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This subject demonstrates how the mathematical modelling process naturally gives rise to certain classes of ordinary and partial differential equations in many contexts, including the infectious diseases, the flow of traffic and the dynamics of particles and of fluids. It advances the student’s knowledge of the modelling process, as well addressing important mathematical ideas in deterministic modelling and the challenges raised by system nonlinearity.
- Infectious disease models and other contexts leading to systems of autonomous first-order differential equations; initial value problem, phase space, critical points, local linearization and stability; qualitative behaviour of plane autonomous systems, structural stability; formulation, interpretation and critique of models.
- Conservation laws and flux functions leading to first-order quasilinear-linear partial differential equations; characteristics, fans, shocks and applications including modelling traffic flow.
- Introduction to continuum mechanics: basic principles; tensor algebra and tensor calculus; the ideal fluid model and potential flow; the Newtonian fluid, Navier-Stokes equations and simple solutions.
Intended learning outcomes
On completion of this subject, students should:
- understand the nature of deterministic mathematical modelling, including model formulation, selection of appropriate mathematical formalism, solution strategies and interpretation of results;
- know contexts in which systems of autonomous ordinary differential equations or quasilinear first-order partial differential equations provide relevant models and appreciate general features of such models and what may be learned from them;
- be able to find and classify critical points in two-dimensional autonomous ODE problems, and be able to infer qualitative behaviour in the phase plane;
- be able to solve quasilinear PDEs in two variables using the method of characteristics, including the construction of weak solutions (fans and shocks);
- understand the fundamental principles of classical continuum mechanics and develop facility in related vector and tensor analysis;
- understand the assumptions underlying the ideal fluid model and the Newtonian fluid model and be able to find and interpret solutions for simple flows.
In addition to learning specific skills that will assist students in their future careers in science, they will have the opportunity to develop generic skills that will assist them in any future career path. These include:
- mathematical modelling skills: the ability to formulate a mathematical model, select an appropriate solution strategy and interpret solutions;
- analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
- time-management skills: the ability to meet regular deadlines while balancing competing commitments.
Last updated: 3 November 2022