def picard_iteration(f, a, c, iterations):
'''
Computes the N-th Picard iterate for the IVP
x' = f(t,x), x(a) = c.
EXAMPLES:
sage: var('x t s')
(x, t, s)
sage: a = 0; c = 2
sage: f = lambda t,x: 1-x
sage: picard_iteration(f, a, c, 0)
2
sage: picard_iteration(f, a, c, 1)
2 - t
sage: picard_iteration(f, a, c, 2)
t^2/2 - t + 2
sage: picard_iteration(f, a, c, 3)
-t^3/6 + t^2/2 - t + 2
'''
if iterations == 0:
return c
if iterations == 1:
x0 = lambda t: c + integral(f(t=s,x=c), s, a, t)
return expand(x0(t))
for i in range(iterations):
x_old = lambda s: picard_iteration(f, a, c, iterations-1).subs(t=s)
x0 = lambda t: c + integral(f(t=s,x=x_old(s)), s, a, t)
return expand(x0(t))
@interact
def picarder(n_iterations = slider(0,20,1,default = 2)):
var('x,t,s')
f = lambda t,x:x
html("Exact solution to $x' = x$, $x(0) = 1$ is $x = \exp(t)$<br>")
pic = picard_iteration(f,0,1,n_iterations)
html('The Picard iteration approximation after ' + str(n_iterations) + ' iterations is:<br>')
html('$'+latex(pic)+'$')
exact = plot(exp(t),(t,0,2))
pic_plot = plot(pic,(t,0,2), rgbcolor = (1,0,0))
show(exact + pic_plot)
