Course specification
The current and official versions of the course specifications are available on the web at http://www.usq.edu.au/course/specification/current.
Please consult the web for updates that may occur during the year.

# MAT3105 Harmony of Partial Differential Equations

 Semester 1, 2012 On-campus Toowoomba Units : 1 Faculty or Section : Faculty of Sciences School or Department : Maths and Computing Version produced : 25 May 2013

## Staffing

Examiner: Dmitry Strunin
Moderator: Oleksiy Yevdokimov

## Requisites

Pre-requisite: MAT2100 or MAT2500

## Rationale

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.

## Synopsis

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-Louiville 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.

This course is offered only in even numbered years.

## Objectives

On completion of this course students will be able to:

1. classify partial differential equations;
2. use separation of variables to solve basic partial differential equations;
3. construct special functions needed to understand differential equations;
4. work with infinite series in one or many dimensions;
5. investigate the convergence of a Taylor series;
6. find approximate power series solutions of differential equations;
7. use Fourier analysis to approximate periodic functions and to help solve differential equations;
8. appreciate the properties of the families of special functions engendered from differential equations;
9. use conservation principles to mathematically model one-dimensional dynamics of car traffic, gas and blood flow.

## 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 eigenfucnction 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

ALL textbooks and materials available to be purchased can be sourced from USQ's Online Bookshop (unless otherwise stated). (https://bookshop.usq.edu.au/bookweb/subject.cgi?year=2012&sem=01&subject1=MAT3105)

• Introductory Book 2010, Course MAT3105 Harmony of Differential Equations, USQ Distance Education Centre, Toowoomba.
• Kreyszig, E 2006, Advanced engineering mathematics, 9th edn, Wiley.
• Roberts, AJ 1994, A one-dimensional introduction to continuum mechanics, New Scientific, Singapore.
• Selected Readings 2010, Course MAT3105 Harmony of Differential Equations, USQ Distance Education Centre, Toowoomba.
• Study Book 2010, Course MAT3105 Harmony of Differential Equations, USQ Distance Education Centre, Toowoomba.

## Reference materials

Reference materials are materials that, if accessed by students, may improve their knowledge and understanding of the material in the course and enrich their learning experience.
• Haberman, R 1998, Elementary applied partial differential equations, 3rd edn, Prentice-Hall.
• Highham, NJ 1998, Handbook of writing for the mathematical sciences, 2nd edn, SIAM.
• Department of Mathematics and Computing CDROM SET 1, 2008 (available from the USQ Bookshop). This CD set contains course material, Windows and Linux Software relevant to this course offering only. For more information about the CD sets and their use, please refer to <http://www.sci.usq.edu.au/cdrom> and the course web site.
• Some electronic resources for this course may be available via its home page: http://www.sci.usq.edu.au/courses/mat3105.

Activity Hours
Assessments 30.00
Examinations 2.00
Lectures or Workshops 48.00
Private Study 89.00

## Assessment details

Description Marks out of Wtg (%) Due Date Notes
ASSIGNMENT 1 100 12 02 Apr 2012
ASSIGNMENT 2 100 12 30 Apr 2012
ASSIGNMENT 3 100 12 28 May 2012
2HR OPEN EXAMINATION 64 64 End S1 (see note 1)

NOTES
1. Examination dates will be available during the Semester. Please refer to the Examination timetable when published.

## Important assessment information

1. Attendance requirements:
There are no attendance requirements for this course. However, it is the students' responsibility to study all material provided to them or required to be accessed by them to maximise their chance of meeting the objectives of the course and to be informed of course-related activities and administration.

2. Requirements for students to complete each assessment item satisfactorily:
Not applicable.

3. Penalties for late submission of required work:
If students submit assignments after the due date without (prior) approval of the examiner then a penalty of 5% of the total marks gained by the student for the assignment may apply for each working day late up to ten working days at which time a mark of zero may be recorded.

4. Requirements for student to be awarded a passing grade in the course:
To be assured of receiving a passing grade a student must achieve at least 50% of the total weighted marks available for the course.

5. Method used to combine assessment results to attain final grade:
A final grade will be allocated as follows: raw marks for the assessments will be summed with weightings specified in the Assessment Details; performance demonstrated in the assessment items will be reviewed with reference to the course's objectives and a scaling decided; the scaled marks then determine the final grade.

6. Examination information:
An open examination is one in which candidates may have access to any printed or written material and a calculator during the examination.

7. Examination period when Deferred/Supplementary examinations will be held:
Any Deferred or Supplementary examinations for this course will be held during the next examination period.

8. University Student Policies:
Students should read the USQ policies: Definitions, Assessment and Student Academic Misconduct to avoid actions which might contravene University policies and practices. These policies can be found at http://policy.usq.edu.au/portal/custom/search/category/usq_document_policy_type/Student.1.html.

## Assessment notes

1. The referencing system to be used in this course is supported by the Department. Information on this referencing system and advice on how to use it can be found in the course materials.