! > 0 U− > 0
u! (x, µ)
1
µ∂x u! + ! (u! − #u! $) = 0 ,
u! (−1, +µ) = U− ,
u! (+1, −µ) = U+ ,
(T )
#u! $(x) =
u! ∈
L∞ ([−1, 1]2 )
%
1
−1
U+ > 0
u! ≡
|x| < 1 , |µ| < 1 ,
0 < µ < 1,
0 < µ < 1,
u! (x, µ)
dµ
.
2
!>0
x &→ u! (x, µ)
C1
[−1, 1]
!
u! (x, µ) =
&
!k uk (x, µ) .
k≥0
u0 u1
u2
u0
U (x) =
U+ + U− U+ − U−
+
x , |x| < 1 .
2
2
v! (x, µ) = u! (x, µ) − U (x) + !µU $
U (x)
x
(v! − #v! $(2L2 ([−1,1]2 ) ≤ C0 !3
U+
U−
[−1, 1]2
C0
v!
S! (x, µ) = 1! (v! (x, µ) − #v! $(x))
√ '% 1
(1/2
2
2
|v! (x, µ)| ≤ !|µ||U | +
S! (x, µ) dx
|µ|
−1
$
U$
v!
(x, µ)
α>0
%
C1
1
−1
%
1
−1
1[α,1] (|µ|)v! (x, µ)2 dxdµ ≤ C1 !2 |U $ |2 +
C2
C0 !
α2
C2
α>0
%
1
−1
%
α
−α
v! (x, µ)2 dxdµ ≤ 2C0 !3 + 8α(v! (2L2 ([−1,1]2 )
! ∈]0, 1]
√
(u! − U (x)(L2 ([−1,1]2 ) ≤ C3 |U $ | !
C3
ν>0
(0, 1)
∂u
∂2u
−ν 2 =0
(x, t) ∈ (0, 1) × R+
∗
∂t
∂x
+
u(t, 0) = u(t, 1) = 0
t ∈ R∗
u(0, x) = u0 (x)
x ∈ (0, 1),
u0 (x)
∆x = 1/(N + 1) > 0
[0, 1]
(tn , xj ) = (n∆t, j∆x)
unj
θ ∈ [0, 1]
u(t, x)
un+1
− unj
j
∆t
n≥1
un0
=
−
n ≥ 0, j ∈ {0, 1, ..., N + 1}.
(tn , xj )
θ
ν ) n+1
θ(uj+1 − 2un+1
+ un+1
j
j−1 )
(∆x)2
*
+(1 − θ)(unj+1 − 2unj + unj−1 ) = 0
unN +1
=0
∆t > 0
N
u0j = u0 (xj )
j ∈ {1, ..., N },
unj
1 ≤ j ≤ N
c = ν∆t/(∆x)2
Un
∆t
AU n+1 = BU n
A
N ×N
B
M
U n+1
A
Un
L∞
L2
U n+1 − U n
+ KU ∗ = 0,
∆t
U ∗ = θU n+1 + (1 − θ)U n
K
2U ∗ = (U n+1 + U n ) + (2θ − 1)(U n+1 − U n ).
2U ∗
N
&
j=1
∆x|un+1
|2 ≤
j
L2
N
&
j=1
1/2 ≤ θ ≤ 1
∆x|u0j |2 .
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