# MTH3011 - Partial differential equations - 2017

## 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

Coordinator(s)

Unit guides

Offered

Clayton

• First semester 2017 (Day)

## Synopsis

Introduction to PDEs; first-order PDEs and characteristics, the advection equation. Finite-difference methods for ODEs, truncation error. The wave equation: exact solution, reflection of waves. The heat equation: exact solution, fixed and insulating boundary conditions. Forward, backward and Crank-Nicholson numerical methods for the heat equation, truncation errors and stability analysis. Types of second-order PDEs; boundary and/or initial conditions for well-posed problems. Exact solutions of Laplace's equation. Iterative methods for Laplace's equation; convergence. Numerical methods for the advection equation; upwind differencing. Separation of variables for the wave and heat equations.

## 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

Examination (3 hours): 70% + Assignments and tests: 25% + Laboratory work: 5%