{{{id=1|
# slightly modified from the interact webpage
def picard_iteration(f, t0, u0, iterations):
'''
Computes the N-th Picard iterate for the IVP
u'(t) = f(t,u(t)), for all t, u(t0) = u0.
'''
if iterations == 0:
u1 = lambda t: u0 + 2*t-t^2
return expand(u1(t))
for i in range(iterations):
u_old = lambda s: picard_iteration(f, t0, u0, iterations-1).subs(t=s)
un = lambda t: u0 + integral(f(t=s,u=u_old(s)), s, t0, t)
return expand(un(t))
@interact
def picarder(n_iterations = slider(0,20,1,default = 2)):
var('y,u,t,s')
f = lambda t,u: u
y = function('y',t) # define x to be a function of that variable
DE = diff(y, t) - f(t,y)
g=desolve(DE,[y,t],[0,1])
exact = plot(g,(t,0,2))
html('The exact solution is: ')
html('$'+latex(g)+'$')
for i in range(n_iterations):
pic = picard_iteration(f,0,1,i)
html('The Picard iteration approximation after ' + str(i) + ' iterations is: ')
html('$'+latex(pic)+'$')
exact = exact+plot(pic,(t,0,2), rgbcolor = (1,0,0))
show(exact)
///