Outline

Overview

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.

Unit details and rules

Unit code	MATH3978
Academic unit	Mathematics and Statistics Academic Operations
Credit points	6
Prohibitions ?	MATH3078 or MATH4078
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)
Corequisites ?	None
Assumed knowledge ?	[MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22]
Available to study abroad and exchange students	Yes

Teaching staff

Coordinator	Florica-Corina Cirstea, florica.cirstea@sydney.edu.au
Lecturer(s)	Florica-Corina Cirstea, florica.cirstea@sydney.edu.au

Type	Description	Weight	Due	Length
Supervised exam ?	Final exam Written exam, including computational and proof-based questions.	60%	Formal exam period	2 hours
Outcomes assessed:
Small test	Quiz 1 Written calculations and arguments	15%	Week 05 Due date: 31 Aug 2023 at 23:59 Closing date: 31 Aug 2023	45 minutes
Outcomes assessed:
Assignment	Assignment Written assignment	10%	Week 09 Due date: 08 Oct 2023 at 23:59 Closing date: 18 Oct 2023	2 weeks
Outcomes assessed:
Small test	Quiz 2 Written calculations and arguments	15%	Week 12 Due date: 26 Oct 2023 at 23:59 Closing date: 26 Oct 2023	45 minutes
Outcomes assessed:

Assessment summary

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: If a second replacement exam is required, this exam may be delivered via an alternative assessment method, such as a viva voce (oral exam). The alternative assessment will meet the same learning outcomes as the original exam. The format of the alternative assessment will be determined by the unit coordinator.

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.

For more information see 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.

Academic integrity

The Current Student website provides information on academic integrity 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 integrity breaches seriously.

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

You may only use artificial intelligence and writing assistance tools in assessment tasks if you are permitted to by your unit coordinator, and if you do use them, you must also acknowledge this in your work, either in a footnote or an acknowledgement section.

Studiosity is permitted for postgraduate units unless otherwise indicated by the unit coordinator. The use of this service must be acknowledged in your submission.

Learning support

Simple extensions

If you encounter a problem submitting your work on time, you may be able to apply for an extension of five calendar days through a simple extension.  The application process will be different depending on the type of assessment and extensions cannot be granted for some assessment types like exams.

Special consideration

If exceptional circumstances mean you can’t complete an assessment, you need consideration for a longer period of time, or if you have essential commitments which impact your performance in an assessment, you may be eligible for special consideration or special arrangements.

Special consideration applications will not be affected by a simple extension application.

Using AI responsibly

Co-created with students, AI in Education includes lots of helpful examples of how students use generative AI tools to support their learning. It explains how generative AI works, the different tools available and how to use them responsibly and productively.

