u
L2
u ˆ(ξ ) = ( u)(ξ ) =
F
∞
e−iξx u(x)dx
∀ξ ∈ R
−∞
x
ξ
∀ξ
M
M
→ ∞
−M
L2
u
u ˆ
u = √ 2π uˆ ,
L2
√ 2π 1 u(x) = (F −1 u ˆ)(x) =
∞
2π −∞
u(x) u ˆ(ξ )
eiξx u ˆ(ξ )dξ
∀x ∈ R. ξ
∈ L2
u, v
F{u + v}(ξ ) = uˆ(ξ ) + ˆv(ξ ) F{cu}(ξ ) = cˆu(ξ ). ∈ R
F{u(x + x0)}(ξ ) = eiξx uˆ(ξ )
∈ R
F{eiξ xu(x)}(ξ ) = uˆ(ξ − ξ 0)
x0
0
ξ 0 c
0
∈R
F{u(cx)}(ξ ) = uˆ(ξ/c) |c|
c =0
F{u}(ξ ) = uˆ(−ξ ). ∈ L2
ux
F{ux}(ξ ) = iξ ˆu(ξ ).
F{F −1{u}}(ξ ) = 2π1 ˆu(−ξ ). ∗
u, v
∗
∗
(u v)(x) = (v u)(x) =
∞
u(x
−∞
u v
− y)v(y)dy =
∞
u(y)v(x
−∞
uL1
− y)dy,
vL2
u vL2
∗
F{u ∗ v}(ξ ) = uˆ(ξ )vˆ(ξ ).
h > 0
hZ
xj = jh
{ }
v = vj
2h
v = 2h
−
| | h
∞
vj
1/2
2
j=
−∞
2h = v : v <
{ ∞} .
,
∀ j ∈ Z
v
vˆ(ξ ) = (
2π h xj
∈ 2h
v
F hv)(ξ ) = h
∞
j=
∀ξ ∈ − πh , πh
e−iξjh vj
−∞
vˆ(ξ ) 2π h exp(iξ 1 xj ) = exp(iξ 2 xj ) j
−
ξ 1 ξ 2
∀ ∈ Z
h = 0,25
.
sin(πx ) sin(9πx) 0, 0,25, 0,5, 0,75, 1
±
± ±
Efecto alias del sin
±
x y sin 9
π
π x
1
0.5
0
−0.5
−1 −1
−0.75
−0.5
−0.25
0
0.25
0.5
h =
1 4
0.75
1
x
ξ
L2h
vˆ(ξ )
vˆ =
| | π h
−
1/2
vˆ(ξ ) 2 dξ
,
π h
2h
√
v =
1 vj = ( F −1 vˆ)(xj ) = h
L2h
v
2π vˆ .
π h
2π −
eiξxj vˆ(ξ )dξ
π h
∀ j ∈ Z
x [0, 2π] h = 2π N xj = jh, j = 1,...,N 2N
⇒ πh = N 2
N
| | N
v =
h
vj
1/2
2
j=1
N
vˆξ = (
F N v)(ξ ) = h
e−iξxj vj
j=1
ξ =
<
∞
− N 2 + 1, . . . . . . , N 2 eiξx
2π ξ
vˆ
discreto
acotado
acotado
1 vj = ( F −1 vˆ)j = N
2π
∈ {h, 2h , . . . , 2π} N } ξ ∈ {− 2 + 1, . . . , N 2 x
discreto
N/2
eiξxj vˆξ
j = 1, . . . , N .
ξ= N/2+1
−
eiξx xj = jh
h =
2π N
N
N
h
e−ipxj eiqxj = 2πδ pq
j=1
v(x)
2π xj
1 I N v(x) = 2π
N/2
N
iξx
vˆξ e
vˆξ = h
e−iξxj v(xj ),
j=1
ξ= N/2+1
−
N 1 P N v(x) = 2π
v
N/2
2π
iξx
vˆ(ξ )e
vˆ(ξ ) =
e−iξx v(x)dx,
0
ξ= N/2+1
−
vˆ(ξ )
v v(x) ∈ L2
v
[0, 2π] x0
v(x0 )
+ 0
−
v(x )+v(x0 ) 2
x0
v
[0, 2π]
v
[0, 2π]
vˆ(ξ )
∞
vˆξ = vˆ(ξ ) +
vˆξ
vˆ(ξ + N m),
−∞,m=0
m=
ξ, ξ N, ξ 2N,... ξ
±
±
N/2
I N v = P N v + RN v,
RN v =
ξ= N/2+1
−
∞
m=
−∞,m=0
RN u
v − P N v ≤ v − I N v,
vˆ(ξ + mN )
eiξx ,
v
∈ L2
p
L2
−1
p
≥ 0
u( p) vˆ(ξ ) = O ( ξ − p−1 ),
|ξ | → ∞.
||
L2
v vˆ(ξ ) = O( ξ −m ),
||
m
|ξ | → ∞,
≥ 0
v
∈ L 2
p
L2
−1
p
u( p)
|vˆ(ξ ) − vˆξ | = O(h p+1),
h
→ 0. L2
v
|vˆ(ξ ) − vˆξ | = O(hm), m
h
→ 0,
≥ 0 v(x) v(x)
d v(x) dx
≈
d d 1 (I N v(x)) = ( dx dx 2π
N/2
ξ= N/2+1
−
iξx
vˆξ e
1 )= 2π
N/2
(iξ vˆξ )eiξx ,
ξ= N/2+1
−
v w
≥ 1
ν
ν
ν
v
xj vL2
p
−1
L2
p
|wj − v(ν ) (xj )| = O(h p−ν ),
≥ ν + 1
h
→ 0.
L2
v
|wj − v(ν ) (xj )| = O(hm),
h
→ 0,
≥ 0
m
2N 2
vˆξ N = 2n
O(N log N )
2
O(N )
4
×3
u( p)
(ξ, j)
v
2π
vˆξ
w ˆξ = iξ ˆ vξ
w ˆN/2 = 0 wj xj
p w ˆξ = (iξ ) p vˆξ
w ˆN/2 = 0
p
{ − |x −2 π| },
f (x) = m´ax 0, 1
g(x) = e sin x ,
function
spectral derivative
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
0
2
4
−1
6
3
0
2
4
6
2 max error = 9.5679e−013 1
2
0 1 0
−1 0
2
4
ut + c(x)ux = 0, x[0, 2π] u(x, 0) = exp( 100(x
−
−2
6
c(x) =
0
2
1 + sin2 (x 5
4
− 1),
− 1)2)
t x (n+1)
uj
− u(nj −1) = −c(x )(Du(n) ) ,
2∆t
j
j
ut = u x .
u(x, 0) = u 0 (x).
j = 1, . . . , N .
