units

MTH3011

Faculty of Science

18 September 2017
23 February 2024

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## 6 points, SCA Band 0 (NATIONAL PRIORITY), 0.125 EFTSLRefer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.
## SynopsisIntroduction 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. ## Objectives
On the completion of this unit students will: - understand the role of partial differential equations in the mathematical modelling of physical processes;
- be able to solve a range of first-order partial differential equations, including using the 'method of characteristics';
- be aware of the properties of the three basic types of linear second-order partial differential equations and recognise which types of initial and/or boundary conditions are appropriate;
- be able to solve the diffusion equation, wave equation and Laplace's equation exactly for some simple types of initial and boundary conditions, and understand the mathematical properties of these equations;
- analyse and interpret some simple applications that are modelled by the advection equation, diffusion equation and Laplace's equation;
- understand the principles of finite-difference approximation of ordinary and partial differential equations;
- appreciate the advantages and disadvantages of a range of useful numerical techniques for determining an approximate solution to each type of partial differential equation, including understanding how to identify when a technique is susceptible to numerical instability;
- have practical experience in determining an approximate numerical solution of partial differential equations using computers, including the graphical display of the results.
## Assessment
Examination (3 hours): 70% ## Chief examiner(s)
Associate Professor Michael Page ## Contact hoursThree 1-hour lectures and one 2-hour laboratory class per week ## Prerequisites |