def picard_iteration(f, a, c, N):
'''
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
sage: var('x t s')
(x, t, s)
sage: a = 0; c = 2
sage: f = lambda t,x: (x+t)^2
sage: picard_iteration(f, a, c, 0)
2
sage: picard_iteration(f, a, c, 1)
t^3/3 + 2*t^2 + 4*t + 2
sage: picard_iteration(f, a, c, 2)
t^7/63 + 2*t^6/9 + 22*t^5/15 + 16*t^4/3 + 11*t^3 + 10*t^2 + 4*t + 2
'''
if N == 0:
return c*t**0
if N == 1:
#print integral(f(s,c*s**0), s, a, t)
x0 = lambda t: c + integral(f(s,c*s**0), s, a, t)
return expand(x0(t))
for i in range(N):
x_old = lambda s: picard_iteration(f, a, c, N-1).subs(t=s)
#print x_old(s)
x0 = lambda t: c + integral(f(s,x_old(s)), s, a, t)
return expand(x0(t))
v=var('x t s')
a = 0; c = 4; N=10; b=2;
f = lambda t,x: sin(t)-2*x; assume(t>0)
z=[picard_iteration(f, a, c, i) for i in range(N+1)]
for i in range(N+1):
show(z[i])
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,-b+1,b]
p=plot(-1/5*(cos(t)*e^(2*t) - 2*e^(2*t)*sin(t) - 21)*e^(-2*t),where,color='gray',gridlines=True) #SoluciĆ³n exacta.
#p+=plot(z[0],where,gridlines=True)
for i in range(7,N+1):
p+=plot(z[i],where,color=c[i])
show(p)
N=50;b=1
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,0,b]
p=plot(x^0,where,color='gray',gridlines=True)
for i in range(1,N+1):
p+=plot(x^i,where,color=c[i])
show(p)
x = var('x')
y = function('y')(x)
show(desolve(diff(y,x) - exp(x+y), y))
x = var('x')
y = function('y')(x)
f = desolve(diff(y,x) -exp(x+y), y, ics=[0,1]); show(f)
t = var('t')
x = function('x')(t)
f = desolve(diff(x,t) -sin(t) + 2*x, x, ics=[0,4]); f
t = var('t')
x = function('x')(t)
f = desolve(diff(x,t) -sin(t) + 2*x, x, ics=[0,4]); show(f)
def picard_iteration(f, a, c, N):
'''
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
sage: var('x t s')
(x, t, s)
sage: a = 0; c = 2
sage: f = lambda t,x: (x+t)^2
sage: picard_iteration(f, a, c, 0)
2
sage: picard_iteration(f, a, c, 1)
t^3/3 + 2*t^2 + 4*t + 2
sage: picard_iteration(f, a, c, 2)
t^7/63 + 2*t^6/9 + 22*t^5/15 + 16*t^4/3 + 11*t^3 + 10*t^2 + 4*t + 2
'''
if N == 0:
return c*t**0
if N == 1:
#print integral(f(s,c*s**0), s, a, t)
assume(s>0)
x0 = lambda t: c + integral(f(s,c*s**0), s, a, t)
return expand(x0(t))
for i in range(N):
x_old = lambda s: picard_iteration(f, a, c, N-1).subs(t=s)
#print x_old(s)
x0 = lambda t: c + integral(f(s,x_old(s)), s, a, t)
return expand(x0(t))
v=var('x t s')
a = 0; c = 1; N=2; b=.5;
f = lambda t,x: exp(x+t); assume(t>0)
z=[picard_iteration(f, a, c, i) for i in range(N+1)]
for i in range(N+1):
show(z[i])
from sage.plot.colors import rainbow
c=rainbow(N+1)
where = [x,-b,b]
p=plot(-log(abs(-1-exp(-1)+exp(t))),where,color='gray',gridlines=True) #SoluciĆ³n exacta.
#p+=plot(z[0],where,gridlines=True)
for i in range(N+1):
p+=plot(z[i],where,color=c[i])
show(p)