# Lecture plan

This section contains descriptions of the material covered in lectures, all of which are examinable unless explicitly exempted.

**Part I: Laplace transforms and Fourier analysis.**

This part introduces you to the* Laplace transform* (a smart way to transform differential equations into algebraic ones, that might be easier to solve). After Laplace transforms you will learn about *Fourier series*, which express simple functions as sums of sinus- and cosinus signals, and their extension, the *Fourier transforms*. These topics have applications in signal processing, image compression and many other areas in applied mathematics and the mathematical sciences. We will also discuss functions of several variables, and how to solve *partial differential equations (PDEs)*, e.g., the heat equations, by Fourier series.

The material of this part of the lecture is based on *Advanced Engineering Mathematics* by Erwin Kreyszig, 10th edition, John Wiley & Sons, 2011, and the chapter numbers in the table below refer to this textbook. Note that this lecture plan is not set in stone and might be subject to (smaller) modifications.

The .ipynb files can be run by Jupyter Notebook. If Jupyter Notebook is not already installed, we recommend using the anaconda distribution, a detailed installation guide for which can be found here. The css file called at the beginning of the notebooks can be found here: Cascade Stylesheet file for Jupyter notebooks.

Week | Chapter | Content | Lecture Notes | Jupyter notebooks | |
---|---|---|---|---|---|

Markus Grasmair | Helge Holden & Elisabeth Köbis | ||||

34 | 6.1, 6.2 | Laplace transforms, transform of derivatives, ODE | lecture1.pdf lecture1_withnotes.pdf lecture2.pdf lecture2withnotes.pdf (last line on page 10 is corrected in this version) | ||

35 | 6.3 - 6.5 | Heaviside function, delta function, convolution | lecture3.pdf lecture2withnotesupdated24-08-20.pdf lecture3withnotes.pdf lecture3withnotesupdated.pdf lecture4.pdf lecture4withnotes.pdf | ||

36 | 6.6, 6.7, 11.1 | systems of ODEs; Fourier series | lecture5withnotes.pdf lecture6withnotes.pdf | ||

37 | 11.2 - 11.4 | Fourier series: representations and convergence | lecture7withnotes.pdf lecture8withnotes.pdf | ||

38 | 11.7, 11.9 | Fourier integral and transform | lecture9.pdflecture9withnotes.pdffouriertransform.pdf lecture10.pdflecture10withnotes.pdf | ||

39 | 12.1 - 12.4 | Wave equation | lecture_11.pdf lecture12.pdf some | ||

40 | 12.5 - 12.7 | Heat equation | lecture13.pdf lecture14.pdf |

**Part II: Numerical methods.**
The curriculum is covered by the notes found in Jupyter notes, which will be made available below. For now, you can have a look at the notes from 2018. There are also pdf-versions of the notes available. The .ipynb files can be run by Jupyter Notebook. If Jupyter Notebook is not already installed, we recommend using the anaconda distribution, a detailed installation guide for which can be found here.

Alternatively, you can also use the Jupyter hub. There you can upload the .ipynb files and run them on a dedicated server. In order to be able to use the Jupyter hub, you have to be logged on to your Feide account.

Instructions for how to log in can also be found on the course-related blackboard page.

About programming: You are supposed to be able to read and understand simple python code, and to do small modifications on a given code. Possible small syntax errors will have no influence on the grade.

There are video lectures from 2011, which should only be considered as support, and not as replacement of the class lectures. Note that the order in which the different topics were discussed in 2011 differs from the lectures, the curriculum has also been altered, in particular the numerics part.