Undergraduate - UnitMTH3011 - Partial differential equations

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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 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%
Assignments and tests: 25%
Laboratory work: 5%

Chief examiner(s)

Associate Professor Michael Page

Contact hours

Three 1-hour lectures and one 2-hour laboratory class per week

Prerequisites

MTH2010 or MTH2015, and MTH2032, or equivalent