Cours DF

ρ(A)6kAk
í
í
∂u
∂t −∂2u ∂x2=f
í
F(u, ux, uy, uxx, uxy, uyy, . . . , x, y) = 0
F
(x, y)
uxx + 3uxy +uyy +uxuexy = 0
sin(xy)uxx + 3x2uxy +uyy +uxu= 0
uxx + 3uxy + (ux)2uexy= 0
auxx + 2buxy +cuyy +dux+euy+fu =g
α β
P(α, β) = aα2+ 2bαβ +cβ2+dα +eβ +f
P(α, β)b−ac
íb2−ac > 0
íb2−ac = 0
íb2−ac < 0
ía=b= 1
ía= 1, b =c=1
2
ía=b= 1, c = 2

z=P(x, y)
ut−∆u=f
utu x
t
−∆u=f
−∆u=∂2
1u+∂2 2u ∂i
−∆2u=f
ut−ux= 0, t ∈R, x ∈R
u(x, 0) = u0(x)

u0
u(x, t) = u0(x+t)
u0
Ω⊂Rd
íu=u0∂Ω
í∂u
∂n =a ∂Ω∂u ∂n =
∇u.−→ n−→ n ∂Ω
Ω⊂RdΩ
Ωd= 2
d= 3
Ω = ]0,1[ ×]0,1[
ín∈Nh=1
n+1
Ωh={(ih, jh),16i, j 6n}
x=ih y =jh 16i, j 6n

í[0,1]
(xi)16i6nx(yj)16j6ny
Ωh={(xi, yj) 1 6i6nx,16j6ny}
x=xiy=yj16i6nx,16j6ny
x0=y0= 0 xnx+1 =yny+1 = 1
Ω
Au =b
ΩhUhRNN
t∈[0, T ]x y
tn∈[0, T ]
tn=n.δt δt > 0
u: [0, T ]×Ω→Ru(tn, xi, yj)
un
i,j
Cn+1 x∈R
h
f(x+h) =
n
X
k=0
hk
k!f(k)(x) + Ohn+1
k f x)
f(k)(x)ki`eme f, O (hn+1) 0
hn+1 h0
ϕ(h)O(hn+1)ϕ(h)
hn+1 
h= 0.
f(x+h), f
x.
f x,
f x.
f
x
f0(x) = f(x+h)−f(x)
h+O(h)
f0(x) = f(x)−f(x−h)
h+O(h)
f0(x) = f(x+h)−f(x−h)
2h+Oh2
f00 (x) = f(x+h)−2f(x) + f(x−h)
h2+Oh2
x−h x +h,

(1.6)
x+h x −h
f(x+h) = f(x) + hf0(x) + h2
2f00 (x) + Oh3
f(x−h) = f(x)−hf0(x) + h2
2f00 (x) + Oh3
2hf0(x) = f(x+h)−f(x−h) + Oh3
O(h3)
h=O(h2),
f: Ω ⊂Rd→R∂f
∂xi
∂2f
∂xi∂xj
16i, j 6d
d= 2
∂2f
∂x2(x, y) = 1 h2(f(x+h, y)−2f(x, y) + f(x−h, y)) + Oh2
∂2f
∂y2(x, y) = 1 h2(f(x, y +h)−2f(x, y) + f(x, y −h)) + Oh2
∆f(x, y) = 1
h2(f(x+h, y)−4f(x, y) + f(x−h, y) + f(x, y +h) + f(x, y −h)) + Oh2
h
∆h=1
h2
1
1−4 1
1
Au =b
A
−u00 +cu0=f]0,1[
u(0) = 0
u(1) = 0
f, c c >0 [0,1] n+ 1
h=1
n+1
Ωh={ih, 06i6n+ 1}
íu0=un+1 = 0
íih
u00
i=1
h2(ui−1−2ui+ui+1) + Oh2