MTH3011 - Partial differential equations

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Overview

Introduction to PDEs; first-order PDEs and characteristics, the advection equation, nonlinear equations. Classes of second-order PDEs; boundary and/or initial conditions for well-posed problems. The wave equation: exact solutions on infinite and finite spatial domains, other hyperbolic PDEs, reflection of waves. The heat equation: exact solutions on infinite domain, separation of variables for fixed and/or insulating boundary conditions. Finite-difference methods for ODEs, truncation error. Forward, backward and Crank-Nicolson numerical methods for the heat equation, truncation errors and stability analysis. Numerical methods for the advection equation; upwind differencing. Exact solutions of Laplace's equation in various domains. Numerical methods for Laplace's and Poisson's equation.

Offerings

S1-01-CLAYTON-ON-CAMPUS

Rules

Enrolment Rule

Contacts

Chief Examiner(s)

Dr Kengo Deguchi

Unit Coordinator(s)

Dr Kengo Deguchi

Learning outcomes

On successful completion of this unit, you should be able to:

Understand the role of partial differential equations in the mathematical modelling of physical processes;

Solve a range of first-order partial differential equations including using the 'method of characteristics';

Appreciate the properties of the three basic types of linear second-order partial differential equations, including suitable initial and/or boundary conditions;

Understand the mathematical properties of the diffusion equation, wave equation and Laplace's equation and solve them exactly under some simple conditions;

Analyse and interpret simple applications modelled by the advection equation, diffusion equation and Laplace's equation;

Understand the principles of finite-difference approximation of ordinary and partial differential equations and appreciate the advantages and disadvantages of a range of useful numerical techniques, including their stability;

Evaluate numerical solutions of some partial differential equations using computers, and display those results graphically.

Assessment

1 - Continuous assessment

2 - Examination (3 hours and 10 minutes)

Scheduled and non-scheduled teaching activities

Applied sessions

Lectures

Workload requirements

Workload

Learning resources

Required resources

A First Course in the Numerical Analysis of Differential Equations by A. Iserlies, Cambridge, 1996.

An Introduction to Partial Differential Equations by Y. Pinchover & J. Rubinstein, Cambridge, 2005.

Applied Partial Differential Equations by R. Haberman, Pearson, 2004.

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Partial Differential Equations for Scientists and Engineers by S.J. Farlow, Dover, 1993.

Partial Differential Equations with Numerical Methods by S. Larsson & V. Thomas, Springer, 2005.

Partial Differential Equations: An Introduction by W.A. Strauss, Wiley, second edition 2008.

Partial Differential Equations: Theory and Technique by G.F. Carrier & C.E. Pearson, Academic, 1976.

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Other unit costs

Costs are indicative and subject to change.
Miscellaneous items required (unit course reader, printing, stationery) - $120.

Availability in areas of study

Applied mathematics
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Mathematical statistics
Mathematics
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