# Approximation of ODE solution using Taylor series methods

This is my first post on here, so please excuse mistakes if any.

I am trying to plot out the difference between two ODE solvers based on Taylor series:

• 1st order acccurate: $$x(t_0 + h) = x(t_0) + h x'(t_0)$$ and
• 2nd order accurate: $$x(t_0 + h) = x(t_0) + h x'(t_0) + \frac{h^2}{2} x''(t_0)$$

ODE: x'= -x + e^-t x0 = 0; t0 = 0 x'' = x - 2e^-t

For some reason real solution and both 1st and 2nd order approximation are the same. Any ideas?

from math import exp
from numpy import arange
from numpy import zeros
from matplotlib import pyplot as plt

def f(t,x):
dx = -x + exp(-t)
return dx

def df(t,x):
df = x - 2*exp(-t)
return df

def real(h):
t = 0
xk_list = []
tk_list = []
def func(t):
x = t*exp(-t)
return x

while(True):
xk_list.append(func(t))
tk_list.append(t)
t += h
if (t > 3.0):
return tk_list, xk_list

def forward_euler_1(x0, t0, h):
xk = x0
tk = t0
tk_list = []
xk_list = []
counter = 0
while(True):
xk1 = xk + h*f(tk,xk)
xk_list.append(xk)
tk_list.append(tk)
if(tk > 3.0):
print(counter,tk)
return tk_list, xk_list
xk = xk1
tk = tk + h
counter += 1

def forward_euler_2(x0,t0,h):
xk = x0
tk = t0
tk_list = []
xk_list = []
counter = 0

while(True):
xk1 = xk + h*f(tk, xk) + (h**2)/2*df(tk, xk)
xk_list.append(xk1)
tk_list.append(tk)
if(tk > 3.0):
print(counter,tk)
return tk_list, xk_list
xk = xk1
tk = tk + h
counter += 1

x, y = forward_euler_1(0.0, 0.0,0.001)
x2, y2 = forward_euler_2(0.0, 0.0, 0.001)
x3, y3 = real(0.001)

plt.plot(x,y)
plt.plot(x2,y2,'--')
plt.plot(x3, y3)
plt.show()

• I'm an idiot, there is nothing wrong with the code, I am just using an h resoution that is too high, if one put's the time discretization at 0.1 you see what I expected to see. Jan 19 '19 at 13:03
• Hi welcome to scicomp.stackexchange. If your comment answers (and resolve) your question you can post an answer and accept it. Jan 19 '19 at 14:37
• But delete the "I'm an idiot" part first ;-) Welcome to scicomp SE! Jan 22 '19 at 20:18
• done and thank you very much, it has been a most gracious welcome ;D Jan 23 '19 at 14:36