Units

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Sample subject outline - Semester 1 2012

Note: Subject outlines often change before the semester begins. Below is a sample outline.

Rationale

Many of the problems facing applied mathematicians involve such difficulties as nonlinear governing equations, variable coefficients and nonlinear boundary conditions at complex known or unknown boundaries that preclude solving them exactly (Nayfeh, 1981). As a consequence, solutions are approximated using numerical techniques, analytical techniques and combinations of both. Foremost among the analytical techniques are the systematic perturbation methods. These methods constitute a valuable tool for the applicable mathematician. In this unit you will be introduced several of the perturbation methods and be shown how they have been utilised to solve problems of significance in industrial, medical and biological applications.

Aims

This unit which is available to students majoring in Mathematics in the SC60 Bachelor of Applied Science Honours course and the postgraduate MA65, MA75, MA85, SC71, SC80 and IF49 courses aims to provide a sound understanding and appreciation of the advanced theories, concepts and techniques of Perturbation Theory. You will gain expertise in problem formulation, problem solving and critical thinking and written communication.

Objectives

On successful completion of this unit you will:

1. Have developed problem-solving skills in the types of extended problems that arise in the highly quantitative area of Perturbation Methods;

2. Have consolidated your knowledge and skills in handling the manipulative aspects, both algebraic and conceptual, which are adopted in advanced applications;

3. Have developed an understanding of the role of analytic techniques in 'real world' applications;

4. Have been introduced to some representative advanced applied mathematics methodology of importance in scientific, financial, engineering and industrial settings;

5. Have further developed your written communication skills so as to be able to articulate and communicate ideas using high-level mathematical notation and language;

6. Be able to formulate problems in mathematical terms, and perform the subsequent analysis, starting experimental and/or observational information.

Content

Perturbation Expansions:
Algebraic equations; Iteration and expansion, singular perturbations and rescaling, non-integral powers. Initial value problems.

Asymptotic Approximations:
Order symbols, asymptotic sequences and expansions, uniform and non-uniform expansions, regions of non-uniformity, diverging series, optimal truncation.


Multiple Scales:
Second order systems, limitation of renormalisation, the method of multiple-scales. Industrial and biological applications.

Matched Asymptotic Expansions:
The boundary layer problem, outer approximations, inner approximations, matching, composite approximations, boundary layer location, boundary layer thickness, linear and nonlinear examples. Industrial and biological applications.

Asymptotic approximation to integrals:
Integration by parts, Laplace's method, method of stationary phase, method of steepest descents, saddle point method.

Note: It is possible that not all of the above material will be covered in a given semester.

Approaches to Teaching and Learning

You are required to attend a weekly 3-hour lecture/tutorial session where both the exposition of course content and demonstration of suitable skills will occur. You will have opportunities to apply your knowledge in problem-solving exercises with guidance from the lecturer (either in class or in informal consultation outside of class times).

Assessment

The assessment items in this unit are designed to determine your level of competency, measured against criteria, in meeting the unit outcomes while providing you with a range of tasks with varying levels of skill development and difficulty.Formative feedback will be provided for the in-semester assessment items by way of written comments on the assessment items, student perusal of the marked assessment piece and informal interview as required.

Summative feedback will be provided throughout the semester with progressive posting of results via Blackboard.

Assessment name: Problem Solving Task
Description: This assessment will consist of a number of individual assignments that will generally consist of traditional problem-solution based exercises. Formative and summative.
Relates to objectives: 1, 2, 3, 4, 5 and 6.
Weight: 40%
Internal or external: Internal
Group or individual: Individual
Due date: Progressive

Assessment name: Examination (Theory)
Description: A 3 hour examination that covers the core material of the unit with a range of clearly defined and not open-ended questions.
Relates to objectives: 1, 2, 4 and 5.
Weight: 60%
Internal or external: Internal
Group or individual: Individual
Due date: End Semester

Academic Honesty

Academic honesty means that you are expected to exhibit honesty and act responsibly when undertaking assessment. Any action or practice on your part which would defeat the purposes of assessment is regarded as academic dishonesty. The penalties for academic dishonesty are provided in the Student Rules. For more information you should consult the QUT Library resources for avoiding plagiarism.

Resource materials

Texts:

There is no nominated text for this unit.

References:

Appropriately detailed, fully referenced reading material will be issued via Blackboard.

Risk assessment statement

While you are working on this project you should be mindful of the Health and Safety requirements of the university, your workplace and other relevant locations as appropriate. Further information on health and safety at QUT can be found at the site
http: //www.hrd.qut.edu.au/healthsafety/healthsafe/index.jsp

Disclaimer - Offer of some units is subject to viability, and information in these Unit Outlines is subject to change prior to commencement of semester.

Last modified: 30-Sep-2011