In [1]:
# import the necessary modules
import numpy as np
import matplotlib.pyplot as plt
%matplotlib inline
# for pretty plot
import seaborn as sns
sns.set()
Below, we define our parameters
In [2]:
# parameters for our ODE
N_0 = 1
k = 0.03 # min^-1
# parameters for the our integration
dt = 0.1 # min
total_time = 120 # mins
Set up the integration
In [3]:
# determine number of time steps that will be taken
num_steps = int(total_time/dt)
# intilize an array of length num_steps
# into which we will store our values of N
N_t = np.zeros(num_steps)
N_t[0] = N_0
Actually do the integration
In [4]:
# numerically integrate by looping through N_t
for t in range(1, num_steps):
# calculate dN, using previous N_t entry
dN = k * dt * N_t[t-1]
# update current N_t entry
N_t[t] = N_t[t-1] + dN
Plot the results
In [5]:
# get x values for time
times = np.arange(num_steps)*dt # in units of minutes
# plot
plt.plot(times,N_t)
plt.xlabel("time (mins)")
plt.ylabel("N");
Let's compare to a "real" exponential growth curve
In [6]:
# compute the known solution
soln = N_0 * np.exp(k*times)
# plot known solution and our solution
plt.plot(times, N_t)
plt.plot(times, soln)
plt.xlabel("time (mins)")
plt.ylabel("N")
plt.legend(["numerical integration","known solution"])
Out[6]:
