TP-euler

f, y0, t0, tf n
Lt = [t0, . . . , tn]Ly = [y0, . . . , yn]
(tk, yk)
(Lt, Ly)
(∀t∈[0,1], y0(t) = y(t)
y(0) = 1 n
n= 3 n= 10 n= 50
x7→ x
(∀t∈[0,20], y0(t) = tsin(y(t))
y(0) = 1 n= 50 n= 80
t
•[t0, tf]n[tk, tk+1]tk=t0+k×tf−t0
n
h=tf−t0
n
•(yk)06k6n(y0=y(t0)
∀k∈J0, n −1K, yk+1 =yk+h×f(tk, yk+1)
h

f, y0, t0, tf n
(∀t∈[0,20], y0(t) = tsin(y(t))
y(0) = 1
n= 50 n= 80
(E)
∀t∈[t0, tf], x00(t) = f(t, x(t), x0(t))
x(t0) = x0
x0(t0) = v0
(E)
(x, x0)tk+1 tk
x, v, t
x(t)v(t)t t 0tf(n+ 1)
(H)
∀t∈[0,30], x00(t) + sin(x(t)) = 0
x(t0) = x0
x0(t0) = 0
(H)n= 1000 x0=1
•
•
•
t EM(t) = 1
2(v2(t) + (1 −cos(t))
n= 1000
x v

x= 0
n= 10000
(K)
∀t∈[0,30], x00(t)+2λx0(t) + ω2
0sin(x(t)) = 0
x(t0) = x0
x0(t0)=0
λ > 0ω0>0