Unit of study_

# MATH3078: PDEs and Waves

## 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 MATH3078 Mathematics and Statistics Academic Operations 6 MATH3978 or MATH4078 6cp from (MATH2X21 or MATH2X65 or MATH2067) and 6cp from (MATH2X22 or MATH2X61) None [MATH2X61 and MATH2X65] or [MATH2X21 and MATH2X22] Yes

### Teaching staff

Coordinator Florica-Corina Cirstea, florica.cirstea@sydney.edu.au Florica-Corina Cirstea

## Assessment

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.

### 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.

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 Learning outcomes
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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)
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.

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

• LO1. demonstrating the ability to recognize different types of partial differential equations: "linear" or "nonlinear", "order of the given equation", "homogeneous" or "inhomogeneous", and if it concerns 2nd-order equations, whether they are of "elliptic", "parabolic", or "hyperbolic" type
• LO2. demonstrating the conceptional understanding of how to apply different methods for solving different types of partial differential equations. Those methods include the use of classical ODE-concepts to solve PDEs
• LO3. understanding the definitions, main theorem, and corollaries of Green's functions and Poisson kernel
• LO4. be fluent with "change of variable" into polar, cylindrical and spherically coordinates and to be able to compute partial derivatives in these coordinates
• LO5. develop an appreciation and strong working knowledge of the theory and application of elementary partial differential equations
• LO6. be fluent in using generalized Fourier transforms to solve parabolic and hyperbolic initial boundary value problems where the spatial variable might be of more than one variable

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

GQ1 GQ2 GQ3 GQ4 GQ5 GQ6 GQ7 GQ8 GQ9

## 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.