# MTH3011 - Partial differential equations - 2019

## 6 points, SCA Band 2, 0.125 EFTSL

Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.

Faculty

Science

Organisational Unit

School of Mathematical Sciences

Chief examiner(s)

Coordinator(s)

Unit guides

Offered

Clayton

• First semester 2019 (On-campus)

Prerequisites

Students must be enrolled in the Master of Financial Mathematics or have passed one unit from MTH2010, MTH2015 or ENG2005 and one unit from MTH2032, or MTH2040

## Synopsis

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.

## Outcomes

On completion of this unit students will be able to:

1. Understand the role of partial differential equations in the mathematical modelling of physical processes;
2. Solve a range of first-order partial differential equations including using the 'method of characteristics';
3. Appreciate the properties of the three basic types of linear second-order partial differential equations, including suitable initial and/or boundary conditions;
4. Understand the mathematical properties of the diffusion equation, wave equation and Laplace's equation and solve them exactly under some simple conditions;
5. Analyse and interpret simple applications modelled by the advection equation, diffusion equation and Laplace's equation;
6. 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;
7. Evaluate numerical solutions of some partial differential equations using computers, and display those results graphically.

## Assessment

End of semester examination (3 hours): 60% (Hurdle)

Continuous assessment: 40% (Hurdle)

Hurdle requirement: To pass this unit a student must achieve at least 50% overall and at least 40% for both the end-of-semester examination and continuous assessment components.