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During 2021 we will continue to support students who need to study remotely due to the ongoing impacts of COVID-19 and travel restrictions. Make sure you check the location code when selecting a unit outline or choosing your units of study in Sydney Student. Find out more about what these codes mean. Both remote and on-campus locations have the same learning activities and assessments, however teaching staff may vary. More information about face-to-face teaching and assessment arrangements for each unit will be provided on Canvas.

Unit of study_

MATH4078: PDEs and Applications

The aim of this unit is to introduce some fundamental concepts of the theory of partial differential equations (PDEs) arising in Physics, Chemistry, Biology and Mathematical Finance. The focus is mainly on linear equations but some important examples of nonlinear equations and related phenomena re introduced as well. After an introductory lecture, we proceed with first-order PDEs and the method of characteristics. Here, we also nonlinear transport equations and shock waves are discussed. Then the theory of the elliptic equations is presented with an emphasis on eigenvalue problems and their application to solve parabolic and hyperbolic initial boundary-value problems. The Maximum principle and Harnack's inequality will be discussed and the theory of Green's functions.

Details

Academic unit Mathematics and Statistics Academic Operations
Unit code MATH4078
Unit name PDEs and Applications
Session, year
? 
Semester 2, 2021
Attendance mode Normal day
Location Camperdown/Darlington, Sydney
Credit points 6

Enrolment rules

Prohibitions
? 
MATH3078 or MATH3978
Prerequisites
? 
[A mark of 65 or greater in 6cp from (MATH2X21 or MATH2X65 or MATH2067) and a mark of 65 or greater 6cp from (MATH2X22 or MATH2X61)] or [12cp from (MATH3061 or MATH3066 or MATH3063 or MATH3076 or MATH3961 or MATH3962 or MATH3963 or MATH3968 or MATH3969 or MATH3971 or MATH3974 or MATH3976 or MATH3977 or MATH3979)]
Corequisites
? 
None
Assumed knowledge
? 

(MATH2X61 and MATH2X65) or (MATH2X21 and MATH2X22)

Available to study abroad and exchange students

Yes

Teaching staff and contact details

Coordinator Florica Corina Cirstea, florica.cirstea@sydney.edu.au
Lecturer(s) Florica Corina Cirstea , florica.cirstea@sydney.edu.au
Type Description Weight Due Length
Final exam (Take-home short release) Type D final exam Final exam
Written exam
60% Formal exam period 2 hours
Outcomes assessed: LO1 LO7 LO6 LO5 LO4 LO3 LO2
Assignment Quiz 1
Quiz (short take-home test)
15% Week 05
Due date: 09 Sep 2021

Closing date: 09 Sep 2021
45 minutes
Outcomes assessed: LO2 LO3 LO5
Assignment Assignment
Canvas assignment
10% Week 09
Due date: 17 Oct 2021
2 weeks
Outcomes assessed: LO1 LO7 LO5 LO4 LO3 LO2
Assignment Quiz 2
Quiz (short take-home test)
15% Week 12
Due date: 04 Nov 2021

Closing date: 04 Nov 2021
45 minutes
Outcomes assessed: LO1 LO2 LO3 LO4 LO5 LO6 LO7
Type D final exam = Type D final exam ?
  • Quizzes: Two quizzes will be held online through Canvas. Each quiz is 45 Minutes and has to be submitted by the closing time of 23:59 on the due date. The quiz can be taken any time during the 24 hour period before the closing time. The better mark principle will be used for the quiz so do not submit an application for Special Consideration or Special Arrangements if you miss a quiz. The better mark principle means that the quiz counts if and only if it is better than or equal to your exam mark. If your quiz mark is less than your exam mark, the exam mark will be used for that portion of your assessment instead.
  • Assignment: one assignment to provide written solutions to questions. The assignment must be submitted electronically, as one single typeset or scanned PDF file only, via Canvas by the deadline. Note that your assignment will not be marked if it is illegible or if it is submitted sideways or upside down. It is your responsibility to check that your assignment has been submitted correctly and that it is complete (check that you can view each page). Late submisions will receive a penalty.

  • Final exam: The exam will cover material covered in lectures and tutorials including the theory and proofs, and not just problems to solve.

Detailed information for each assessment can be found on Canvas.

Assessment criteria

The University awards common result grades set out in the Coursework Policy 2014 (Schedule 1).

