| Semester 1, 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: ENM2600 or MAT2100 or MAT2500
Overview
This course develops methods needed to apply the mathematics of partial differential equations. An understanding of their qualitative behaviour provides a structure for the analysis of wide-ranging problems. The methods of systematic approximation introduced with Fourier series and power series. The application of conservation principles in mechanics enable the modelling of physical problems as partial differential equations. Nonlinear partial differential equations (PDEs) are important for modelling numerous real-life processes; some basic nonlinear PDEs are introduced.
This course establishes properties of the basic partial differential equations (PDEs) that arise commonly in applications such as the heat equation, the wave equation and Laplace's equation. It also develops the mathematical tools of Fourier transforms and special functions necessary to analyse such PDEs. The theory of infinite series is used to introduce special functions for solutions of ODEs and the general Sturm-Liouville theory. A modelling part introduces the use of partial differential equations to mathematically model the dynamics of cars, gases and blood. The analysis is based upon conservation principles, and also emphasises mathematical and physical interpretation. Nonlinear PDEs are introduced and discussed.
The on-campus 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:
- apply different techniques to a range of real-world problems described by differential equations;
- select and develop appropriate models for a range of problems and their solution methods;
- interpret and communicate the results of analyses in terms of classification of partial differential equations and the properties of the families of special functions;
- develop an awareness of how conservation principles are used in mathematical models of one-dimensional dynamics.
Topics
| Description | Weighting(%) | |
|---|---|---|
| 1. | Fourier Analysis: Fourier series for functions with arbitrary period; half-range expansions; Fourier integrals; approximation by eigenfunction expansions; computer algebra; evaluates integrals. | 16.00 |
| 2. | Classify Partial Differential Equations: PDE's model physical systems; the wave equation; the heat equation; Laplace's equation; classification of PDE's; waves on a membrane. | 16.00 |
| 3. | Series Solutions of Differential Equations: power series, radius and interval of convergence; Power series method leads to Legendre polynomials; Frobenius methods is needed for Bessel functions; orthogonal solutions to second order differential equations; orthogonal eigenfunction expansions. | 20.00 |
| 4. | Methods for PDEs: circular membranes and Bessel functions; Laplacian in polar and spherical coordinates. | 16.00 |
| 5. | Describing the conservation of material: the motion of a continuum, Eulerian description, the material derivative, conservation of material, car traffic & nonlinear characteristics. | 18.00 |
| 6. | Dynamics of momentum: conservation of momentum, sound in ideal gases, dynamics of quasi-one-dimensional blood flow; nonlinear effects. | 14.00 |
Text and materials required to be purchased or accessed
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 | 1,2 |
| Problem Solving 2 | No | 30 | 1,2,3 |
| Problem Solving 3 | No | 40 | 1,2,3,4 |