6
Onda de velocidad variable: leap−frog y espectral 5 4 3 u
2 1 0
8 7 6 5 4 3 2 t
1 0 0
1
3
2
6
5
4
x
u(x) = u 0 (x + t),
−1 sech(x)
t = 10 x = 10
−
ut = uxx .
∞ 2 1 u(x, t) = eiξx−ξ t u ˆ0 (ξ )dξ = 2π −∞ e−ξ t = 10
2
∞ 2 1 e−(x−s) /4t u0 (s)ds, 4πt −∞
√
ξ t
1
(a)
0
−1 −50
−40
−30
−20
−10
0
10
20
30
40
50
−40
−30
−20
−10
0
10
20
30
40
50
−40
−30
−20
−10
0
10
20
30
40
50
−40
−30
−20
−10
0
10
20
30
40
50
1
(b)
0
−1 −50 1 (c)
0
−1 −50 1
(d)
0
−1 −50
ut = iu xx .
∞ 2 1 u(x, t) = eiξx−iξ t u ˆ0 (ξ )dξ = 2π −∞
∞ 2 1 e−i(x−s) /4t u0 (s)ds. 4πit −∞
√
u(x, t) = e i(ξx+wt) ,
∈ R w ∈ C,
ξ
w ut = ux ut = u xx
ξ
⇒ ⇒
w = ξ, w = iξ 2 ,
⇒
ut = iuxx
w =
−ξ 2.
dw dξ
ξ
ut + uux + uxxx = 0, uux
uxxx
a u(x, t) = 3a2 sech2 ( (x 2
− x0) − a3t),
∀a, x0, 3a2
sech(x) = 2/(ex + e−x ) 2a2 u 2
x0 + 2 a t
1 ut + ( u2 )x + uxxx = 0, 2 i u ˆt + k u2 2
−
ˆ = e −ik U
3
ik3 u ˆ = 0. e−ik t u ˆ
3
t
ˆt + i e−ik3 t k U 2 = 0, U 2
ˆt + i e−ik3 t k F U 2
·
F −1 (eik
3
t
ˆ · U )
2
= 0,
F
2000
0 6 5 4 −3
x 10
3 2
3 2 1
1
0 −1
t
0
−2 −3 x
iut + uxx + q u 2 u = 0,
||
−∞ < x < ∞,
u(x, t)
q
|
d ∞ u(x, t) 2 dx = 0, dt −∞
u(x, t) =
{
2α 1 exp i c(x q 2
c
− x0) −
|
1 ( c2 4
− α)t
}
α 2
|u|
iˆ ut
− k2uˆ + q |u|2uˆ = 0,
√ − x − ct)), 0
sech( α(x
u ˆ(k, t) = e i(−k
2
+q u 2 )t
||
u(x, t) = F −1 (ˆu(k, t)) = F −1 e−ik
2
t
· F
δt
u(x, t + δt) = F −1 e−ik
2
eiq|u|
δt
2
δt
u(x, 0) = exp
i(x
− 20)
2
· F
eiq|u|
2
t
· u(x, 0) t
2
eiq|u|
t + δt
· u(x, t)
i(x + 20) 2
δt
.
,
u(x, t)
e−ik
−
· ˆu(k, 0),
sech(x
2
δt
− 20) + exp
uL2 u(x)
u ˆ(ξ )
u(x)
u ˆ(ξ )
−
u(x) = u( x)
u(x) =
1 2
0
−1 ≤ x ≤ 1,
sech(x + 20).
2
1
0 140 120
60 100
50 80
40 60
30 40
20 20
10 0
0
∗
∗ ∗
u u u, u u u
u u u
−
L1 L2 1h
⊆ 2h ⊆ ∞h v (v w)m = h
∗
∞
j=
−∞
L∞
w
vm−j wj = h v
2h
∗ ∈
v w
u, u
[ 3, 3]
∗ ∗
∞
j=
−∞
∈ 1h
F h(v ∗ w)(ξ ) = vˆ(ξ )w(ξ ˆ ).
vj wm−j . w
∈ 2h
∗
v
w
N/2
∗
(v w)m = h v, w
j= N/2+1
−
N/2
vm−j wj = h
j= N/2+1
−
vj wm−j .
∈ 2N F N (v ∗ w)(ξ ) = vˆ(ξ )w(ξ ˆ ).
(n+1)
uj
(n) − u(nj −1) = −c(x ) u(n) j+1 − uj −1 , j
2∆t
N = 128, 256
(kG) (kdV ) (RLW )
utt = uxx u. ut = uxxx . ut + ux = u xxt .
−
sech(x)
2h
j = 1, . . . , N .
RN RN .
ut ∆u = 0, u(x, 0) = ϕ(x),
× (0, ∞)
−
u(x, t) = (4πt)−N/2
− | − | exp
x
RN
y 4t
2
ϕ(y)dy,
ϕ(y) (x, t)
u
ϕ
≥ 0
ϕ
≡ 0
u(x, t)dx =
RN
u(x, t)L
y
u
≥ 0
(RN )
u
∀t > 0.
ϕ(x)dx,
RN
∞
∈ RN
≤ Ct−N/2ϕ(x)L (
),
1 RN
∀t > 0.
−
ut uxx = 0, 0 0 u(x, 0) = ϕ( ϕ(x), 0 < x < π.
t > 0
π ϕ(x)
∞
u(x, t) = ωj (x) = 2
2
ϕj e−j t ωj (x),
j =1
2 jx ) π sin( jx)
ϕj
ϕL (0, (0, π )
π
ϕj =
ϕ(x)ωj (x)dx.
