Third year course, Amsterdam University College, Fall 2016.

This is a course for third year students in the Science major in AUC that I have developed in 2013 and also taught in 2014 and 2016. There are two lectures per week and four written tests.

- Introduction
- Examples
- Speed of light from Maxwell's equations

- ODEs
- First order
- Basic concepts
- First order differential equations
- Geometrical interpretation as slope fields
- Existence and uniqueness for Cauchy problem
- Separable equations
- Exact equations
- Integrating factor
- Linear equations

- Second order
- Equations reducible to first order
- Linear equations
- Existence and uniqueness
- Principle of superposition
- Fundamental sets of solutions and Wronskian
- General solution of homogeneous equation and inhomogeneous equation
- Linear second order equations with constant coefficients

- First order
- First order PDEs
- Basic definitions
- Linear operators
- Homogeneous and inhomogeneous linear equations
- First order linear PDEs with constant and variable coefficients
- Characteristic curves
- Derivation of simple transport equation

- The wave equation
- General solution to the wave equation on the line
- Initial value problem and d'Alembert formula
- Vibrating string
- Causality, dependence and influence domains, characteristic lines.
- Waves in 3D and Huygens principle
- Energy conservation
- Boundary value problems
- Dirichlet and Neumann coundary conditions
- Even and odd functions
- Semi-infinite vibrating string
- Reflection of waves
- Wave equation on the interval
- Separated solutions
- Dirichlet and Neumann boundary conditions on the interval
- Well-posed problems
- Stability of initial value problem

- Fourier series
- Eigenvalue problems
- Sine and cosine Fourier series
- Sine and cosine Fourier series
- Orthogonality and coefficient formulas
- Solution of the wave equation initial value problems with Dirichlet, Neumann boundary conditions.
- Periodic boundary conditions
- Full Fourier series
- Symmetric boundary conditions and orthogonality
- Pointwise, uniform and mean-square convergence
- Convergence theorems

- Heat equation
- Derivation of heat equation
- Weak and strong maximum principle
- Proof of weak maximum principle
- Uniqueness and stability using maximum principle
- Energy method
- Uniqueness and stability using energy method
- Greenâ€™s function
- Poisson formula for heat equation on the line
- Invariance properties
- The heat equation on the half-line
- Separation of variables for the heat equation on the interval
- Smoothing property of the heat equation

- Fourier transform
- Heuristic derivation of Fourier transform inversion formula
- Definition of Fourier and inverse Fourier transforms
- Examples of Fourier transforms
- Fourier transform of the gaussian function
- Properties of the Fourier transform
- Fourier transform of the derivative
- Convolution and its Fourier transform
- Solution formula for the heat equation in terms of Fourier transform
- Inversion theorems
- Fourier transform as a unitary transformation on L^2(R)
- Cauchy-Schwarz and triangle inequalities
- Parseval's equality

- Laplace equation
- Laplace and Poisson equations
- Complex analytic functions, Cauchy-Riemann equations
- Maximum principle, proof of the weak case
- Uniqueness and stability of Dirichlet problem
- Separation of variables on a rectangle
- Laplacian operator in polar coordinates
- Separation of variables on a circle
- Poisson formula
- Mean value property
- Neumann b.v.p.
- Proof of strong maximum principle

- Laplace transform
- Definition of Laplace transform
- Main properties of Laplace transform

- Second order linear PDEs
- Comparison of heat, wave and Laplace equations
- Classification of second order linear PDEs in two variables