Lecture Plan for Mathematics 4N

The following plan is preliminary and subject to change.

For the analytical parts of the lecture, we will mostly follow the book Advanced Engineering Mathematics by Erwin Kreyszig, 10th edition, John Wiley & Sons, 2011, and the chapter numbers in the table below refer to this textbook. The numerical parts are based on our own notes; we will publish them during the course of the semester.

For numerics we will heavily use Jupyter Notebooks (.ipynb-files). There is two ways you can work with these.

Jupyterhub

The fastest way to get started with Python and the Jupyter ecosystem is to use our Jupyterhub, which provides a cloud-based Jupyter environment which you can simply use in your browser. Just click on the link, enter your NTNU/Feide credantials and voilà, a Jupyter notebook application will show up in your browser tab.

Anaconda

If you want to work on your computer and not in the browser, we recommend using the Anaconda distribution, a detailed installation guide for which can be found here.

In order to work with jupyter notebooks in Anaconda, follow the following steps:

  • (Download and install Anaconda on your computer.)
  • Download the jupyter notebook on your computer. We recommend to set up a dedicated working directory for Maths 4N/D.
  • Start the Anaconda Navigator, and launch Jupyter Notebook. This should open a new window in your web-browser, from which you can navigate to your stored notebooks or open new ones.

Estimated Weekly lecture plan

For more corresponding Jupyter notebooks see also the Exercises.

Week Topic Additional material Literature
Numerical mathematics
2 Introduction and Numerical mathematics
Example: chemical reactor
Two-point boundary value problems
Numerical differentiation
Slides: 01_preliminaries_bvp_web.pdf
Notebook 01-preliminaries-bvp.ipynb
Preliminaries: preliminaries.pdf
Boundary value problems: bvp.pdf
3 Polynomial interpolation
Spline interpolation
Numerical integration
Simple and composite quadrature rules
Adaptive integration
Slides: 02-interpolation-quadrature-web.pdf
Notebook: 02-interpolation-quadrature.ipynb
Interpolation: interpolation.pdf (Sec. 1, Sec. 2 – only direct approach, Sec. 3)
Numerical Integration: quadrature.pdf

Kreyszig, Ch. 19.3-19.5
4 Python with numpy
Rounding errors
Numerical solution of nonlinear equations
Fixed-point iterations
Slides: 03b-numerical-errors.pdf
Notebook: 03b-numerical-errors.ipynb
Kreyszig, Ch. 19.1
5 Newton's method
Newton's method in several variables
Slides: 04-Newton.pdf
Notebook: 04-numerical-solution-of-nonlinear-equations.ipynb
Numerical Solution of Nonlinear Equations: nonlinearequations.pdf

Kreyszig Ch. 19.2
Fourier analysis
Periodic functions
Fourier series
Euler's formulas
Convergence of Fourier series
Slides: 05-fourier-series.pdf Kreyszig, Ch. 11.1, 11.2, 11.4
additionally: Plonka, Potts, Steidl, Tasche: Numerical Fourier Analysis, Birkhäuser, 2018, Ch. 1.
6 Changes of the period
Half-range expansions
Approximation with trigonometric polynomials
Complex Fourier series
7 Fourier transform
Properties of the Fourier transform
Convolution
Discrete Fourier Transform (DFT)
Slides: 06-fourier-transforms-web.pdf
Notebook: 06-discrete-fourier-transform.ipynb
Kreyszig Ch. 11.9
additionally: Plonka, Potts, Steidl, Tasche: Numerical Fourier Analysis, Birkhäuser, 2018, Ch. 3 and 5
Partial differential equations
8 Partial differential equations (PDEs)
Wave equation
Separations of variables
Slides: 07-partial-differential-equations-web.pdf
A Desmos Sketch: String example
Kreyszig, Ch. 12.1, 12.2, 12.3
9 Numerics for the wave equation
d'Alembert's solution method
Heat equation
Slides: 08a_numerics_wave_eq.pdf
08a_dalembert_heat_eq.pdf
Numerical solution to PDEs: numpde.pdf
Kreyszig, Ch. 12.4, 12.6
10 Numerics of the heat equation
Fourier methods for PDEs
Slides: 08b-numerics-for-pdes-ii-web.pdf
Notebook: 08b-heat_equation_numerics.ipynb
Kreyszig, Ch. 12.7
Laplace transform
11 Laplace transform Slides: 09a-laplace-transform-i-web.pdf
09a-laplace-transform-ii-web.pdf
Kreyszig, Ch. 6.1, 6.2
12 Laplace transform Slides: 09b-laplace-transform-iii-web.pdf
09b-laplace-transform-iv-web.pdf
Kreyszig, Ch. 6.3, 6.4, 6.5, 6.7
Numerics of ordinary differential equations
13 Numerics of initial value problems Slides: 10-numerics-for-odes-web.pdf (V2, 31/03)
Notebook: 10-numerical-methods-for-odes.ipynb
Numerical solution of ordinary differential equations: numode.pdf
and an accompanying notebook with examples: numode.ipynb
14 Easter break
15 Numerics of initial value problems Slides: 10b-numerics-for-odes-stepsize-and-stability-web.pdf (V2, 17/04)
Notebook: 10b-adaptivity.ipynb (V2, 14/04)
Numerical solution of ordinary differential equations (Part 2): numode_part2.pdf
Summary and revision
16 Revision Slides: 11-summary-web.pdf
2024-01-16, Ronny Bergmann