0
u(t)
2 L2 (0,π (0,π)) =
∞
| j =1
2
ϕ(x)| e−2j
2
t
≤ e−2t
π
E (t) =
∞
|
ϕ(x) 2 = e −2t ϕ
j =1
|
2L (0,π (0,π)) , 2
t
u2 (x, t)dx,
0
d E (t) dt
≤ −2E (t) M
M
uM (x, t) =
2
ϕj e−j t ωj (x).
j =1
2
−M t/2 t/2 u(x, t) − uM (x, t)L (0,π ϕ(x)L (0,π ∀t > 0 (0,π)) ≤ e (0,π)) , → ∞ L2 M → 2
2
→ ∞
x
h =
π M +1 +1
M
(t) 2u (t)+u )+uj
−u
−
j +1 j u j (t) h2 u0 (t) = u M +1 +1 (t) = 0, uj (0) = ϕ = ϕ(( jh) jh ),
u =
u1 (t)
(t)
= 0, 0 , j = 1, . ...., M t > 0 j = 1, .. ..., M
t > 0 ,
Ah u(t) = 0, u(0) = = ϕ. d u(t) + dt
−
2
,
uM (t)
1
−
Ah =
1 h2
−1
1
−1
−
.
1 2
u(t) = e −tAh ϕ. x
Ah w = λ = λw,
4 h λl (h) = 2 sin 2 l h 2
w l (h) =
2 π
h
sin(lx sin(lx1 ) sin(lx sin(lxM )
→ 0
,
l = 1,...,M,
,
l = 1,...,M,
t
M
uh (t) =
ϕl e−λl (h) w l (h),
l=1
L2 uh (t)
− u(t)
t
h E h (t) = 2
d E h (t) = dt
δ > 0
→ 0
M
| |
uj 2 .
j =1
− M
h
j =0
−
uj +1 uj h
h0 > 0
a0 , a1 ,...,a M +1 +1
u(t)
2
.
0 < h < h 0
a0 = aM +1 +1 = 0
M
h
j =0
aj +1 aj h
−
≥
M
2
(1
− δ )h
| |
aj 2 .
j =1
uj uj
{ }j=1,...,M =1,...,M ,
e = ej
− uj .
ej = u j
ej
−
e
−2e
+e
j +1 j j ej h2 e0 = e M +1 +1 = 0, ej (0) = 0, 0,
εj
1
−
εj = uxx (xj , t)
= ε j , j = 1, . ...., M t > 0 j = 1, .. ..., M
− uj+1 − 2hu2j + uj−1
t > 0 ,
|εj (t)|2 ≤ Ch4u(t)2C ([0,π]), 0 ≤ h ≤ h 0 0 ≤ t ≤ T 4
1
≤ j ≤ M,
− | | − | | ≤− | | ≤ | | d dt
M
h 2
M
ej
2
=
h
j=1
j=0
ej+1 ej h
M
2
+h
εj ej ,
j=1
δ = 1/2
d dt
h 2
M
ej
j=1
h
M
M
ej (t)
j=1
2
T
h
2
+h
j=1
e
|
ej (t)
j=1
h k = xj jh t = nk u(x, t)
0 < h < h 0 ,
|
M
2 h = h
h 2
εj ej
j=1
εj (t) 2 dt,
0
j=1
M
ej
| ≤ |
M
|
h 2
2
2
4
| ≤ C h
x
T
0
M
εj 2 ,
j=1
0
≤ t ≤ T .
2C ([0,π])dt,
u(t)
4
(xj , tn )
t
n
U jn ≈ u(xj , tn ) ut (x, t)
ut (x, t) = uxx (x, t) =
u(x
u(x, t + k) k
− u(x, t) + O(k),
− h, t) − 2u(x, t) + u(x + h, t) + O(h2), h2
uxx (x, t)
O(k )
U jn+1 U jn U jn−1 = k
−
r =
O(h2 )
n − 2U jn + U j+1 ,
h2
k h2
U jn+1 = (1
n − 2r)U jn + r(U j+1 + U jn−1 ).
ϕ(x) =
− x)
x(π
h = π/5 k = 3/20
0
≤ t ≤ 3
k = 3/10
r r = 0,38
r
≈ 0,75
≤ 1/2
inestable
estable
k
h
k 2
x, t + k ut (x, t + ) = 2 k uxx (x, t + ) = 2
u(x, t + k) u(x, t) + O(k 2 ), k 1 [u(x h, t + k) 2u(x, t + k) 2h2 +u(x + h, t + k) + u(x h, t) 2u(x, t) +u(x + h, t)] + O(h2 ),
−
−
−
−
−
O(k 2 ) n+1 n+1 n n n −rU jn+1 −1 + (2 + 2r)U j − rU j+1 = rU j−1 + (2 − 2r)U j + rU j+1 , r = hk j = 1,...,N − 1 N − 1
h =
2
n+1
O(h2 )
π N
n
AUn+1 = b ,
A =
Un+1 =
U 2n+1 U 3n+1 n+1 U N −1
−
2 + 2r r
,
b =
−r
−r
2 + 2r
−r
2 + 2r
,
U 3n + (2 2r)U 2n U 2n + (2 2r)U 3n + U 4n
−
n U N −2 + (2
−
− 2r)U N n −1
h
B
t
ut (t) = Au(t) 0 u(0) = u 0 ,
.
k
u(t)
≤ t ≤ T ,
∈B
A:
B → B
u0
∈ B
A
B B = L2
u0
∈ B
A = ∂ x2
S k :
B → B,
k t U n+1 = S k U n
U n = S kn U 0 ,
⇒
S kn = (S k )n
n A
S k
h
h = h(k)
hj = h j (k)
{S k }
p
u(t + k) − S k u(t) = O(h p+1), t
∈ [0 , T ]
u(t) p > 0
{S k } − (S k )nu0
en = u(nk) k
{S k } S kn u(0) − u(t) = 0, k→0,nk=t l´ım
t [0, T ]
∈
→ 0,
k
{S k } C > 0
S kn ≤ C, n
k
≥ 0
0
≤ nk ≤ T
α
S k ≤ 1 + αk,
S kn ≤ S k n ≤ (1 + αk)n ≤ eαhk ≤ eαT = C,
{S k } {S k }
u((n + 1)k) = S k u(nk) + T n+1 T n+1
en+1 = S k en + T n+1
en+1 ≤ S k en + T n+1 ≤ S k n e0 + T n+1 .
