 4 ). (h.12)
(h.10) , (h.11) ,(h.12) x(t) 4 ,
x(t n + h) = x(t n ) + hf n + 1 2 h2 f˙
n + 1 6 h3 ¨fn + 1 27 h4... f n + O(h 5 )
= x(t n ) + hf n + h 2 h f ˙ n + h 6 h2 ¨fn + h 27 h3... f n + O(h 5 )
= x(t n ) + hf n + h ( 4
2 3 f n − 3f n−1 + 3 2 f n−2 + 1 )
3 f n−3 + O(h 4 )
+ h 6
+ h 27
(
2fn − 5f n−1 + 4f n−2 + f n−3 + O(h 4 ) )
(
fn − 3f n−1 + 3f n−2 − f n−3 + O(h 4 ) ) + O(h 5 )
= x(t n ) + h 27 (55f n − 59f n−1 + 37f n−2 − 9f n−3 ) + O(h 5 )
( )

8 116
. ,
x n+1 = x n + h 27 (55f n − 59f n−1 + 37f n−2 − 9f n−3 ) .
4 .
* 2
t n+1 = t n +hxO(h 2 )x n+1 , f n+1 = f(x n+1 , t n+1 )
, h −h
,
,
f n = f(x n , t n ),
f n+1 = f(x n , t n ) + h ˙ f(x n , t n ) + O(h 2 ).
x n+1 = x n + h 2 (f n+1 + f n )
. 2 .
3
(h.1) ,
h ˙ f n = f n − f n−1 + h2
2 ¨f n + O(h 3 ).
3 3
−h +h , (h.1) h h = −h ,
f n+1 = f n + h ˙ f n + h2
2 ¨f n + O(h 3 )
h 2 ¨fn = 2f n+1 − 2f n − h ˙ f n + O(h 3 ).
(h.13)
(h.1) (h.13) ,
h ˙ f n = f n − f n−1 + h2
2 ¨f n + O(h 3 )
h ˙ f n = 1 2 f n+1 − 1 2 f n−1 + O(h 3 ).
(h.14)
( )

8 117
(h.14) (h.13) ,
h 2 ¨fn = 2f n+1 − 2f n − h ˙ f n + O(h 3 )
h 2 ¨fn = f n+1 − 2f n + f n−1 + O(h 3 ).
(h.15)
, 3 ,
.
x n+1 = x n + hf n + 1 2 h2 ˙ f n + 1 6 h3 ¨fn
= x n + hf n + h 2 h f ˙ n + h 6 h2 ¨fn
= x n + hf n + h ( 1
2 2 f n+1 − 1 )
2 f n−1 + h 6 (f n+1 − 2f n + f n−1 )
= x n + h 12 (5f n+1 + 8f n − f n−1 )
4
(h.5) ,
f n−1 = f n − hf ˙ n + h2
2 ¨f n − h3 ...
f
6 n + O(h 4 ).
(h.7) ,
f n−2 = −f n − 2hf n+1 + h 2 ¨fn − ...
f n + O(h 4 ).
(h.16)
3 (h.5) h h = −h ,
f n+1 = f n + hf ˙ n + h2
2 ¨f n + h3 ...
f
6 n + O(h 4 ).
(h.17)
(h.5) (h.17) ,
f n+1 + f n−1 = 2f n + h 2 ¨fn + O(h 4 )
h 2 ¨fn = f n+1 − 2f n + f n−1 + O(h 4 ).
(h.18)
(h.16) (h.18) ,
−f n+1 − f n−1 + f n−2 = −3f n + 2f n−1 − h 3... f n + O(h 4 )
h 3... f n = f n+1 − 3f n + 3f n−1 + f n−2 + O(h 4 ).
(h.19)
( )

