# Learning material

All Jupyter notebooks can be beautified using the css file caculus4N.css used found here.

## Week 34 - Introduction and Preliminaries

Note that topics from week 34 won't be relevant for the exam.

## Week 35 - Polynomial interpolation

### Key concepts

• Interpolation problem
• Lagrange Interpolation using Lagrange polynomials (Ccardinal functions)
• Newton Interpolation using Newton polynomials
• Estimate for interpolation error
• Runge's example and optimal distribution of interpolation nodes using Chebychev nodes/interpolation

## Week 37 - Numerical methods for nonlinear equations

### Key concepts

• Idea of numerical integration/quadrature rules
• Simple quadrature rules (QR), such as endpoint QR, midpoint QR, trapezoidal QR, Simpson's QR
• Degree of exactness
• Quadrature error estimate for simple QRs
• Design of composite quadrature rules (CQR)
• Error estimates for CQR
• Convergence/approximation order and experimental order of convergence (EOC)
• Gauss quadrature (not relevant for exam)

## Week 38-39 Laplace transform

### Key Concepts

• s-shifting
• t-shifting
• existence of the Laplace transform
• Laplace transform of derivatives and integrals
• solving ODEs with the Laplace transform
• unit step function, Dirac's delta function
• Convolution
• Integral Equations
• Differentiation and Integration of Transforms
• Systems of ODEs

## Week 40-41 Numerical methods for ordinary differential equations

### Key concepts

• Initial value problem for ordinary differential equations (ODE)
• Euler's method and Heun's method, derivation of the methods and implementation
• General one-step methods
• Error theory including consistency/convergence error, convergence order, EOC (again!)
• Explicit Runge-Kutta methods: Motivation, description via Butcher tables, implementation
• Adaptive/embedded Runge-Kutta methods including computable error estimates and adaptive time-step selection

## Week 42 - 43 - Fourier series

### Key Concepts

• Fouries series representation of functions with period 2 \pi and arbitrary period; Euler formulas
• even, odd functions and their Fourier series representation
• half-range expansions
• approximation by trigonometric polynomials; minimum square error
• Fourier integral

## Week 44 Fourier transform

### Key Concepts

• Fourier integral
• Gibbs phenomenon
• Fourier cosine and Fourier sine integral
• complex form of the Fourier integral
• Fourier transform and its properties; convolution#
• Discrete Fourier transform

## Week 45 Heat Equation

### Key Concepts

• Fundamental theorem on superposition
• Solution of the heat equation by Fourier series
• Steady two-dimensional heat equation: Laplace's equation
• Solving the Dirichlet problem in a rectangle
• Heat equation: Modeling very long bars

## Week 46 Wave Equation

### Key concepts

• Formulation of wave equation with boundary and initial conditions
• Solution formula for a wave equation on a bounded interval using separation of variables
• d'Alembert's solution formula for wave equation on real line

## Week 47 Numerical methods for partial differential equations

### Key concepts

• Formulation of general two-point value problems
• Finite difference operators for first and second order derivatives and their approximation/convergence order
• Finite difference methods for the 1 dimensional Poisson problem -u'' = f with Dirichlet boundary conditions including resulting linear system and convergence order
• Finite difference methods for general two-point value problem with Dirichlet boundary conditions
• Finite difference methods for general two-point value problem with Neuman/Robin boundary conditions
• Finite difference methods for the 2-dimensional Poisson problem

## Important!

All learning material below this line stems from the spring edition of Calculus 4N.` For your convenience, we include it here in case want to get an idea of what lies ahead of you. The material will continuously be updated during the course and whenever this happens, this "marking line" will be moved downward accordingly and you will receive an email notification via blackboard.

### 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.