Partial Differential Equations (MAST90133)

Graduate courseworkPoints: 12.5Not available in 2020

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As part of the University’s response to COVID-19 and the associated Government restrictions and guidelines, most subjects will continue to be delivered online in Winter and Semester 2.
For information about the University’s phased return to campus and in-person activity in Winter and Semester 2, please refer to the on-campus subjects page.

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Overview

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This subject offers a wide ranging introduction to the modern theory of partial differential equations (PDEs) in pure mathematics. Thus we will study questions of existence, uniqueness, regularity, and long time behaviour (e.g.\ energy dispersion) for solutions to PDEs. We will discuss these questions first for the classical equations (Laplace's equation, the heat equation, and the wave equation) which will lead us to the broader theory of elliptic, parabolic, and hyperbolic equations. The course covers mostly linear equations, but exposes the student also to some of the most interesting non-linear equations arising in physics and geometry.

Further topics may include: Calculus of variations, Hamilton-Jacobi equations, Systems of Conservation laws; Non-linear elliptic equations, Schauder theory; Quasi-linear hyperbolic equations, propagation of singularities, blow up phenomena.

Intended learning outcomes

After completing this subject, students will gain an understanding of: 

  • Elements of the general theory of PDE's: Principal symbol, solvability.
  • The basic theory of elliptic equations: Regularity, Dirichlet's problem, maximum principle.
  • The basic theory of hyperbolic equations: Cauchy problem, energy estimates.
  • Existence theory for weak solutions, Sobolev spaces.
  • Examples of non-linear equations

Generic skills

  • Problem-solving skills: the ability to engage with unfamiliar problems and identify relevant solution strategies;
  • Analytical skills: the ability to construct and express logical arguments and to work in abstract or general terms to increase the clarity and efficiency of analysis;
  • Collaborative skills: the ability to work in a team;
  • Time-management skills: the ability to meet regular deadlines while balancing competing commitments.

Last updated: 3 November 2022

Eligibility and requirements

Prerequisites

Code Name Teaching period Credit Points
MAST30021 Complex Analysis
Semester 1 (On Campus - Parkville)
Semester 2 (On Campus - Parkville)
 12.5
 
Code Name Teaching period Credit Points
MAST20030 Differential Equations
Semester 2 (On Campus - Parkville)
 12.5

Corequisites

None

Non-allowed subjects

None

Inherent requirements (core participation requirements)

The University of Melbourne is committed to providing students with reasonable adjustments to assessment and participation under the Disability Standards for Education (2005), and the Assessment and Results Policy (MPF1326). Students are expected to meet the core participation requirements for their course. These can be viewed under Entry and Participation Requirements for the course outlines in the Handbook.

Further details on how to seek academic adjustments can be found on the Student Equity and Disability Support website: http://services.unimelb.edu.au/student-equity/home

Last updated: 3 November 2022

Assessment

Due to the impact of COVID-19, assessment may differ from that published in the Handbook. Students are reminded to check the subject assessment requirements published in the subject outline on the LMS

DescriptionTimingPercentage

Up to 40 pages of assignments (4 assignments worth 10% each spread evenly)

  • 40 pages
Throughout the teaching period40%

A written examination

  • 3 hours
During the examination period60%

Last updated: 3 November 2022

Dates & times

Not available in 2020

Last updated: 3 November 2022

Further information

  • Texts

    Prescribed texts

    Recommended texts and other resources

    L. C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19, AMS (2010)

  • Available through the Community Access Program

    About the Community Access Program (CAP)

    This subject is available through the Community Access Program (also called Single Subject Studies) which allows you to enrol in single subjects offered by the University of Melbourne, without the commitment required to complete a whole degree.

    Please note Single Subject Studies via Community Access Program is not available to student visa holders or applicants

    Entry requirements including prerequisites may apply. Please refer to the CAP applications page for further information.

  • Available to Study Abroad and/or Study Exchange Students

    This subject is available to students studying at the University from eligible overseas institutions on exchange and study abroad. Students are required to satisfy any listed requirements, such as pre- and co-requisites, for enrolment in the subject.

Last updated: 3 November 2022