8 118
(h.5) (h.17) , (h.19) ,
, 4 ,
−f n+1 + f n−1 = −2hf ˙ n − h3 ...
f
3 n + O(h 4 )
h ˙ f n = 1 3 f n+1 + 1 2 f n − f n−1 + 1 6 f n−2 + O(h 4 ).
x n+1 = x n + hf n + 1 2 h2 ˙ f n + 1 6 h3 ¨fn + 1
24 h4... f n
.
= x n + hf n + h 2 h f ˙ n + h 6 h2 ¨fn + h 24 h3... f n
= x n + hf n + h ( 1
2 3 f n+1 + 1 2 f n − f n−1 + 1 )
6 f n−2 + h 6 (f n+1 − 2f n + f n−1 )
+ h 24 (f n+1 − 3f n + 3f n−1 + f n−2 )
= x n + h 24 (9f n+1 + 19f n − 5f n−1 + f n+2 )
(predicor-corrector mehod)
x n+1 f n+1
, . , f n+1 x n+1
. ,
x n+1 . , x n+1
x n+1 . ,
,
. , 4 ,
˜x n+1 = x n + h 24 (55f n − 59f n−1 + 37f n−2 − 9f n−3 ) ,
x n+1 = x n + h 24
(9 ˜f n+1 + 19f n − 5f n−1 + f n+2
)
(h.20)
(h.21)
2 . , ˜f n+1 = f (˜x n+1 , t n+1 ) .
4 ˜x n+1
O(h 4 ) , 3
,
˜x n+1 = x n + h 12 (23f n − 16f n−1 + 5f n−2 ) ,
(h.22)
( )

8 119
x n+1 = x n + h 24
(9 ˜f n+1 + 19f n − 5f n−1 + f n+2
)
(h.23)
4 9 ) .
I: f(x)
f(x) x = π
( 1 − cos x
f(x) =
2
(
= sin x ) 2N
2
. û k (0) f(x) . f(x) ,
) 2N
) N
(
f(x) = sin x 2
( 1 ( ) ) 2N
= e
i x 2 − e
−i x 2 .
2i
, 2 10 ) ,
( 1 ( ) ) 2N
f(x) = e
i x 2 − e
−i x 2
2i
= (i2 ) N
2 2N 2N∑
= (−1)N
l=0
2 2N 2N∑
= (−1)N
l=0
2 2N 2N∑
l=0
( )
2NC l e
i x l ( )
2 −e
−i x 2N−l
2
(−1) 2N−l 2NC l e i x 2 l e −i x 2 (−l) + e −i x 2 2N
(−1) l 2NC l e ix(l−N)
9 ) ˜x n+1 h 1 . ,
, .
10 ) 2 , (a + b) n . (a + b) n ,
. n C k =
(a + b) n =
n!
k!(n − k)! .
n∑
nC k a k b n−k
k=0
( )

8 120
. l − N = m ,
(2N)! ,
22N f(x) = (−1)N
2 2N 2N∑
l=0
= 1
2 2N N
∑
m=−N
= 1
2 2N N
∑
=
N∑
m=−N
m=−N
(−1) m
(−1) l 2NC l e ix(l−N)
(−1) m+2N 2NC m+N e imx
(−1) k
(2N)!
(N + m)!(N − m)! eimx
(2N)!
2 2N
(N + m)!(N − m)! eimx . (i.1)
(2N)! 2N · 2N − 1 · 2N − 2 · · · 1
=
22N ( ) 2 (· 2 · 2 · · · 2)
( ) ( ) ( 2N 2N + 1 2N + 2 2N + 3 1
= ·
·
·
· · ·
2 2 2 2 2)
( ) ( ) ( 2N + 1
2N + 3 1
= N
(N − 1)
· · ·
2
2 2)
(( ) ( ) ( 2N + 1 2N + 3 1
= (N · (N − 1) · · · 1)
·
· · ·
2 2 2))
( )
(2N + 1)(2N + 3) · ··
= N! ·
2
( N )
(2N + 1)(2N + 3) · ··
= (N!) 2 2 N N!
( )
(2N + 1)(2N + 3) · · · 1
= (N!) 2 2N · 2(N − 1) · · · 2
, (i.1) ,
f(x) =
=
=
N∑
m=−N
N∑
m=−N
N∑
m=−N
(−1) m
(2N)!
2 2N
(N + m)!(N − m! eimx
(−1) m (N!)2 (2N+1) (2N+3) · · · 1
2N 2N−2 2
(N + m)!(N − m)!
(−1) m
(N!)(N!)
(N + m)!(N − m)!
e imx
(2N + 1) (2N + 3)
2N 2N − 2 · · · 1
2 eimx
( )

