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.pdf
Notebook: 01-preliminaries.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: 03-rounding-errors-nonlinear-equations-web.pdf
Notebooks: 03-numerical-errors.ipynb, 04-numerical-solution-of-nonlinear-equations.ipynb
Nonlinear Equations: nonlinearequations.pdf
Kreyszig, Ch. 19.1
5 Newton's method
Newton's method in several variables
Slides: 04-newton-web.pdf
Notebook: 04-numerical-solution-of-nonlinear-equations.ipynb
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-web.pdf
Notebook: 05-fourier-series.ipynb
The computation for the Fourier coefficients on Slide 26: 05-fourier-heaviside-comp.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
Notebook: Pluto-notebook as a pdf
A Desmos Sketch of the same example: String example
Kreyszig, Ch. 12.1, 12.2, 12.3
9 d'Alembert's solution method
Numerics for the wave equation
Heat equation
Slides: 08-numerics-for-pdes-web.pdf (V3, 13/03)
Notebook I: 08-numerical-methods-pdes-i-wave-equation.ipynb
Notebook II: 08-numerical-methods-pdes-ii-heat-equation.ipynb
Numerical solution to PDEs: numpde.pdf
Kreyszig, Ch. 12.4, 12.6
10 Numerics of the heat equation
Fourier methods for PDEs
Kreyszig, Ch. 12.7
Laplace transform
11 Laplace transform 09-laplace-transform-web.pdf Kreyszig, Ch. 6.1, 6.2
12 Laplace transform Kreyszig, Ch. 6.3, 6.4, 6.5, 6.7
13 Easter break
Numerics of ordinary differential equations
14 Numerics of initial value problems Slides: 10-numerics-for-odes-web.pdf (V2, 16/04)
Notebooks: 10a-numerical-methods-for-odes.ipynb
10b-adaptivity.ipynb
numode.pdf
15 Numerics of initial value problems numode_part2.pdf
Summary and revision
16 Summary and Exam practice 11-summary-web.pdf (V2, 17/04)
2024-04-17, Ronny Bergmann