# MTH2040 - Mathematical modelling - 2017

## 6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Faculty

Science

Organisational Unit

School of Mathematical Sciences

Coordinator(s)

Unit guides

Offered

Clayton

• Second semester 2017 (Day)

## Synopsis

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 ordinary 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 ordinary differential equations include: advanced Laplace transforms, Fourier transforms, matrix solutions of constant coefficient systems, conservative systems, phase-planes of simple non-linear ODEs, applications. Students will be introduced to the Mathematica computer package, and learn how to use it for analytic and numerical calculations and graphics. It will be integrated into most activities.

## Outcomes

On completion of this unit students will be able to:

1. Apply differential equations and linear algebra to the modelling of real-world systems.
2. Solve linear systems and calculate the eigenvalues and eigenvectors of square matrices.
3. Calculate the solution of difference and differential equations using matrix functions.
4. Apply optimisation techniques to the solution of real-world problems.
5. Solve constant coefficient ordinary differential equations using Laplace and Fourier transforms and matrices.
6. Understand the phase-planes for second-order differential equations describing oscillating systems and interacting populations.
7. Use the Mathematica software package for the solution and presentation of mathematical problems.
8. Present clear mathematical arguments in both written and oral forms.

## Assessment

Examination (3 hours): 60% (Hurdle)

Continuous assessment: 40%

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for the end-of-semester exam.