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 |