→ 0
e0
C 0 [0, π]
ξ r
U jn+1
= SU jn
=
αµ U j +µ ,
µ= l
−
{αµ2 }
S
n
n+1
h
∗
SU = a U,
aµ =
1 (α−µ ). h
ˆ ), SU (ξ ) = a U (ξ ) = ˆa(ξ )U (ξ a ˆ(ξ ) U
∗
ξ
S g(ξ )
g = g(ξ )
U jn = gn eiξjh
≤
SU = SU S ≤ g∞
ˆ = gU
ˆ , g ∞ U
= g . S = sup SU ∞ U U ∈ 2
h
n S n U 0
U n (ξ ) = (g(ξ ))n U 0 (ξ )
g(ξ )
(g(ξ ))n∞ = m´ξax(|g(ξ )|n) = m´ξax(|g(ξ )|)n = ( g∞)n, S n = (g∞)n = S n ,
S : 2h
→ 2h S n = (g∞)n, ∀n > 0.
U n =
|g(ξ )| = 1 + O(k), → 0
∈
k
ξ [π/h, π/h]
g(ξ ) = 1
− 4r sin2 ξh2
−1 ≤ 1 − 4r sin2 ξ2h ≤ 1 ⇒ r
≤ 12 .
r
r
n+1 β µ U j+µ
=
µ= l
n αµ U j+µ ,
µ= l
−
−
{αµ} {β µ}
β 0 = 0 β µ = 0 S : U n U n+1 U n+1
U n
→
β µ = 0 µ =0
BU n+1 = AU n , A
B b U n+1 = a U n ,
∗
aµ = n U
1 hα µ
−
bµ =
1 h β µ
−
∗
U n+1
n+1 n+1 n n n −rU jn+1 −1 + (2 + 2r)U j − rU j+1 = rU j−1 + (2 − 2r)U j + rU j+1 ,
µ = 0
U jn = 0, j
∀
n+1 n+1 −rU jn+1 −1 + (2 + 2r)U j − rU j+1 = 0,
U jn+1 = 0, j
∀
U jn+1 = κ j
κ
−rκ2 + (2 + 2r)κ − r = 0,
U n
2h
U n+1
ˆb(ξ ) U n+1 (ξ ) = ˆa(ξ )U n (ξ ),
∈ −
ξ [ π/h, π/h]
ˆb(ξ ) = 0, ξ [ π/h, π/h]
∀ ∈ −
a ˆ(ξ ) n U n+1 (ξ ) = g(ξ )U n (ξ ) = U (ξ ),
g(ξ )
ˆb(ξ )
g (ξ )
−
[ π/h, π/h] a ˆ(ξ ) g∞ = ξ∈[−m´ ax < ∞, π/h,π/h] ˆ b(ξ )
U jn = gn eiξjh
1 2r sin2 a ˆ(ξ ) g(ξ ) = = ˆb(ξ ) 1 + 2r sin2
−
ξh 2 ξh 2
< 1,
U jn
× N
N
N A
αµ
β µ
B N
× N
G(ξ ) = [ˆb(ξ )]−1 [ˆa(ξ )]. C
G(ξ )n ≤ C, n, k 0 ≤ nk ≤ T
ξ [ π/h, π/h]
∈ −
≤ G(ξ )n ≤ G(ξ )n ,
ρ(G(ξ )n ) ρ(G)
G
G(ξ ) ρ(G(ξ ))
≤ 1 + O(k)
G(ξ ) ≤ 1 + O(k) k
→ 0
ξ [ π/h, π/h]
∈ −
n U jn+1 = U jn−1 + 2r(U j+1
− 2U jn + U jn−1).
r= k/h2
−
W jn
W jn+1 =
2r
0
0
0
W jn−1 +
U jn
=
U jn−1
4r
1
1
0
,
W jn +
2r
0
0
0
n W j+1 ,
n W jn+1 = α−1 W jn−1 + α0 W jn + α1 W j+1 ,
α−1 =
− − 2r
0
0
0
, α0 =
G(ξ ) =
4r
1
1
0
, α1 =
ξh 2
8r sin2 1
λ2 + 8rλ sin2 γ = 8r sin2
1
2r
0
0
0
.
,
0
ξ h 2
− 1 = 0,
ξh 2
−γ ± λ =
γ 2 + 4
2
,
√ − γ − 4 λ< < −1, 2
U jn = g n eiξjh g 2 = 1 + 2rg(eiξh
− 2 + e−iξh ) = 1 − 8rg sin2 ξ2h ,
g
≤ 21
r
Kv jn
n = v j+1
vjn = v(xj , tn ) n µ+ vjn = 21 (vj+1 + vjn )
µ− vjn = 21 (vjn + vjn−1 ) n µ0 vjn = 21 (vj+1 + vjn−1 ) n δ + vjn = h1 (vj+1
− vjn )
δ − vjn = h1 (vjn
− vjn−1) 1 n − vjn−1) δ 0 vjn = 2h (vj+1 n δ × vjn = h1 (vj+1 − 2vjn + vjn−1) 2
x
µ+ vjn = 21 (vjn+1 + vjn ) µ− vjn = 21 (vjn + vjn−1 ) µ0 vjn = 21 (vjn+1 + vjn−1 ) δ + vjn = k1 (vjn+1
− vjn )
− vjn−1) 1 δ 0 vjn = 2k (vjn+1 − vjn−1 ) δ × vjn = k1 (vjn+1 − 2vjn + vjn−1 ) δ − vjn = k1 (vjn
2
t
δ 0 = 21 (δ + + δ − )
I
δ × =
1 k2 (Z
− 2I + Z −1)
Z
δ + v = δ × v δ + v = µ + δ × v δ 0 v = δ × v
( 56 I + 61 µ0 )δ + v = µ+ δ × v δ + v = µ + [ 43 δ × (h) 31 δ × (2h)]v δ + v = h12 (K 2µ0 + K −1 )v
−
δu c x,t,u, δx
≤ x ≤ b
a f
δu δ = x −m δt δx t0
≤ t ≤ tf
δu x f x,t,u, δx m
−
δu + s x,t,u, δx
,
m = 0, 1, 2
c
s u(x, t0 ) = u 0 (x)
x = a
δu pa (x,t,u) + q a (x, t)f x,t,u, δx x = b
pb (x,t,u) + q b (x, t)f x,t,u,
δu δx
=0
=0
[a, b]
(tj , xk )
δu δ 2 u = 2 + (k δt δx
− 1) δu − ku, δx
k = r/(σ2 /2) r = 0,065 σ = 0,8 a = log(2/5) b = log(7/5) t0 = 0 tf = 5 u(x, 0) = m´ax(ex 1, 0) u(a, t) = 0 7 5e−kt u(b, t) = . 5
−
−
ut = u xx + uyy , (x,y,t) Ω, u(x,y, 0) = u 0 (x, y), (x, y) ∂ Ω Ω, u(x,y,t) = 0, (x,y,t) ∂ Ω,
Ω = ( a, b)
∈ ∈ ∪ ∈
× (a, b) × (0, T )
∂ Ω
x, y
uxx (x,y,t)
≈ δ xu(x,y,t) uyy (x,y,t) ≈ δ y u(x,y,t)
= =
h =
b a N +1
−
− 2u(x,y,t) + u(x − h,y,t) , h2 u(x, y + h, t) − 2u(x,y,t) + u(x, y − h, t) , u(x + h,y,t)
N 2 d U(t) = A U(t), dt
h2
U
= (U 1,1 , . . . , UN,1 , U 1,2 , . . . , UN,2 , . . . . . . U1,N , . . . , UN,N )T ,
A = B + C
B =
1 h2
B1 B1 B1
C =
B
1 h2
−
2
,
B1 =
−
2I
I
I
−2I
1
−
,
1 2
−1
I 2I
I
y
−
−1
.
