| Semester 2, 2022 Toowoomba On-campus | |
| Units : | 1 |
| Faculty or Section : | Faculty of Health, Engineering and Sciences |
| School or Department : | School of Mathematics, Physics & Computing |
| Grading basis : | Graded |
| Course fee schedule : | /current-students/administration/fees/fee-schedules |
Staffing
Examiner:
Requisites
Pre-requisite: MAT2100 or MAT2500 or ENM2600
Overview
Mathematical modelling is a process of fundamental importance to the practising researcher. Differential equations and an understanding of their qualitative behaviour provide a structure for the analysis of a wide variety of practical problems. This course uses mathematical tools developed so far and introduces dimensional analysis, the phase-plane concept, elements of bifurcation theory and theory of catastrophe, the calculus of variations and other contemporary methods to explore many problems of practical applications.
The course uses mathematical tools introduced in pre-requisite studies to model a variety of realistic phenomena surrounding us in everyday life and introduces calculus of variations for optimisation problems. The course emphasises the importance of the dimensional analysis and demonstrates the close connection between phase-plane concept and qualitative analysis of solutions of ODE. The basics of technical communication in the mathematical sciences are developed throughout the course. The oncampus offering of this course is normally available only in even-numbered years. The external offering of this course is available yearly.
Course learning outcomes
On completion of this course students will be able to:
- solve systems of linear differential equations
- analyse the dynamics of systems of differential equations to determine stability of solutions
- illustrate solutions by sketching phase portraits; deduce qualitative conclusions
- apply mathematical equations, modelling processes and principles to a range of authentic and real-life problems
Topics
| Description | Weighting(%) | |
|---|---|---|
| 1. | Systems of differential equations: solution of linear ODE's, the conversion of higher-order linear ODE's to first-order systems; fixed points and phase portraits for second order ODEs, qualitative solution of nonlinear ODE in the vicinity of critical points. | 15.00 |
| 2. | Potentials, bifurcations, catastrophes. | 15.00 |
| 3. | Dimensions, scaling, dimensional analysis. | 10.00 |
| 4. | Growth and relaxation: exponential growth and decay, autoregulation. | 10.00 |
| 5. | Vibrations in complex systems: free vibrations, mechanical vibrations, nonlinear oscillations, forced vibrations, linear response, resonance, nonlinear response; coupled oscillators. | 25.00 |
| 6. | Dynamic and chaotic oscillations and waves. Simple and strange attractors. Auto-oscillations and auto-waves. | 10.00 |
| 7. | Calculus of variations: challenge problems and functionals; Euler-Lagrange equation, comparison functions, fundamental lemma; special cases; straight lines minimise arc length; geodesics; brachistochrone; the Lagrangian of dynamical systems. | 15.00 |
Text and materials required to be purchased or accessed
(Available on course 精东传媒appDesk.)
(Available on course 精东传媒appDesk.)
Student workload expectations
To do well in this subject, students are expected to commit approximately 10 hours per week including class contact hours, independent study, and all assessment tasks. If you are undertaking additional activities, which may include placements and residential schools, the weekly workload hours may vary.
Assessment details
| Description | Group Assessment |
Weighting (%) | Course learning outcomes |
|---|---|---|---|
| Problem Solving 1 | No | 30 | 1 |
| Problem Solving 2 | No | 30 | 2,3 |
| Report | No | 40 | 4 |