Introduction to 06-623 Mathematical Modeling of Chemical Engineering Processes

numpy is a Python library for arrays. We have to import this library to access the functionality in it. The conventional way to import this library is:

<div class="alert alert-warning"> Remember this cell. You will use it almost every time. </div>

import numpy as np

To see help on the numpy library, run this cell:

?np

Now, we can access functions in the numpy module using "dot notation". For example, let us start by creating an array of linearly spaced points using the linspace function. First, we access the help to see how to use it.

?np.linspace

<div class="alert alert-warning"> np.pi is a constant for the number \(\pi\) </div>

x = np.linspace(0, 2 * np.pi)
x

array([0.        , 0.12822827, 0.25645654, 0.38468481, 0.51291309,
       0.64114136, 0.76936963, 0.8975979 , 1.02582617, 1.15405444,
       1.28228272, 1.41051099, 1.53873926, 1.66696753, 1.7951958 ,
       1.92342407, 2.05165235, 2.17988062, 2.30810889, 2.43633716,
       2.56456543, 2.6927937 , 2.82102197, 2.94925025, 3.07747852,
       3.20570679, 3.33393506, 3.46216333, 3.5903916 , 3.71861988,
       3.84684815, 3.97507642, 4.10330469, 4.23153296, 4.35976123,
       4.48798951, 4.61621778, 4.74444605, 4.87267432, 5.00090259,
       5.12913086, 5.25735913, 5.38558741, 5.51381568, 5.64204395,
       5.77027222, 5.89850049, 6.02672876, 6.15495704, 6.28318531])

Most mathematical operations are element-wise on arrays.

2 * x

array([ 0.        ,  0.25645654,  0.51291309,  0.76936963,  1.02582617,
        1.28228272,  1.53873926,  1.7951958 ,  2.05165235,  2.30810889,
        2.56456543,  2.82102197,  3.07747852,  3.33393506,  3.5903916 ,
        3.84684815,  4.10330469,  4.35976123,  4.61621778,  4.87267432,
        5.12913086,  5.38558741,  5.64204395,  5.89850049,  6.15495704,
        6.41141358,  6.66787012,  6.92432667,  7.18078321,  7.43723975,
        7.69369629,  7.95015284,  8.20660938,  8.46306592,  8.71952247,
        8.97597901,  9.23243555,  9.4888921 ,  9.74534864, 10.00180518,
       10.25826173, 10.51471827, 10.77117481, 11.02763136, 11.2840879 ,
       11.54054444, 11.79700098, 12.05345753, 12.30991407, 12.56637061])

We can define new variables

y1 = np.sin(x)
y2 = np.cos(x)

We can make plots using matplotlib. First we need these two lines. The first one makes the plots appear inline in the notebook. The second imports the plotting library. These should be used in this order.

<div class="alert alert-warning"> Remember this cell. You will use it almost every time you make a plot. </div>

%matplotlib inline
import matplotlib.pyplot as plt

You call functions in the plt library to create plots. These are automatically saved in the notebook.

plt.plot(x, y1, x, y2)
plt.xlabel('x')
plt.ylabel('y')
plt.legend(['y1', 'y2'])
# Always include axis labels and legends when appropriate

<Figure size 432x288 with 1 Axes>

scipy contains numerous libraries for a broad range of scientific computing needs.

Suppose we want to perform the following integral: \(I = \int_0^{4.5} J_{2.5}(x) dx\). The function \(J_{2.5}\) is a special function known as a Bessel function. scipy provides both the integration function, and an implementation of the special function we can use.

from scipy.integrate import quad
from scipy.special import jv

?quad

?jv

To evaluate this integral, we have to define a function for the integrand, and use the quad function to compute the integral. The quad function returns two values, the value of the integral, and an estimate of the maximum error in the integral.

# This is how we define a function. There is a function name, and arguments
# The function returns the output of the jv function.
def integrand(x):
    return jv(2.5, x)

I, err = quad(integrand, 0, 4.5)

I, err

(1.1178179380783253, 7.866317216380692e-09)