x
C
A
−4 λi,j = h2
iπ jπ sin + sin 2 2(N + 1) 2(N + 1) 2
,
i, j = 1, . . . , N .
U(t) = e t(B+C ) U(0),
U(t + k) = e k(B+C ) U(t).
ek(B+C )
≈ I + k(B + C )
Un+1 = (I + k(B + C ))Un ,
I + k(B + C ) k h2
≤ 14 . B
n+1 n U i,j = U i,j +
k (U n h2 i+1,j
1 + kλ i,j
C
n − 2U i,jn +U in−1,j )+ hk2 (U i,j+1 − 2U i,jn +U i,jn −1).
O(k + h2 ) n U i,j = g n eiξ1 xi eiξ2 yj
− − −
k I (B + C ) 2
I
k B 2
I
k C 2
n+1
=
n+1
=
I +
k B 2
I +
k (B + C ) 2
I +
k C 2
B
+
,
k2 BC ( 4
C n+1
U
− − k I B 2
n
n
k I C 2
n
−U
n+1
− −
k I B Vn+1 2 k I C Un+1 2
=
= I +
k I B Vn+1 2 k I C Un+1 2
− −
= O (k)
= =
k B 2
I +
.
k C Un , 2 k I + B Vn+1 . 2 I +
I +
b a
n
k C 2
I +
= Vn+1 .
a c
k C 2
b c
a
.
k B Un , 2
n+1
−
n
).
λs = a + 2
√
·
A = Aij
·
bc cos
sπ , N + 1
s = 1
A11
·· ·
A1m
Am1
·· ·
Amm
· ·· N.
n
n
(k)
λij
k
(k)
· ··
λ1m
(k)
· ··
λmm
λ11
λm1
(k)
(k)
Aij
ut = uxx 0 < x < 1, u(x, 0) = sin(πx) + sin(3πx) u(0, t) = u(1, t) = 0.
− 1
1 (k 2
2
− h6 )δ ×
1 U jn+1 k
−
U jn
k = 1
1 h 2 U jn+1 = 1 + (k + )δ × U jn , 2 6
a n + 3 U j+2 h
−
a
n 3U j+1 + 3U jn
−
···n
0 < t < 0,1,
ut + auxxx = f
U jn 1
−
= f jn .
ut h, k
uxxx
−
k
U jn 2
→ 0 ut + auxxx = f
1 U n+1 k j
h
1 n a n (U j+1 + U jn−1 ) + 3 U j+2 2 2h
ut
−
n 2U j+1 + 2U jn 1
−
−
−
= f jn .
≤ x ≤ 1/2, t > 0 1 x(1 − x), 0 ≤ x ≤ , 2
= xuxx ,
u(x, 0) =
a
0
u(0, t) = 0, t > 0, 1 1 1 ux ( , t) = u( , t), 2 2 2
−
t > 0.
x t
h
k
≤ 2+2
r
h
r =
2
x = 1/2
δu δt δv δt 0
≤ x ≤ 1
0
= =
1 δ 2 u 1 + , 2 2 δx 1 + v2 1 δ 2 v 1 + , 2 2 δx 1 + u2
≤ t ≤ 0,2 1 u(x, 0) = 1 + cos(2πx), 2 1 v(x, 0) = 1 cos(2πx), 2
−
k h2
δu δu δv δv (0, t) = (1, t) = (0, t) = (1, t) = 0. δx δx δx δx
1 E (t) = 2
δu δt δv δt F (y) = exp (5,73y)
1
0
δu δx
2
+
δv δx
δ 2 u F (u δx2 δ 2 v = 0,170 2 + F (u δx = 0,024
−
− exp(−11,46y) u(x, 0) v(x, 0) δu (0, t) δx v(0, t) u(1, t) δv (1, t) δx
2
dx,
− v), − v), 0 ≤ x ≤ 1 0 ≤ t
= 1, = 0, = 0, = 0, = 1, = 0.
n+1 n U i,j = 1 + kδ x 1 + kδ y U i,j ,
F (x,t,u,
a(x, t)
∂u ∂u , ) = 0. ∂t ∂x
∂u ∂u + b(x, t) = c(x, t)u + d(x, t). ∂x ∂t
u(x, t) P (x,t,u)
∂u ∂u + Q(x,t,u) = R(x,t,u), ∂x ∂t
≡
γ
s (s0 , s1 )
∈
x = x0 (s), t = t 0 (s), u = u0 (s)
v(x,t,u) = P (x,t,u)i + Q(x,t,u) j + R(x,t,u)k,
u = u(x, t) ∂u ∂ u i+ j ∂x ∂t
n =
− k,
·
v n = 0,
u
v
γ (x0 (s), t0 (s), u0 (s))
dx dr dt dr du dr
v
= P (x,t,u), = Q(x,t,u), = R(x,t,u),
(x0 (s), t0 (s), u0 (s))
r = 0
x = x(s, r), t = t(s, r), u = u(s, r),
γ
i
j
k
P
Q
R
dx0 ds
dt0 ds
du0 ds
x
= 0 .
∂u ∂u 2 + t = u, ∂x ∂t 3 t = z 3 , t > 0, x = 1
dx dr
= x,
dt dr
= t,
du dr
= 32 u,
r = 0 x = 1, t = s 3 , u = s
x(s, r) = e r , t(s, r) = s 3 er , u(s, r) = se
s
r
2r 3
,
x
t
1
u = (xt) 3 .
s t
r
x
u(x, t)
∂ ∂ u(x, t) + f (u(x, t)) = 0, ∂t ∂x u
m f
−∞ < x < ∞ u(x, 0) = u 0 (x),
−∞ < x < ∞.
t > 0
ρ(x, t)
v(x, t) x1
x2
x2
[x1 , x2 ]
t =
ρ(x, t)dx.
x1
(x, t) (x, t) = ρ(x, t)v(x, t). [x1 , x2 ]
d dt
x2
ρ(x, t)dx = ρ(x1 , t)v(x1 , t)
x1
ρ(x, t)
v(x, t)
−
x2
ρ(x1 , t)v(x1 , t)
− ρ(x2, t)v(x2, t) =
− ρ(x2, t)v(x2, t),
x1
∂ (ρ(x, t)v(x, t))dt, ∂x
ρt + (ρv)x = 0. v ρv = f (ρ)
v(x, t) = a ρt + aρx = 0,
t
ut + aux = 0.
−∞ < x < ∞, t ≥ 0 u(x, 0) = u 0 (x).
dx dr dt dr du dr
= 1, = a, = 0,
r = 0 t = 0, x = x0 , u = u o (x0 )
u(x, t) = u 0 (x t
≥ 0
a
− at),
a > 0
a<0 x at = cte
−
1 0.9 0.8 0.7 0.6 ) t , x ( 0.5 u
0.4 0.3 0.2 0.1 0
0
0.5
1
1.5
2
2.5 x
3
3.5
4
4.5
5
u(x, t)
xtu
ut + (a(x)u)x = 0, a(x)
∂ ∂ + a(x) ∂t ∂x
u(x, t) =
−a (x)u(x, t),
u
x (t) = a(x(t)), x(0) = a 0 ,
d u(x(t), t) = dt
f (u) =
−a (x(t))u(x(t), t).
1 2 2u
ut + uux = 0. u(x, 0) = u0 (x) u(x, t) = u0 (x
− tu(x, t)), u
u0 (x) =
−x u(x, t) =
−x , 1−t
t = 1 x(t) = (1 t = 1
− t)x(0)
t = 1
(0, 1)
h x
k
x
U jn+1 = U jn λ =
t
t
k h
n − λa(U j+1 − U jn ).
u
T hk = c hk
chk = k h
a (ak 2
∂ 2 u ∂ 3 u + d + O(k 3 + h3 ), hk ∂x 2 ∂x 3
− h),
dhk =
a 2 (h 6
− a2k2),
= a
chk = 0 ∂u ∂u ∂ 2 u ∂ 3 u + a + chk 2 + dhk 3 = 0, ∂t ∂x ∂x ∂u
U jn = g n eiξjh g(ξ ) = (1
− aλ) + aλeiξh ,
|g(ξ )|2 = 1 − 2aλ(1 − aλ)(1 − cos ξh) ≤ 1, |a|λ ≤ 1
U jn+1 = U jn
n − a2 λ(U j+1 − U jn−1),
T hk =
k2 a2 ∂ 2 u 2 ∂x 2
3 3
− ( k 6a
U jn+1 = U jn
+
aλh 3 ∂ 3 u ) 3 + O(k 4 ) + O(h4 ). 6 ∂x
2
n n − a2 λ(U j+1 − U jn−1) + a2 λ2(U j+1 − 2U jn + U jn−1),
g(ξ ) = 1 + iaλ sin ξh
− 2a2λ2 sin2 ξ2h ,
|g(ξ )|2 = 1 + 4(a4λ4 − a2λ2)sin4 ξ2h ≤ 1, |a|λ ≤ 1 (xj , tn ) u(xj , tn ) = u0 (xj atn ) x xj = a(t
− −
λ =
−
xj n t )
− atn
k h
(xj , tn ) [xj 1λ tn , xj +
−
1 n λt ]
− atn ∈ [xj − λ1 tn, xj + λ1 tn)], |a| ≤ λ1
xj
|a|λ ≤ 1
ut + aux = b ut + Aux = b ,
u(x, t)
b(x, t)
m
A(x, t)
m
A P Λ = diag(λ1 ,...,λ m ) = P −1 AP, λj
A = P ΛP −1
A
ut + P ΛP −1 ux = b ,
(P −1 u)t + Λ(P −1 u)x = P −1 b, v = P −1 u
m
(vj )t + λj (vj )x = (P −1 b)j , vj (x, t) = (P −1 u0 )j (x
− λj t)
j = 1,...,m
m=2 A =
∂u 1 ∂t ∂u 2 ∂t
− 0
1 0
−1
− ∂∂xu2 − ∂∂xu1
= 0, = 0. t
∂ 2 u1 ∂t 2 ∂ 2 u2 ∂x∂t
2
∂ u2 − ∂t∂x 2 − ∂ ∂xu21
∂ 2 u1 ∂t 2
x
2
= 0, = 0,
− ∂ ∂xu21 = 0,
×m
c
utt c2 uxx = 0, x u(x, 0) = ϕ(x), x ut (x, 0) = ψ(x), x
−
1 u(x, t) = [ϕ(x + ct) + ϕ(x 2
±c
−
∈ R, t > 0, ∈ R, ∈R 1 ct)] + 2
u(x, t) = F (x + ct) + G(x
F (x) = G(x) = x
1 1 ϕ(x) + 2 2 1 1 ϕ(x) + 2 2
x+ct
ψ(s)ds,
x ct
−
− ct),
s
ψ(σ)dσ,
0
0
ψ(σ)dσ.
s
± ct = cte
(x, t)
− ct,x + ct]
[x
u ϕ
ψ
1 E (t) = 2
|
ux (x, t) 2 + ut (x, t) 2 dx,
| |
R
|
dE = 0, t dt
∀ ≥ 0.
