MTH2040 · Mathematical modelling

Overview

The mathematical modelling of physical systems is based upon differential equations and linear algebra. This unit will introduce fundamental techniques for studying linear systems and differential equations, focusing on applications to physical systems. The topics in linear algebra to be considered include eigenvalues and eigenvectors, diagonalisation of square matrices, matrix functions, LU-decomposition, applications. The topics in optimisation include Lagrange multipliers, the method of least-squares, linear programming, applications. Finally, the topics in differential equations include matrix solutions of constant coefficient systems of ordinary differential equations, conservative systems and phase-planes of simple non-linear ordinary differential equations. Solutions of first order partial differential equations will be analysed and applied to real world problems. You will be introduced to the Mathematica computer package, and learn how to use it for analytical and numerical calculations and graphics. Mathematica will be integrated into most activities.

Areas of study: Astrophysics Applied mathematics Financial and insurance mathematics Mathematics Physics Pure mathematics

Offerings

Assessment

The Handbook lists an examination for this unit.

Assessment details may change. Please refer to the assessment information in Moodle closer to the start of the teaching period.

Requisites

Learning outcomes

1. Apply differential equations, linear algebra, and optimisation methods to model and analyse real-world systems;
2. Solve linear systems and ordinary differential equations, and use eigenvalues, eigenvectors, and matrix functions in applied contexts;
3. Use analytical and numerical techniques, including Mathematica, to investigate and visualise mathematical models;
4. Interpret and explain the behaviour of dynamical systems using tools such as phase planes, bifurcation analysis, and the method of characteristics;
5. Communicate mathematical reasoning and modelling results clearly and effectively in both written and oral forms, following best practice in scientific communication.

Workload

• Three 1-hour seminars; • One 2-hour applied class and • 7 hours of independent study per week.

Student experience, not official rules. Nothing here changes what the Handbook says.