no10 :
R2
x0= 2(x−ty)
y0= 2y
(x0, y0)
t= 0
[0, T ]h=T
Nt0= 0 (xn, yn)
tn=nh n ∈[0, N]
(xn+1, yn+1) (xn, yn)
(xn, yn)n h x0y0
zn=xn
(1 + 2h)n
lim
N→+∞max({|xn−x(tn)|,0≤n≤N}) et lim
N→+∞max({|yn−y(tn)|,0≤n≤N})
[0,1]
y0(t) = −150y(t) + 50
y(0) = 1
3
100 y0= 0.3
y(1)
y(1)
(y0(t) = f(t, y(t)), t ∈[t0, t0+T],
y(t0)∈Rmfix .(1)
y∈C1([t0, t0+T],Rm)
f[t0, t0+T]×Rm
Rm
LRm
∀t∈[t0, t0+T],∀(y1, y2)∈(Rm)2,||f(t, y2)−f(t, y1)|| ≤ L||y2−y1||
N∈ ||| ∆T=T
N
∆T L < 1 (yn)0≤n≤N
(y0∈Rmfix ,
yn+1 =yn+ ∆T f(t0+ (n+ 1)∆T, yn+1),0≤n≤N−1.
tn=t0+n∆T en=yn−y(tn)
||en+1|| ≤ (1 + ∆T L1)(||en|| +||n||)
L1=L
1−L∆Tn
εn=y(tn+1)−y(tn)−∆T f(tn+1, y(tn+1))
en≤eL1n∆Te0+
n−1
X
i=0
eL1(n−1−i)∆T(1 + ∆T L1)εi
lim
N→+∞,y0→y(t0)( max
0≤n≤Nen) = 0
f C1
nO(∆T)2
1
/
2
100%