∈
u C ([0,
∞); H 1(R))
(ϕ, ψ) H 1 (R) L2 (R) C 1 ([0, ); L2 (R))
∞
∈
×
π
utt c2 uxx = 0, u(x, 0) = ϕ(x), ut (x, 0) = ψ(x), u(0, t) = u(π, t) = 0,
−
0 < x < π,t > 0, 0 < x < π, 0 < x < π, t > 0,
ϕ(x) =
ϕˆl wl (x),
ψ(x) =
l 1
ψˆl wl (x),
0 < x < π,
l 1
≥
≥
2 π sin(lx)
wl (x) =
u(x, t) =
| ∞
l=1
E (t) =
1 2
ψˆl ϕˆl cos(lt) + sin(lt) wl (x). l
π
ux (x, t) 2 + ut (x, t) 2 dx.
| |
0
|
H 01 (0, π) L2 (0, π)
×
0 < x < 1 ψ(x) = 0 0
ux (0, t) = 0 x
h =
π M +1
M
u
(t) 2u (t)+uj
−
j uj (t) c2 j+1 h2 uj (0) = ϕj , uj (t) = ψj , u0 (t) = uM +1 (t) = 0,
−
1
−
(t)
= 0, t > 0
t > 0,
j = 1,...,M
u(t) =
u1 (t)
uM (t)
u (t) Ah u(t) = 0 , u(0) = ϕ, u (0) = ψ.
−
−
2
,
Ah =
c2 h2
1
1
−AhW = λW,
1
− 1 2
.
4c2 h λl (h) = 2 sin2 l h 2
Wl (h) =
j = 1,...,M
2 π
,
sin(lx1 ) sin(lxM )
h
,
→ 0
M
ϕ =
ϕˆl (h)Wl (h),
l=1 M
ψ
=
ψˆl (h)Wl (h),
l=1
ϕˆl (h) = ψˆl (h) =
e, f h = h
M
u(t) =
l=1
µl (h) =
M j=1 ej f j
ϕ, W(h)h , ψ, W(h)h
ψˆl (h) ϕˆl (h)cos(µl (h)t) + sin(µl (h)t) Wl (x), µl (h)
λl (h) h E h (t) = 2
M
j=0
uj+1 uj h
−
dE h dt
| | 2
+ uj
2
,
=0
uj (t) = g(t)eiξjh g (t) + 4
c2 ξ h sin 2 g(t) = 0, 2 h 2
g (t)
|
|
d 1 2 c2 ξ h g (t) + 2 2 sin 2 g(t) 2 = 0, dt 2 h 2
|
|
n n n U jn+1 2U jn +U jn 1 2 U j+1 2U j +U j = c 2 2 k h U j0 = ϕj , n U 0n = U M +1 = 0, −
−
j = 1,...,M
−
≥ 0
1
−
, ,
n
U j0 = ϕ j
U j1
U j1 = ϕ j + kψj ,
2 1.5 1 0.5 ) t , x ( u
0 −0.5 −1 −1.5 −2
20
0 2
15 4
10 6 5
8 10
0
t
x
U jn+1 = (2 λ = c hk
n − 2λ2)U jn + λ2(U j+1 + U jn−1 ) − U jn−1 ,
U jn = g n eiξjh 2
−
g = 2
−
ξh 4λ sin ( ) g 2 2
2
λ
utt 4uxx = 0, u(x, 0) = sin πx + sin 2πx, ut (x, 0) = 0, u(0, t) = u(π, t) = 0,
0 0 0 0
−
x
∂u ∂x
1,
≤
1
≤ x ≤ 1, 0 ≤ t ≤ 0,5, ≤ x ≤ 1 ≤ x ≤ 1 ≤ t ≤ 0,5,
− t ∂u = 0, ∂t x = 0, z = t 2
xux + tut
= 2u, x2 + t2 = 1, γ u = 1.
≡
x x = 1, t = 0
ut + ux γ
= u,
≡
t = 1, x =
−1
−
x = s, t = s, u = 1.
U jn
U jn+1 =
1 n 1 n (U j+1 + U jn−1 ) + aλ(U j+1 2 2
U jn+1 U jn 3U jn +a k
−
− U jn−1).
− 4U jn−1 + U jn−2 − a 2k U jn − 2U jn−1 + U jn−2 = 0, 2h
h2
2
ut + aux = 0 ut + ux = 0 (1
n+1 n n − λ)U jn+1 −1 + (1 + λ)U j = (1 + λ)U j−1 + (1 − λ)U j
λ =
k h
λ
∂u 1 ∂u 2 ∂u 1 ∂ u2 2 +5 ∂t ∂t ∂x ∂x ∂u 1 ∂u 2 ∂ u1 ∂u 2 2 +4 +5 ∂t ∂t ∂x ∂x 4
−
−
−
−
utt c2 uxx = 0, u(x, 0) = f (x), ut (x, 0) = g(x), u(0, t) = α(t), u(L, t) = β (t).
−
u(x, t) = P (x + ct) + Q(x
− ct)
0
≤ x ≤ L,
= 0 = 0
0
≤ t,
utt = u xx 0 < x < π, u(x, 0) = 0, ut (x, 0) = 5 sin 5x, u(0, t) = 0, u(π, t) = 0 .
−
utt 4uxx = 0, u(x, 0) = x u(x, 0) = 1,5 1,5x ut (x, 0) = 0, u(0, t) = u(π, t) = 0,
−
t
≥ 0,
≤ x ≤ 1,3 0 ≤ t ≤ 0,5, ≤ x ≤ 5 , 3 ≤ x ≤ 1, 5 0 ≤ x ≤ 1 0 ≤ t ≤ 0,5, 0 0
u(x)
v(x) = u(x, t)
N k=0 ak φk (x)
v(x, t) =
N k=0 ak (t)φk (x)
v(x) = u(x) ak d dx
N k=0 ak φk (x)
bk
N
N
ak φk (x)
=
k=0
bk φk (x).
k=0
ak v(x)
xi , i = 0,...,N
φk (x)
ak
ak (t)
v(x) R(x)
R(x)
uxx + ux
− 2u + 2 = 0, −1 ≤ x ≤ 1,
−
u( 1) = u(1) = 0,
−2x
x
u(x) = 1
+ sh(1)e − sh(2)e sh(3)
.
4
v(x) =
ak T k (x),
k=0
x
4
vx (x) =
bk T k (x),
k=0
b0 b1 b2 b3 b4
=
0 1 0 3 0 0 4 0 8 0 6 0 0 8 0
a0 a1 a2 a3 a4
= A
a0 a1 a2 a3 a4
4
R(x) = v xx + vx
− 2v + 2 =
k=0
Ak T k (x),
.