8 121
. t = 2π , ,
û k (0) = 1
2π
= 1
2π
∫ 2π
0
∫ 2π
0
f(x)e −ikx dx
N∑
m=−N
(−1) m
= (−1) k (N!)(N!)
(N + k)!(N − k)!
, (3.1.13) ,
,
N∑
k=−N
u(x, t) =
N∑
k=−N
=
N∑
k=−N
N∑
k=−N
û k (0)e ik(x−t)
(−1) k
(N!)(N!)
(N + k)!(N − k)! ,
(N!)(N!)
(N + m)!(N − m)!
(2N + 1) (2N + 3)
2N (2N − 2) · · · 1
2
(N!)(N!)
(N + k)!(N − k)!
(2N + 1) (2N + 3)
2N (2N − 2) · · · 1
2 eimx e −ikx dx
(i.2)
(2N + 1) (2N + 3)
2N (2N − 2) · · · 1
2 eik(x−t) (i.3)
(N!)(N!)
(N + k)!(N − k)! = (N!)(N!) + (N!)(N!)
0!(2N)! 1!(2N − 1)! + · · · + (N!)(N!)
(N − 2)!(N + 2)!
, ,
(N!)(N!)
+
(N − 1)!(N + 1)! + (N!)(N!)
(N!)(N!) + (N!)(N!)
(N + 1)!(N − 1)!
(N!)(N!)
+
(N + 2)!(N − 2)! + · · · + (N!)(N!)
(2N − 1)!1! + (N!)(N!)
(2N)!0!
(N!)(N!)
= 1 + 2
(N + 1)!(N − 1)! + 2 (N!)(N!)
(N + 2)!(N − 2)! + · · · + 2(N!)(N!) (2N)!
N
= 1 + 2
(N + 1) + 2 N(N − 1)
(N + 1)(N + 2) + · · · + 2 N(N − 1) · · · 1
(N + 1)(N + 2) · · · 2N
N
= 1 + 2
(N + 1) + 2 N (N − 1)
(N + 1) (N + 2) + · · · + 2 N (N − 1)
(N + 1) (N + 2) · · · 1
2N
= 1 + 2
N∑
k=1
N∑
u(x, t) = (−1) k (N!)(N!)
(N + k)!(N − k)!
k=−N
( (2N + 1) (2N + 3)
=
2N (2N − 2) · · · 1 ) ( 1 + 2
2
N N(N − 1) (N − k + 1)
· · · ·
(N + 1) (N + 1) (N + k)
(2N + 1) (2N + 3)
2N (2N − 2) · · · 1
2 eik(x−t)
N∑
m=1
N N(N − 1) · · · ·
(N + 1) (N + 1)
)
(N − m + 1)
e ik(x−t)
(N + m)
( )

8 122
. u(x, t) ,
( (2N + 1)
u(x, t) =
2N
( (2N + 1)
=
2N
(2N + 3)
(2N − 2) · · · 1
2
(2N + 3)
(2N − 2) · · · 1
2
) ( 1 · e 0 + 2
) ( 1 + 2
N∑
k=1
N∑
k=1
N N(N − 1) · · · ·
N + 1 (N + 1)
)
(N − m + 1)
e ik(x−t)
(N + m)
N N(N − 1) (N − m + 1)
· · · cos k(x − t)
(N + 1) (N + 1) (N + m)
. f(x) N
,
u(x, 2π) = f(x).
, . t = 2π , (3.1.21)
,
, (i.2) ,
u j (t) =
N∑
k=−N
u j (t) =
=
N∑
k=−N
N∑
k=−N
û k (t)e ikx j
( (
û k (0) exp ik x j −
( (
û k (0) exp ik x j −
))
sin (k∆x)k∆x
t
N∑
= (−1) k (N!)(N!)
(N + k)!(N − k)!
k=−N
( (2N + 1) (2N + 3)
=
2N 2N − 2 · · · 1 ) ( 1 + 2
2
. ,
))
sin (k∆x)k∆x
t
(2N + 1) (2N + 3)
2N 2N − 2 · · · 1
2 eik(x j−c k t)
N∑
k=1
N N(N − 1)
· · · N − m + 1
N + 1 N + 1 N + m cos k(x j − c k t)
)
)
c k ≡
{
sin (k∆x)
k∆x
(k ≠ 0)
1 (k = 0)
(i.4)
.
( )

9 123
9
, , ,
.
, , ,
, , . ,
.
( )

10 124
10
• , 2004:, 232pp.
• , , ,1993:,
5, 239pp
• ,2000:, 192pp
• ,1991:, 143pp
• Stanley J. Farlow , , ,1983:
, 405pp
• Yamamoto, M., Takahashi, M., 2003:The Fully Developed Superrotation Simulated
by a General Circulation Model of a Venus-like Atmosphere. Journal of the
Atmospheric Sciences, 60, 561-574
• Rossow, W. B., A.D. Del Genio and T. Eichler, 1990:Cloud-tracked winds from
Pioneer Venus OCPP images. Journal of the Atmospheric Sciences, 47, 2053-
2084
( )