1.5. ODEs Exercises

Answer the following exercises based on the content from this chapter. The solutions can be found in the appendices.

Exercise 1.1

Solve the following IVP using the Euler method with a step length of \(h = 0.5\). Write down your solutions correct to 4 decimal places.

\[\begin{align*} y' + y = 1 - e^{-t} \qquad t\in [0,4], \qquad y(0) = 1. \end{align*}\]

Exercise 1.2

The exact solution to the IVP from Exercise 1.1 is \(y = 1 - te^{-t}\). Write a Python or MATLAB program to compute the solution to this IVP and calculate the absolute error for each value. Present your results in the form of a table and a plot of the numerical and exact solutions.

Exercise 1.3

Repeat Exercise 1.2 using values of \(h = 0.4, 0.2, 0.1, 0.05\) to solve the IVP.

(a) Produce a plot of the numerical solutions on the same axes.

(b) Calculate the global truncation error for \(y(1)\) and present your results as a table and a plot of the global truncation error against the step length \(h\). (Hint: you can use the NumPy command idx = np.argmin(abs(t - t0)) or the MATLAB command [~,idx] = min(abs(t - t0)) to determine the index of the value in the array t which is closest to t0).

(c) Comment on your results. What do they tell you about the Euler method?

Exercise 1.4

The motion of a pendulum can be modelled by the following ODE

\[ \begin{align*} \ddot{\theta} + \frac{g}{L} \sin(\theta) = 0, \end{align*} \]

where \(\theta\) is the angle between the pendulum and the vertical, \(L\) is the length of the pendulum and \(g=9.81\text{ms}^{-2}\) is the acceleration due to gravity.

Write a program that solves this IVP using the Euler method with \(h=0.001\) over the interval \(t\in [0, 5]\) for a pendulum of length \(L = 1\) set at an initial angle \(\theta = \frac{\pi}{2}\). Produce a plot of the displacement angle \(\theta\) against \(t\).