Course overview
Differential equation models describe a wide range of complex problems in biology, engineering, physical sciences, economics and finance. This course focuses on ordinary differential equations (ODEs) and develops students' skills in the formulation, solution, understanding and interpretation of coupled ODE models. A range of important biological problems, from areas such as resource management, population dynamics, and public health, drives the study of analytical and numerical techniques for systems of nonlinear ODEs. A key aim of the course is building practical skills that can be applied in a wide range of scientific, business and research settings.
Topics covered are: analytical methods for systems of ODEs, including vector fields, fixed points, phase-plane analysis, linearisation of nonlinear systems, bifurcations; general theory on existence and approximation of ODE solutions; biological modelling; explicit and implicit numerical methods for ODE initial value problems, computational error, consistency, convergence, stability of a numerical method, ill-conditioned and stiff problems.
Course learning outcomes
- Understand how to model time-varying systems using ordinary differential equations
- Be able to identify and analyse stability of equilibrium solutionsbe able to numerically solve ordinary differential equations
- Be able to analyse how the structure of solutions can change depending on a parameter
- Understand the analytical solution theory for linear systems of ordinary differential equations
- Appreciate the necessity of numerical and qualitative methods for analysing solutions for nonlinear systems
- Have a detailed understanding of several ordinary differential equations models arising in physics, engineering, biology and other applications
- Be able to apply the calculus of variations to find optimal solutions to problems
- Appreciate the derivation of many physical laws from variational principles