As a general guide, a high distinction indicates work of an exceptional standard, a distinction a very high standard, a credit a good standard, and a pass an acceptable standard.

Result name

Mark range

Description

High distinction

85 - 100

Representing complete or close to complete mastery of the material;

Distinction

75 - 84

Representing excellence, but substantially less than complete mastery;

Credit

65 - 74

Representing a creditable performance that goes beyond routine knowledge

and understanding, but less than excellence;

Pass

50 - 64

Representing at least routine knowledge and understanding over a spectrum of topics and important ideas and concepts in the course.

Fail

0 - 49

When you don’t meet the learning outcomes of the unit to a satisfactory standard.

For more information see sydney.edu.au/students/guide-to-grades.

Late submission

In accordance with University policy, these penalties apply when written work is submitted after 11:59pm on the due date:

  • Deduction of 5% of the maximum mark for each calendar day after the due date.
  • After ten calendar days late, a mark of zero will be awarded.

Special consideration

If you experience short-term circumstances beyond your control, such as illness, injury or misadventure or if you have essential commitments which impact your preparation or performance in an assessment, you may be eligible for special consideration or special arrangements.

Academic integrity

The Current Student website provides information on academic honesty, academic dishonesty, and the resources available to all students.

The University expects students and staff to act ethically and honestly and will treat all allegations of academic dishonesty or plagiarism seriously.

We use similarity detection software to detect potential instances of plagiarism or other forms of academic dishonesty. If such matches indicate evidence of plagiarism or other forms of dishonesty, your teacher is required to report your work for further investigation.

WK Topic Learning activity Learning outcomes
Week 01 Introduction into PDEs - What is a PDE? The four most important PDEs and classes of PDEs. Lecture (3 hr) LO1 LO2 LO4 LO5
Week 02 The principle of Superposition and classification of 2nd-order linear PDEs. Lecture (3 hr) LO1 LO4 LO5
Introduction into PDEs - What is a PDE? The four most important PDEs and classes of PDEs. Tutorial (1 hr) LO1 LO4 LO5
Week 03 1st-order linear PDEs and the method of characteristics Lecture (3 hr) LO2 LO4 LO5
The principle of Superposition and classification of 2nd-order linear PDEs. Tutorial (1 hr) LO1 LO5
Week 04 Conservation laws, standing waves, traveling waves, waves trains, and general transport equations with uniform velocity vector. Lecture (3 hr) LO2 LO5
1st-order linear PDEs and the method of characteristics Tutorial (1 hr) LO2 LO4 LO5
Week 05 Linear and nonlinear transport equations Lecture (3 hr) LO2 LO4 LO5
Conservation laws, standing waves, traveling waves, waves trains, and general transport equations with uniform velocity vector. Tutorial (1 hr) LO2 LO4 LO5
Week 06 Shock waves Lecture (3 hr) LO2 LO4 LO5
Linear and nonlinear transport equations Tutorial (1 hr) LO2 LO4 LO5
Week 07 Laplace's equation on various symmetric regions in the plane and space, and the fundamental solution. Lecture (3 hr) LO2 LO4 LO5
Shock waves Tutorial (1 hr) LO2 LO4 LO5
Week 08 Harmonic functions, mean-value property, maximum principles, and Harnack's inequality Lecture (3 hr) LO2 LO4 LO5
Laplace's equation on various symmetric regions in the plane and space, and the fundamental solution. Tutorial (1 hr) LO2 LO4 LO5
Week 09 Sturm-Liouville operator on a bounded interval and eigenvalue problems. Lecture (3 hr) LO2 LO5
Harmonic functions, mean-value property, maximum principles, and Harnack's inequality Tutorial (1 hr) LO2 LO5
Week 10 The Schrödinger operator on regions in the plane and space, Eigenvalue problems on the disc and ball - spherical harmonics. Lecture (3 hr) LO2 LO4 LO5
Sturm-Liouville operator on a bounded interval and eigenvalue problems. Tutorial (1 hr) LO2 LO5
Week 11 Application: Solving initial boundary-value problems for parabolic and hyperbolic equations. Lecture (3 hr) LO2 LO4 LO5 LO6
The Schrödinger operator on regions in the plane and space, Eigenvalue problems on the disc and ball - spherical harmonics. Tutorial (1 hr) LO2 LO4 LO5
Week 12 Poisson's equation and Green's function Lecture (3 hr) LO2 LO3 LO4 LO5
Application: Solving initial boundary-value problems for parabolic and hyperbolic equations. Tutorial (1 hr) LO2 LO4 LO5 LO6
Week 13 Green's function on the ball and half-space Lecture (3 hr) LO2 LO3 LO4 LO5
Poisson's equation and Green's function Tutorial (1 hr) LO2 LO3 LO4 LO5

Study commitment

Typically, there is a minimum expectation of 1.5-2 hours of student effort per week per credit point for units of study offered over a full semester. For a 6 credit point unit, this equates to roughly 120-150 hours of student effort in total.

