# Learning material

All Jupyter notebooks can be beautified using the css file tma4125.css used found here.

#### Week 2 - Introduction and Preliminaries

- Lecture notes Introduction.pdf and corresponding Jupyter notebook Introduction.ipynb
- Regarding vector spaces, inner products, orthogonal projections etc. it might be a good idea to review the corresponding lecture notes 9-projeksjon.pdf from last autumn's Calculus 3 course.

#### Week 2 and 3 - Polynomial interpolation

- Lecture notes LagrangeInterpolation.pdf and corresponding Jupyter notebook LagrangeInterpolation.ipynb
- Lecture notes NewtonInterpolation.pdf and corresponding Jupyter notebook NewtonInterpolation.ipynb
- Lecture notes ErrorTheoryInterpolation.pdf and corresponding Jupyter notebook ErrorTheoryInterpolation.ipynb
- The python code used in the Jupyter notebooks above is can be also found in polynomialinterpolation.py

#### Week 4 - Numerical integration

- Preliminaries: Some numerical integration methods were already considered in Calculus 1, Week 42, please review this material, e.g. by watching video recording of lecture 08 (in Norwegian).
- Lecture notes SimpleQuadrature.pdf and corresponding Jupyter notebook SimpleQuadrature.ipynb
- Lecture notes CompositeQuadrature.pdf and corresponding Jupyter notebook CompositeQuadrature.ipynb
- Lecture notes AdaptiveQuadrature.pdf and corresponding Jupyter notebook AdaptiveQuadrature.ipynb
- Lecture notes GaussQuadrature.pdf and corresponding Jupyter notebook GaussQuadrature.ipynb

#### Week 5 - Numerical methods for nonlinear equations

- Hand-written lecture notes for Fixed-point iteration (pre-written, including extra material) fixed_point_iteration.pdf.
- Copy of the notes from the Fixed-point lecture (written during the lecture) fixed_point_lecture.pdf
- Hand-written lecture notes for Fixed-point iteration (pre-written, including extra material) newton_method.pdf
- Copy of the notes from the Newton lecture (written during the lecture) newton_lecture.pdf
- Notebook used during plenary seminar nonlinearequations.ipynb

#### Week 6 - 8 - Fourier series

- Lecture notes on Fourier series by Marten Nome (in Norwegian) fourierrekker.pdf
- To motivate the importance of understanding periodic functions, we watched a video on atrial fibrillation made by NHI (Norsk Helseinformatikk) in the beginning of lecture 10.
- My handwritten lecture notes on Fourier series containing both my English prepared ahead of the lecture and the Norwegian version written during the lecture. fourierseries_hwn.pdf . Note that this one lacks the long motivational discussion we had in lecture 10 (but I refer you to the video recording). The handwritten notes will be constantly updated while we progress with the topic.
- Jupyter notebook fourier_series_supplemental_material.ipynb provides supplemental material, in particular plots of some (partial sums of) Fourier series. You can also simply pull the notebook into your JupyterHub instance by clicking this link.
- Finally, here is a short video by Steve Brunton explaining/illustrating the Gibb's phenomena very nicely! Extremely nice setup for digital teaching and I can only encourage you to poke around in Steve Brunton's well-known and famous YouTube channel covering a lot of computational engineering/mathematics topics!

#### Week 8 Fourier transform

- My handwritten lecture notes on the Fourier transform containing both my English prepared ahead of the lecture and the Norwegian version written during the lecture fouriertransform_hwn.pdf.

#### Week 9-10 Laplace transform

- My handwritten lecture notes on the Laplace transform laplace_transform_hwn.pdf (10th of March, 21:37).
- The motivation for definition of the Laplace transform was inspired by Steve Brunton's short video lecture on the Laplace transform, so make sure to look the video as it also derives a formula for the inverse Laplace transform!
- Her is a short reminder on (a simplified version of) partial fraction decomposition (delbrøksoppspaltning), including an example.
- Lecture notes on Laplace transform by Marten Nome (in Norwegian) 01-laplacetransform.pdf

#### Week 11-12 Numerical methods for ordinary differential equations

- Introduction to numerical methods for ODEs: Jupyter notebook IntroductionNuMeODE.ipynb, you can also simply pull the notebook into your JupyterHub instance by clicking this link. There is also a pdf version IntroductionNuMeODE.pdf.
- Error analysis of numerical methods for ODEs: Jupyter notebook ErrorAnalysisNuMeODE.ipynb, you can also simply pull the notebook into your JupyterHub instance by clicking this link. There is also a pdf version ErrorAnalysisNuMeODE.pdf.
- Introduction to Runge-Kutta methods:RungeKuttaNuMeODE.ipynb, RungeKuttaNuMeODE.pdf. You can also simply pull the notebook into your JupyterHub instance by clicking this link.
- Adaptive Runge-Kutta methods:AdaptiveODE.ipynb, AdaptiveODE.pdf. You can also simply pull the notebook into your JupyterHub instance clicking this link.
- You will find some background information on the SIR model for disease spreading in in chapter 8.3 of this book , in particular sections 8.3.1 (modelling) and 8.3.7 (impact of vaccination on spreading curve).
- Notebook for the interactive lecture on a simple Covid 19 model and its numerical solution: SimpleCovidModel.ipynb,SimpleCovidModel.pdf or pull it directly into your JupyterHub instance
- Lecture notes on stiff problems stiff_ode_lecture.pdf. Notebook connected with the lecture lecture_stiffode.ipynb. In order to use the adaptive step-size methods you also need to make sure that the Python file ode.py is in the same folder.

#### Week 13-14 Heat Equation

- My handwritten lecture notes on the heat equation: English prepared ahead of the lecture HeatEquation.pdf and the Norwegian version written during the lecture HeatEquationLiveNotes.pdf (12th of April, 19:52)
- Lecture notes on heat equation by Marten Nome (in Norwegian varmelikningen.pdf

#### Week 14 Wave Equation

- My handwritten lecture notes on the wave equation: English prepared ahead of the lecture WaveEquation.pdf and the Norwegian version written during the lecture WaveEquationLiveNotes.pdf (18th of April, 14:00). In the lecture we didn't discuss d'Alembert presentation for the wave equation on the real line, so please study the 2 last pages in the English version or in Marten's notes (see below).
- Lecture notes on wave equation by Marten Nome (in Norwegian bolgelikningen.pdf

#### Week 15 Numerical methods for partial differential equations

- Finite difference methods for two-point boundary problems: Jupyter notebook TwoPointBoundaryProblems.ipynb, you can also simply pull the notebook into your JupyterHub instance by clicking this link. There is also a pdf version TwoPointBoundaryProblems.pdf.
- Numerical methods for the heat equation: Jupyter notebook NumericalMethodsHeatEquation.ipynb, you can also simply pull the notebook into your JupyterHub instance by clicking this link. There is also a pdf version NumericalMethodsHeatEquation.pdf.
- Seminar covering how to write down discretizations and how to numerically solve some other PDE's. IPython Notebook numerical_solution_nonlin_and_wave.ipynb and hand-written notes. numerical_pde_seminar.pdf
- Lecture notes numerical methods for partial differential equations by Morten Nome (in Norwegian) 11-numerikk-for-pde.pdf

#### Week 16 Revision

- Handwritten lecture notes: Revisjonsforelesning.pdf

### Supplemental material

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.