Di érentiabilité et intégrabilité en Coq Application à la formule de d

∗
1 2
∗H

(x, u(x, t))
∀(x, t)∈R2,
∂2u
∂t2(x, t)−c2∂2u ∂x2(x, t) = f(x, t)
u(x, 0) = u0(x)
∂u
∂t (x, 0) = u1(x)
u(x, t) = 1
2(u0(x+ct) + u0(x−ct))
| {z }
α(x,t)
+1
2cZx+ct
x−ct
u1(ξ)dξ
| {z }
β(x,t)
+1
2cZt
0Zx+c(t−τ) x−c(t−τ)
f(ξ, τ)dξ dτ
| {z }
γ(x,t)
u
x t α β γ u
γ
α β γ

P
f x
f x
f `
f x `
f:R→Rx, ` ∈R
f x `
fR
f x `
R

∀(x, t)∈R2, α(x, t) = 1
2(u0(x+ct) + u0(x−ct))
α
+×
f, g :R→Rx, l ∈R
∀y∈R, f(y) = g(y)f0(x) = l⇒g0(x) = l
∀(x, t)∈R2, β(x, t) = 1
2cZx+ct
x−ct
u1(ξ)dξ
β
1
2cZx+ct
0
u1(ξ)dξ −Zx−ct
0
u1(ξ)dξ.
β
P f a b
R
x7→ Rx
af(t)dt
fR
f:R7→ Ra∈RFa:x7→ Zx
a
f(t)dt
∀x∈R, F 0
a(x) = f(x)
β
β

fR
Rint(f):(a, b)7→ Zb
a
f(t)dt.
f(a, b)∈R2(f)
(a, b)
dRint(f)(a, b)=(−f(a), f(b)).
f
∀(x, t)∈R2, γ(x, t) = 1
2cZt
0Zx+c(t−τ) x−c(t−τ)
f(ξ, τ)dξ dτ
x
t t
(t1, t2)7→ 1
2cZt1
0Zx+c(t2−τ) x−c(t2−τ)
f(ξ, τ)dξ dτ.
γ
a, b ∈Ra < b f : (x, t)7→ f(x, t)
f t [a;b]
f x
∂f
∂x R×[a;b]
F:x7→ Zb
a
f(x, t)dt F 0(x) = Zb
a
∂f
∂x (x, t)dt.
R2→R
f S a, b ∈S f a b