Learning outcomes are what students know, understand and are able to do on completion of a unit of study. They are aligned with the University’s graduate qualities and are assessed as part of the curriculum.

At the completion of this unit, you should be able to:

  • LO1. employ foundational techniques to analyze and solve a range of different types of partial differential equations
  • LO2. explain how classical PDEs are derived and their application in different types of problems
  • LO3. solve classical 2nd-order linear differential equations and apply appropriate boundary and initial conditions
  • LO4. apply the theory of orthogonal polynomials to solve a range of different PDEs
  • LO5. calculate solutions or perform analysis of classical nonlinear PDEs
  • LO6. synthesise solution methods and equations analysis to classify complex solutions of nonlinear PDEs
  • LO7. communicate mathematical analysis accurately, completely and correctly using algebraic, computational, or graphical methods

Graduate qualities

The graduate qualities are the qualities and skills that all University of Sydney graduates must demonstrate on successful completion of an award course. As a future Sydney graduate, the set of qualities have been designed to equip you for the contemporary world.

GQ1 Depth of disciplinary expertise

Deep disciplinary expertise is the ability to integrate and rigorously apply knowledge, understanding and skills of a recognised discipline defined by scholarly activity, as well as familiarity with evolving practice of the discipline.

GQ2 Critical thinking and problem solving

Critical thinking and problem solving are the questioning of ideas, evidence and assumptions in order to propose and evaluate hypotheses or alternative arguments before formulating a conclusion or a solution to an identified problem.

GQ3 Oral and written communication

Effective communication, in both oral and written form, is the clear exchange of meaning in a manner that is appropriate to audience and context.

GQ4 Information and digital literacy

Information and digital literacy is the ability to locate, interpret, evaluate, manage, adapt, integrate, create and convey information using appropriate resources, tools and strategies.

GQ5 Inventiveness

Generating novel ideas and solutions.

GQ6 Cultural competence

Cultural Competence is the ability to actively, ethically, respectfully, and successfully engage across and between cultures. In the Australian context, this includes and celebrates Aboriginal and Torres Strait Islander cultures, knowledge systems, and a mature understanding of contemporary issues.

GQ7 Interdisciplinary effectiveness

Interdisciplinary effectiveness is the integration and synthesis of multiple viewpoints and practices, working effectively across disciplinary boundaries.

GQ8 Integrated professional, ethical, and personal identity

An integrated professional, ethical and personal identity is understanding the interaction between one’s personal and professional selves in an ethical context.

GQ9 Influence

Engaging others in a process, idea or vision.

Outcome map

Learning outcomes Graduate qualities
GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9
This year we swapped the theory of first-order and 2nd-order equations.

Work, health and safety

We are governed by the Work Health and Safety Act 2011, Work Health and Safety Regulation 2011 and Codes of Practice. Penalties for non-compliance have increased. Everyone has a responsibility for health and safety at work. The University’s Work Health and Safety policy explains the responsibilities and expectations of workers and others, and the procedures for managing WHS risks associated with University activities.
 

General laboratory safety rules

  • No eating or drinking is allowed in any laboratory under any circumstances 
  • A laboratory coat and closed-toe shoes are mandatory 
  • Follow safety instructions in your manual and posted in laboratories 
  • In case of fire, follow instructions posted outside the laboratory door 
  • First aid kits, eye wash and fire extinguishers are located in or immediately outside each laboratory 
  • As a precautionary measure, it is recommended that you have a current tetanus immunisation. This can be obtained from University Health Service: unihealth.usyd.edu.au/

Disclaimer

The University reserves the right to amend units of study or no longer offer certain units, including where there are low enrolment numbers.

To help you understand common terms that we use at the University, we offer an online glossary.