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

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 1

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] )).

Answer 2

The problem is with your line

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

Here you define x as a single-item array of -200.0. You could do this:

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

or this:

x=np.zeros((n,)) + x0

Note: your imports are quite confused. You import numpy modules three times in the header, and then later import pylab (that already contains all numpy modules). If you want to go easy, with one single

from pylab import *

line in the top you could use all the modules you need.