Example 11-2: User-defined functions

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We will provide another demonstration of Euler’s method, but this time, we’ll write a custom function for the first derivative. You will often have to write your own functions when programming and you’ll have many opportunities to practice doing so in this course.

Summary of commands

In this exercise, we will demonstrate the following:

• Writing our own function in Python!

Euler’s method with custom function

Consider the following IVP:

\[ y' = -y + 10 \sin(3t), \quad y(0) = 0, \quad 0 \le t \le 2 \]

The exact solution is \(y = -3 \cos (3t) + \sin (3t) + 3e^{-t}\).

We’ll numerically solve this ODE using a step size of \(h = 0.1\).

Writing your own function

In Python, the syntax for defining your own function is:

def my_func_name(arg1, arg2, ...):
    """ optional docstring to explain purpose, arguments, etc. """
    do something
    return something   # optional

my_func_name(args)     # calling the function

• The def at the start is required, as is the colon at the end of the first line.

• Just like variables, function names should be descriptive.

• Arguments are optional, but you’ll probably have some input variables.

• Docstrings are optional but very nice to have (for others and future you). No need though if the function is very simple.

• Variables defined in the function body (including arguments) have local scope, i.e., they don’t exist outside the function.

• Returning a value is generally optional, but you’ll probably have something.

Once the ODE function is written, we can call it in the loop.

# import libraries
import numpy as np
import matplotlib.pyplot as plt

# user-defined function
def myODE(t, y):
    return -y + 10 * np.sin(3 * t)

# initialize
h = 0.1
t0 = 0
tf = 2
y0 = 0
y = [y0]
t = np.arange(t0, tf+h, h)

# Euler for loop and exact solution
for n in range(len(t) - 1):
    y.append(y[n] + h * myODE(t[n], y[n]))    # this is where the function goes!

y_exact = -3 * np.cos(3 * t) + np.sin(3 * t) + 3 * np.exp(-t)

# plot and compare with exact
fig, ax = plt.subplots()
ax.plot(t, y, 'k--o', label='Euler')
ax.plot(t, y_exact, 'b', label='exact')
ax.set(xlabel='$t$', ylabel='$y(t)$', title=r"Solution of $y'' = -y + 10 \sin(3t)$",
       xlim=[0, tf], ylim=[-3, 5])
ax.legend()
plt.show()