Course overview
• Topic 1: Nonlinear ordinary differential equations
- Analysing systems of ODEs: existence and uniqueness, equilibria,
- Stability of equilibria, linearisation, Hartman-Grobman theorem.
• Topic 2: Green's functions and Sturm-Liouville theory
- Green’s functions.
- Revision of separation of variables and connection to Sturm-Liouville problems, spaces of functions, inner products.
- Sturm-Liouville eigenvalue problems and their properties, self-adjointness, eigenvalues and eigenfunctions.
• Topic 3: Partial differential equations
- Advection equation, the method of characteristics, upwind numerical schemes.
- Nonlinear first order equations, weak solutions and shocks, Rankine-Hugoniot conditions.
- Classification of higher-order PDEs.
- Parabolic PDEs: use heat equation as an example equation. Boltzmann’s transformation and the error function solution. Finite difference methods for the heat equation, including method of lines, FTCS, BTCS, Crank-Nicolson and their von Neumann stability analysis. Reaction-diffusion equations (e.g. Fisher-KPP), travelling wave solutions, numerical methods.
-Elliptic PDEs: Poisson’s equation.
-Hyperbolic PDEs: Wave equation.
- Nonlinear ordinary differential equations
- Green's functions and Sturm-Liouville Theory
- Partial differential equations
Course learning outcomes
- Analyse systems of ordinary differential equations drawn from physical examples, including nonlinear systems, using techniques such as linearisation and stability analysis
- Use Sturm-Liouville theory to analyse and solve linear boundary value problems, including those arising from separation of variables
- Apply analytical techniques to solve parabolic, elliptic, and hyperbolic partial differential equations and evaluate potential finite difference methods
- Compute numerical solutions to parabolic, elliptic, and hyperbolic partial differential equations using finite-difference numerical methods
- Interpret analytical and numerical solutions for ordinary differential equations and partial differential equations, the main component of continuum mathematical models for physical problems