A0 A1 A2 A3 A4
1 1
= (A2 + A
1 1
−1
1
√
− 2I )
1 1
−1
1
a0 a1 a2 a3 a4
a0 a1 a2 a3 a4
+
2 0 0 0 0
=
0
0
1
R(x)T k (x) dx −1 1 x2
k = 0, 1, 2,
−
A0 = A1 = A2 = 0 a0 = 0,2724,
−0,0444, a2 = −0,2562, a1 =
a3 = 0,0444 a4 =
−0,0162,
− T 0(x), T 3 (x) − T 1 (x), T 4 (x) − T 0 (x),
φ2 (x) = T 2 (x) φ3 (x) = φ4 (x) =
.
√ 1
R(x)φk (x) dx = 0 −1 1 x2
k = 2, 3, 4
−
a0 = 0,2741, a1 =
−0,0370, a2 = −0,2593, a3 = 0,0370, a4 =
−0,0148, R(x) xi = cos
1
1 2
1 2
√
0
− √
1
0
−1
0
1
− √ 12
0
√ 12
− − 1
1
1
A0 A1 A2 A3 A4
iπ 4
=
i = 1, 2, 3
0 0 0
,
a0 = 0,2743,
−0,0371, a2 = −0,2600, a1 =
a3 = 0,0371, a4 =
xj −1
{u1, . . . , uN } u0 = uN
u (xj )
−0,0143,
{x1, . . . , xN } u1 = uN +1
− uj−1 , ≈ wj = uj+1 2h
j = 1, . . . , N
h = x j
−
w1
wN
1 = h
−
0
1 2
1 2
0
− 12 0
1 2
− 12
1 2
0
u1
uN
.
j = 1, . . . , N pj
∈ ℘2
pj (xj ±i ) = uj ±i
i = 0, 1
wj = pj (xj ) j = 1, . . . , N pj ℘4
pj (xj ±i ) = u j ±i
i = 0, 1, 2
wj = pj (xj )
w1
wN
=
1 h
−
1 12
− 121 2 3
0
− 23
1 12 2 3
1 12
− 121
−
2 3
1 12
u1
uN
,
1 D = h
D
1 3
− 12 1 0
−1 1 2
− 13
DT =
p
p(xj ) = uj
.
−D
∀ j
wj = p j (xj ) N
× N
N
DN =
h =
2π N
⇒
π h
1 3h 2 cot 2
− 12 cot 2h2 1 1h 2 cot 2
0
− 12 cot 1h2 1 2h 2 cot 2
− 12 cot 3h2
=
N 2
[0, 2π] N
N
vˆk = h
e−ikxj vj ,
k =
j=1
1 vj = 2π
N/2
− N 2 + 1, . . . . . . , N 2 ,
N/21
ikxj
e
vˆk =
k= N/2+1
eikxj vˆk
j = 1, . . . , N
k= N/2
−
−
vˆ−N/2 = vˆN/2
k =
1 p(x) = 2π
±N/2
1/2
N/21
eikx vˆk
∀x ∈ [0, 2π],
k= N/2
−
N/2
vj p(x) v (xj )
≈ p (xj )
v
δ j =
1, 0,
j = 0 (modN ), j = 0 (modN ),
ˆk = h δ
p(x) =
h 2π
N/21
eikx =
k= N/2
−
h x p(x) = cos( ) 2π 2
h 2π
1 2
N/2 1
−
k= N/2
N/2 1/2
−
eikx =
k= N/2+1/2
−
eikx +
−
1 2
N/2
eikx
(xj ) = S N
0, jh 1 j 2 ( 1) cot( 2 ),
−
,
k= N/2+1
−
h x sin(N x/2) cos( ) = S N (x), 2π 2 sin(x/2)
S N (x) S N (x) =
∀k
sin( πx h ) 2π x h tan( 2 )
j = 0 (modN ), j = 0 (modN ).
N
vj =
m=1
vm δ j −m ,
N
p(x) =
vm S N (x
− xm),
(x vm S N
− xm).
m=1 N
p (x) =
m=1
xj
N
v (xj ) ≈ p(x1 ) = j
(xj vm S N
m=1
− xm),
(xj (S N
− x1), S N (xj − x2), · · · · · · , S N (xj − xN )),
× N
N
DN =
− −
− 12 cot 1h2
0 1 1h 2 cot 2
1 2h 2 cot 2
1 2 h 2 cot 2
− 12 cot 3h2
1 3h 2 cot 2
1 1h 2 cot 2
1 1 h 2 cot 2
S (xj ) = N
−
(2)
−
DN =
0
1 π2 , 3h2 6 ( 1)j 2sin2 (jh/2) ,
− −
j = 0 (modN ), j = 0 (modN ).
− 12 csc2( 2h2 ) 1 2 1h 2 csc2 ( 2 ) 1 π 3h2 6
− −
1 2 1h 2 csc ( 2 )
− 12 csc2( 2h2 )
,
{ − |x −2 π| },
g(x) = e sin x ,
f (x) = m´ax 0, 1
function
spectral derivative
1.5
1
1
0.5
0.5
0
0
−0.5
−0.5
0
2
4
6
3
−1
0
2
4
6
2 max error = 9.6878e−013
1
2
0 1 0
−1 0
2
4
6
−2
0
equispaced points 1.5
1
1
0.5
0.5
0
0
−1
max error = 5.9001
−1
−0.5
0
0.5
4
6
Chebyshev points
1.5
−0.5
2
max error = 0.017523
−0.5
1
−1
−1
u(x) =
−0.5
1 1+16x2
0
0.5
N = 16
1
−
[ 1, 1] xj = 1 +
−
2j N
j = 0, . . . , N xj = cos jN π
j = 0, . . . , N
≥ 1
N
DN N (DN )00 =
2N 2 + 1 , 6
(DN )jj = (DN )ij =
2
− 2N 6+ 1 ,
(DN )NN =
−xj , 2(1 − x2j )
ci ( 1)i+j , cj (xi xj )
−
2 1
N
− 1,
i = j,
−
ci =
j = 1,...,N
i, j = 0,...,N,
i = 0, N, .
3
exp(−x−2)
|x | 0
0
10
10
−5
r o r r
−5
10
r o r r
e
10
e
−10
−10
10
10
−15
−15
10
10
0
10
20
30
40
0
50
10
20
40
50
30
40
50
10
2
x
1/(1+x ) 0
0
10
10
−5
r o r r
30 N
N
−5
10
r o r r
e
10
e
−10
−10
10
10
−15
−15
10
0
10 10
20
30
40
50
0
(2)
2 DN = D N
10
20 N
N
N = 2
N = 4
