6 points, SCA Band 2, 0.125 EFTSL
Undergraduate - Unit
Refer to the specific census and withdrawal dates for the semester(s) in which this unit is offered.
- First semester 2017 (Day)
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.
On completion of this unit students will 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.
Examination (3 hours): 70% + Assignments and tests: 25% + Laboratory work: 5%
Three 1-hour lectures and one 2-hour laboratory class per week
See also Unit timetable information