Mlle Blanche - 9 months ago 95

Python Question

I am numerically solving for x(t) for a system of first order differential equations. The system is:

`dy/dt=(C)\*[(-K\*x)+M*A]`

I have implemented the Forward Euler method to solve this problem as follows:

Here is my code:

`import matplotlib`

import numpy as np

from numpy import *

from numpy import linspace

from matplotlib import pyplot as plt

C=3

K=5

M=2

A=5

#------------------------------------------------------------------------------

def euler (f,x0,t):

n=len (t)

x=np.array ([x0*n])

for i in xrange (n-1):

x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )

return x

#---------------------------------------------------------------------------------

if __name__=="__main__":

from pylab import *

def f(x,t):

return (C)*[(-K*x)+M*A]

a,b=(0.0,10.0)

n=200

x0=-1.0

t=linspace (a,b,n)

#numerical solutions

x_euler=euler(f,x0,t)

#compute true solution values in equal spaced and unequally spaced cases

x=-C*K

#figure

plt.plot (t,x_euler, "b")

xlabel ()

ylabel ()

legend ("Euler")

show()

`

M=2

A=5

#----------------------------------------------------------------------------

def euler (f,x0,t):

n=len (t)

x=np.array ([x0*n])

for i in xrange (n-1):

x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )

return x

#---------------------------------------------------------------------------

if __name__=="__main__":

from pylab import *

def f(x,t):

return (C)*[(-K*x)+M*A]

a,b=(0.0,10.0)

n=200

x0=-1.0

t=linspace (a,b,n)

#numerical solutions

x_euler=euler(f,x0,t)

#compute true solution values in equal spaced and unequally spaced cases

x=-C*K

#figure

plt.plot (t,x_euler, "b")

xlabel ()

ylabel ()

legend ("Euler")

show()

I get following Traceback:

`Traceback (most recent call last):`

File "C:/Python27/testeuler.py", line 50, in <module>

x_euler=euler(f,x0,t)

File "C:/Python27/testeuler.py", line 28, in euler

x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )

IndexError: index 1 is out of bounds for axis 0 with size 1

I don´t understand what is probably wrong.I already looked up after solved questions, but it doesn´t help me along.Can you find my error?

I am using following code as an orientation:

def euler( f, x0, t ):

`n = len( t )`

x = numpy.array( [x0] * n )

for i in xrange( n - 1 ):

x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )

return x

if __name__ == "__main__":

from pylab import *

def f( x, t ):

return x * numpy.sin( t )

a, b = ( 0.0, 10.0 )

x0 = -1.0

n = 51

t = numpy.linspace( a, b, n )

x_euler = euler( f, x0, t )

My goal is to plot the function.

Answer Source

The problem, as the Traceback says, comes from the line `x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )`

. Let's replace it in its context:

- x is an array equal to [x0 * n], so its length is 1
- you're iterating from 0 to n-2 (n doesn't matter here), and i is the index. In the beginning, everything is ok (here there's no beginning apparently... :( ), but as soon as
`i + 1 >= len(x)`

<=>`i >= 0`

, the element`x[i+1]`

doesn't exist. Here, this element doesn't exist since the beginning of the for loop.

To solve this, you must replace `x[i+1] = x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] )`

by `x.append(x[i] + ( t[i+1] - t[i] ) * f( x[i], t[i] ))`

.