# 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.

Week Content Lecture Notes Jupyter Notes
Markus Grasmair Helge Holden & Elisabeth Köbis .ipynb .pdf
Introduction to Jupyter notebooks and python for numerical analysis Introduction I
Introduction II
41 Mathematical preliminaries. Numerical methods for nonlinear equations. lecture_15.pdflecture_16.pdf Preliminaries
Nonlinear equations
Preliminaries
Nonlinear equations
42 Numerical methods for nonlinear equations, polynomial interpolation lecture18.pdflecture18_additional.pdf Polynomial interpolation Polynomial interpolation