Lecture plan and exercise sets

This page contains a description of material covered in lectures, including the exercise sets. All topics of the covered material are examinable unless explicitly exempted.

If you are looking for administrative information regarding exercises, please have a look here.

The course is divided into two parts:

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 the part is based 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.

Week	Chapter	Content	Lecture Notes	Exercises
2	6.1, 6.2	Laplace transforms, transform of derivatives, ODE	lecture_01.pdf, lecture_02_handwritten_notes.pdf	exercise_01.pdf
3	6.3 - 6.5	Heaviside function, delta function, convolution	lecture_03_04_handwritten_notes.pdf	exercise_02.pdf
4	6.6, 6.7, 11.1	More convolution, ODE examples, Fourier series	lecture_05_handwritten_notes.pdf	exercise_03.pdf
5	11.2 - 11.4	Fourier series: representations and convergence	02-fourierrekker.pdf by Marten Nome (in Norwegian), lecture_07_supplemental_material.ipynb, stylesheet for jupyter notebook, lecture_09_hwn.pdf	exercise_04.pdf
6	11.7, 11.9	Fourier integral and transform	03-fouriertransform.pdf by Marten Nome (in Norwegian)	exercise_05.pdf
7	12.5 - 12.7	Heat equation	lecture_11_and_12_hwn.pdf, 05-varmelikningen.pdf by Marten Nome (in Norwegian)	exercise_06.pdf
8	12.1 - 12.4	Wave equation	Morten's and my handwritten notes on the wave equation combined	exercise_07.pdf

Part II: Numerical methods

Most of the mathematical equations and expressions encountered in real life applications can not be solved exactly using pen and paper. Instead, we search for approximate solutions, using numerical algorithms implemented on computers. In this course, we will look at numerical algorithms for solving nonlinear algebraic as well as ordinary and partial differential equations. We will also study how functions can be approximated by polynomials, and how this can be used to find approximations to integrals. Implementation and testing are indispensable elements when studying numerical methods.

The lecture material in this part of the course is given as Jupyter notebooks, which are interactive web based notes containing both mathematical text and executable Python code. The easiest way to obtain a full-fledged Python distribution which also includes the most important scientific computing packages is to install the Anaconda Python Distribution. Detailed instructions for installing Anaconda on Windows, Linux and MacOS can be found in the Anaconda Documentation.

Alternatively, you can also use our Jupyterhub. Instructions for how to log in and to upload own notebooks can be found on the course-related Blackboard page.

Week	Content	Lecture Notes	Jupyter Notes	Python Code	Exercises
9	Introduction to Jupyter notebooks. Numerical methods for nonlinear equations.	Introduction.pdf Preliminaries.pdf, NonLinearEquations.pdf	Cascade Stylesheet file for Jupyter notebooks, Introduction.ipynb, Preliminaries.ipynb, PythonCrashCourse_II.ipynb, NonLinearEquations.ipynb	preliminaries.py, nonlinearequations.py	exercise_08.pdf
10	Numerical methods for nonlinear equations, polynomial interpolation	PolynomialInterpolation.pdf	PolynomialInterpolation.ipynb Combines the (Lagrange/Newton/ErrorTheory)Interpolation notebooks.
11	Polynomial interpolation		LagrangeInterpolation.ipynb, NewtonInterpolation.ipynb, ErrorTheoryInterpolation.ipynb	polynomialinterpolation.py	exercise_09.pdf, exercise_09.ipynb
12	Numerical integration	Quadrature.pdf	SimpleQuadrature.ipynb, CompositeQuadrature.ipynb, GaussQuadrature.ipynb		exercise_10.pdf, exercise_10.ipynb
13	Numerical solution of ordinary differential equations	NuMeODE.pdf	IntroductionNuMeODE.ipynb, ErrorAnalysisNuMeODE.ipynb RungeKuttaNuMeODE.ipynb,		exercise_11.pdf, exercise_11.ipynb
14	Numerical differentiation and numerical solution of partial differential equations
15	Easter vacation
16	Numerical differentiation and numerical solution of partial differential equations	TwoPointBoundaryValueProblem.pdf, PoissonProblem2D.pdf, NuMeStiffODE.pdf, PDE.pdf	TwoPointBoundaryValueProblem.ipynb, PoissonProblem2D.ipynb, NuMeStiffODE.ipynb, PDE.ipynb,
19-20	Revision	repetition_part_1.pdf

Supplemental material

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.

In the 2019 edition of TMA4130, most of the lectures were video recorded video recorded (requires login), and the order and of the covered topics as well as the detail level will be much closer to our course, but of course there will be some variations as well.

Also, I heavily use and reuse Morten Nome's excellent lecture notes from 2019 edition of TMA4215, in particular for Part I of the course. For the second part, the material will be mostly based on Anne Kvœrnø's Jupyter Notebooks.