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Content

# MAT3105 Harmony of Partial Differential Equations

 Semester 1, 2019 Online Short Description: Harmony Part Differential Equa Units : 1 Faculty or Section : Faculty of Health, Engineering and Sciences School or Department : School of Agric, Comp and Environ Sciences Student contribution band : Band 2 ASCED code : 010101 - Mathematics Grading basis : Graded

## Staffing

Examiner: Oleksiy Yevdokimov

## Requisites

Pre-requisite: ENM2600 or 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.

The oncampus offering of this course is normally available only in even numbered years. The external offering of this course is available yearly.

## Objectives

On completion of this course students will be able to:

1. apply different techniques to a range of real-world problems described by differential equations;
2. select and develop appropriate models for a range of problems and their solution methods;
3. interpret and communicate the results of analyses in terms of classification of partial differential equations and the properties of the families of special functions;
4. 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 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://omnia.usq.edu.au/textbooks/?year=2019&sem=01&subject1=MAT3105)

Kreyszig, E 2011, Advanced engineering mathematics, 10th edn, Wiley.
Roberts, A.J 1994, A one-dimensional introduction to continuum mechanics, New Scientific, Singapore.

## 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.
Higham, NJ 1998, Handbook of writing for the mathematical sciences, 2nd edn, SIAM.
Department of Mathematics and Computing DVD-ROM. The DVD contains a complete Debian GNU/Linux distribution ready to install on your computer and can be purchased from the USQ Bookshop. Further information is available at http://www.sci.usq.edu.au/resources/DVD.html.
Some electronic resources for this course may be available via its home page: http://www.sci.usq.edu.au/courses/mat3105.

Activity Hours
Assessments 42.00
Online Lectures 26.00
Online Tutorials 13.00
Private Study 91.00

## Assessment details

Description Marks out of Wtg (%) Due Date Notes
ASSIGNMENT 1 100 12 16 Apr 2019
ASSIGNMENT 2 100 12 13 May 2019
ASSIGNMENT 3 100 12 03 Jun 2019
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:
It is the students' responsibility to participate appropriately in all activities and 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:
To complete each of the assessment items satisfactorily, students must obtain at least 50% of the marks available for each assessment item.

3. Penalties for late submission of required work:
Students should refer to the Assessment Procedure http://policy.usq.edu.au/documents.php?id=14749PL (point 4.2.4)

4. Requirements for student to be awarded a passing grade in the course:
To be assured of receiving a passing grade a student must obtain at least 50% of the total weighted marks available for the course (i.e. the Primary Hurdle), and have satisfied the Secondary Hurdle (Supervised), i.e. the end of semester examination by achieving at least 40% of the weighted marks available for that assessment item.

Supplementary assessment may be offered where a student has undertaken all of the required summative assessment items and has passed the Primary Hurdle but failed to satisfy the Secondary Hurdle (Supervised), or has satisfied the Secondary Hurdle (Supervised) but failed to achieve a passing Final Grade by 5% or less of the total weighted Marks.

To be awarded a passing grade for a supplementary assessment item (if applicable), a student must achieve at least 50% of the available marks for the supplementary assessment item as per the Assessment Procedure http://policy.usq.edu.au/documents/14749PL (point 4.4.2).

5. Method used to combine assessment results to attain final grade:
The final grades for students will be assigned on the basis of the aggregate of the weighted marks obtained for each of the summative items for the course.

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.

## Assessment notes

1. Exam paper presentation: All exam papers should be presented in accurate and clear writing by blue or black pen. Pencil writing is not acceptable. Assignments can be presented using any word processor such as Word or Latex, or can be neatly written by blue or black pen (but not by pencil).

## Other requirements

1. Computer, e-mail and Internet access:
Students are required to have access to a personal computer, e-mail capabilities and Internet access to UConnect. Current details of computer requirements can be found at http://www.usq.edu.au/current-students/support/computing/hardware .

2. Students can expect that questions in assessment items in this course may draw upon knowledge and skills that they can reasonably be expected to have acquired before enrolling in this course. This includes knowledge contained in pre-requisite courses and appropriate communication, information literacy, analytical, critical thinking, problem solving or numeracy skills. Students who do not possess such knowledge and skills should not expect the same grades as those students who do possess them.