Euler's Method.ipynb

Kernel: Python 3 (Anaconda)

import pylab
from cmath import *

def euler_method(f, a, b, ya, h, acterr, y):
    wx.clear()
    wy.clear()
    n = int((b - a) / h)
    t = a
    w = ya
    if acterr:
        print("\nActual Error:")
        print("|y0 - w0| =", abs(y(a) - w))
        ax.append(t)
        ay.append(y(t).real)
        wx.append(t)
        wy.append(w)
    else:
        print("Approximation:")
        print("h =", h)
        print("n =", n)
        print("w0 =", w)
        wx.append(t)
        wy.append(w)
    for i in range(n):
        w += h * f(t, w)
        t += h
        if acterr:
            print("|y%i - w%i| =" % (int(i+1),int(i+1)) , abs(y(t) - w))
            ax.append(t)
            ay.append(y(t).real)
            wx.append(t)
            wy.append(w)
        else:
            print("w%i =" % int(i+1), w)
            wx.append(t)
            wy.append(w)

if __name__ == '__main__':
    def f(t, w):
        return 4/(2-t)

    def y(t):
        return 4*(-log(t-2)+1j*pi+log(2))

    wx = []
    wy = []
    ax = []
    ay = []

    euler_method(f, 0, 0.95, 0, 0.095, False, '')
    euler_method(f, 0, 0.95, 0, 0.095, True, y)

    pylab.plot(wx,wy, label='numerical')
    pylab.plot(ax,ay,'r-', label='exact')
    pylab.xlabel('t')
    pylab.ylabel('v')
    pylab.legend()
    pylab.show()

Approximation:
h = 0.095
n = 10
w0 = 0
w1 = 0.19
w2 = 0.3894750656167979
w3 = 0.5994198169980134
w4 = 0.820994161021337
w5 = 1.055562062255905
w6 = 1.3047423901247575
w7 = 1.5704766558590233
w8 = 1.8551208506155776
w9 = 2.1615724635188034
w10 = 2.493450192776445

Actual Error:
|y0 - w0| = 0.0
|y1 - w1| = 0.004660687925123808
|y2 - w2| = 0.009806275512045715
|y3 - w3| = 0.015516582770155107
|y4 - w4| = 0.021889964241272986
|y5 - w5| = 0.029049019746376636
|y6 - w6| = 0.03714855502775971
|y7 - w7| = 0.04638689897190651
|y8 - w8| = 0.0570223531564209
|y9 - w9| = 0.06939771069616585
|y10 - w10| = 0.08397787278560687