MAT 3105 Partial Differential Equations

SubjectCat-NbrClassTermModeDescriptionUnitsCampus
MAT3105212611, 2003EXTPartial Differential Equations1.00TWMBA
Academic Group:FOSCI
Academic Org:FOS003
HECS Band:2
ASCED Code:010101

Contents


STAFFING:

Examiner: Tony Roberts
Moderator: Dmitry Strunin



PRE-REQUISITES:

Pre-requisite: MAT2100


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. Computer algebra is a necessary tool of modern mathematics which is here introduced to perform routine tedious algebra. The application of conservation principles in mechanics enable the modelling of physical problems as partial differential equations.


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. These methods are implemented in computer algebra. 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. This course is offered only in even numbered years.


OBJECTIVES:

On completion of this course students will be able to:

  • use Fourier analysis to approximate periodic functions and to help solve differentail equations;

  • classify partial differential equations;

  • use separation of variables to solve basic partial differential equations;

  • construct speical functions needed to understand differential equations;

  • work with infinite series in one or many dimensions;

  • investigate the convergence of a Taylor series;

  • find approximate power series solutions of differential equations;

  • use computer packages to perform tedious algebraic manipulations;

  • appreciate the properites of the families of special functions engendered from differential equations;

  • use conservation principles to mathematically model one-dimensional dynamics of car traffic, gas and blood flow.



    • TOPICS:


    DescriptionWeighting (%)
    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 eigenfucntion expansions; computer algebra for repetitive tasks.
    		 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.
    		 14.00

    TEXT and MATERIALS required to be PURCHASED or accessed:

    Books can be ordered by fax or telephone. For costs and further details use the 'Book Search' facility at http://bookshop.usq.edu.au by entering the author or title of the text.

    Study package (purchased from the Bookshop).

    access to computer or internet facilities for computer algebra.

    Kreyszig, E. 1999, Advanced Engineering Mathematics, 8th edition, Wiley.

    Roberts, A.J. 1994, A one-dimensional introduction to continuum mechanics, World Sci.




    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.

    Mathematics and Computing CDROM Set, S1, 2003, Dept Maths & Computing, University of Southern Queensland (purchased from the USQ Bookshop).

    (Available: ) .

    (Some electronic resources for this course may be available via its home page: http://www.sci.usq.edu.au/courses/mat3105)

    Haberman, R. 1987, Elementary applied partial differential equations, Prentice-Hall.

    Highham, N.J. 1998, Handbook of writing for the mathematical sciences, 2nd edition, SIAM.




    STUDENT WORKLOAD REQUIREMENTS:

    ACTIVITYHOURS
    Assessment30
    Examinations3
    Private Study135


    ASSESSMENT DETAILS:

    DescriptionMarks Out ofWtg(%)RequiredDue Date
    ASSIGNMENT12.0012.00Y04 Mar 2003(see note )
    ASSIGNMENT 212.0012.00Y04 Mar 2003(see note )
    ASSIGNMENT 312.0012.00Y04 Mar 2003(see note )
    EXAM 3 HOUR RESTRICTED64.0064.00YEND S1
    NOTES:
    .
    Further details about the due dates will be advised by the Examiner.
    .
    Further details about the due dates will be advised by the Examiner.
    .
    Further details about the due dates will be advised by the Examiner.

    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:
      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:
      If students submit assignments after the due date without prior approval then a penalty of 20% of the total marks gained by the student for the assignment will apply for each working day late.
    4. Requirements for student to be awarded a passing grade in the course:
      To be assured of a passing grade, students must demonstrate, via the summative assessment items, that they have achieved the required minimum standards in relation to the objectives of the course by: (i) satisfactorily completing the examination and assignments; and (ii) obtaining at least 50% of the total weighted marks available for all summative assessment items. Note: in our experience, students who do not achieve over 70% in assignments are unlikely to satisfactorily complete the examination.
    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:
      In a Restricted Examination, candidates are allowed access to specific materials during the examination. The only materials that candidates may use in the restricted examination for this course are: writing materials (non-electronic and free from material which could give the student an unfair advantage in the examination); calculators which cannot hold textual information (students must indicate on their examination paper the make and model of any calculator(s) they use during the examination). With the Examiner's approval, candidates may, take an appropriate non- electronic translation dictionary into the examination. This will be subject to perusal and, if it is found to contain annotations or markings that could give the candidate an unfair advantage, it may be removed from the candidate's possession until the appropriate disciplinary action is completed. A list of the materials candidates may access in the restricted examination will be on the frontispiece of the examination paper.
    7. Examination period when Deferred/Supplementary examinations will be held:
      Any Deferred or Supplementary examinations for this course will be held during the examination period at the end of the semester of the next offering of this course.
    8. University Regulations:
      Students should read USQ Regulations 5.1 Definitions, 5.6. Assessment, and 5.10 Academic Misconduct for further information and to avoid actions which might contravene University Regulations. These regulations can be found at the URL http://www.usq.edu.au/SECARIAT/calendar/Part5/ or in the printed version of the current USQ Handbook.