Weekly schedule

WK	Topic	Learning activity
Week 01	Introduction to PDEs - What is a PDE? Classification of PDEs; Examples of PDEs. Well-posed problems. Harmonic functions.	Lecture (3 hr)
Week 02	PDEs and boundary conditions. Initial conditions for evolution equations. Linear PDEs. The principle of Superposition for linear and homogeneous equations. Nonlinear PDEs: semilinear; quasilinear and fully nonlinear.	Lecture (3 hr)
Week 02	Introduction to PDEs - What is a PDE? Classification of PDEs: stationary or evolution; linear or non-linear; homogeneous or non-homogeneous. Harmonic functions. Properties of the Laplace operator and the fundamental solution of the Laplace equation.	Tutorial (1 hr)
Week 03	1st-order linear PDEs and the method of characteristics.	Lecture (3 hr)
Week 03	Check whether a PDE is linear, semilinear, quasilinear, fully nonlinear and find its order. Solve simple PDEs. Well-posed/Ill-posed problems. The method of characteristics.	Tutorial (1 hr)
Week 04	The method of characteristics for nonlinear PDEs. The wave equation and d'Alembert's Formula. Classification of 2nd order linear PDEs in two variables: hyperbolic, parabolic and elliptic types.	Lecture (3 hr)
Week 04	1st-order linear PDEs and the method of characteristics for solving initial value problems.	Tutorial (1 hr)
Week 05	Linear second-order PDEs with constant coefficients in two variables: canonical form depending on the type (hyperbolic, parabolic, elliptic). Introduction to Fourier series and sufficient conditions for convergence.	Lecture (3 hr)
Week 05	Initial boundary value problems for the heat equation: uniqueness of solutions using the energy method. Green's formulas. The Dirichlet principle for Poisson's equation.	Tutorial (1 hr)
Week 06	Fourier series and the Fourier method of separation of variables. Eigenvalue problems. Self-adjoint operators and positive-definite (positive semi-definite) operators.	Lecture (3 hr)
Week 06	Determine whether a linear second-order PDE in two variables is hyperbolic, parabolic or elliptic. The homogeneous wave equation and d'Alembert's formula.	Tutorial (1 hr)
Week 07	Properties of eigenvalues for self-adjoint and positive-definite (positive semi-definite) operators. The adjoint of a linear operator, properties and examples. Minimisation principle. Sturm-Liouville boundary value problems.	Lecture (3 hr)
Week 07	Determine the type of a second order linear PDE in two variables and bring it to the canonical form using a suitable change of variables. Eigenvalue problems. The Fourier method of separation of variables to solve various PDEs.	Tutorial (1 hr)
Week 08	The regular Sturm-Liouville eigenvalue problem and properties of the eigenvalues. Non-homogeneous equations in the context of Sturm-Liouville theory.	Lecture (3 hr)
Week 08	Sturm-Liouville eigenvalue problems. Solving boundary value problems using generalised Fourier series.	Tutorial (1 hr)
Week 09	The Laplace equation in circular domains. The eigenvalue problem for the Laplacian: The Dirichlet problem in a disk.	Lecture (3 hr)
Week 09	The Helmholtz equation, subject to a homogeneous Dirichlet boundary condition, and the method of separation of variables to determine its eigenvalues and eigenfunctions. Solve the Dirichlet boundary value problems for the Laplace equation in a disk or in a sector. The method of separation of variables to solve non-homogeneous boundary value problems.	Tutorial (1 hr)
Week 10	Fourier-Bessel expansion and application to the vibration of a circular membrane. The eigenvalue problem for the Laplacian in a ball in R^3. The heat equation: scale invariant solutions and the fundamental solution in dimension 1.	Lecture (3 hr)
Week 10	Laplace's equation in a cylinder. The method of separation of variables for elliptic, parabolic and hyperbolic type problems.	Tutorial (1 hr)
Week 11	Heat equation: Fundamental solution and properties; solution to initial-value problem. Elliptic problems: Laplace's equation and Poisson's equation.	Lecture (3 hr)
Week 11	One-dimensional heat equation: self-similar solutions; the fundamental solution. The Cauchy problem for reaction-diffusion equations. Regular Sturm-Liouville eigenvalue problems and non-homogeneous initial boundary value problems.	Tutorial (1 hr)
Week 12	Poisson's equation; Representation formula using Green's function	Lecture (3 hr)
Week 12	Green's function in dimension 1; Sturm-Liouville eigenvalue problems; Solving initial boundary-value problems.	Tutorial (1 hr)
Week 13	Properties of harmonic functions: mean value property, strong maximum principle; Harnack's inequality; Revision	Lecture (3 hr)
Week 13	Green's function and Poisson's formula.	Tutorial (1 hr)

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

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.

Outcomes
Graduate qualities

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 structured PDEs
LO5. calculate solutions or perform analysis of classical nonlinear PDEs
LO6. synthesise solution methods and equations analysis to classify 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
Learning outcomes

Responding to student feedback

This section outlines changes made to this unit following staff and student reviews.

There have been changes regarding the order in which the topics have been introduced. More details and exercises have been added.

Additional information

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.

Current students

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect

Current students

MATH3978: PDEs and Waves (Advanced)

Semester 2, 2023 [Normal day] - Camperdown/Darlington, Sydney

Overview

Overview

Unit details and rules

Teaching staff

Assessment

Assessment

Assessment summary

Assessment criteria

Late submission

Academic integrity

Learning support

Learning support

Simple extensions

Special consideration

Using AI responsibly

Weekly schedule

Weekly schedule

Study commitment

Learning outcomes

Learning outcomes

Graduate qualities

Outcome map

Responding to student feedback

Responding to student feedback

Additional information

Additional information

Disclaimer

On this page

Useful links

Media

Student links

About us

Connect