units

MTH3011

Faculty of Science print version

# Monash University Handbook 2011 Undergraduate - Unit

### 6 points, SCA Band 0 (NATIONAL PRIORITY), 0.125 EFTSL

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

 Level Undergraduate Faculty Faculty of Science Offered Clayton First semester 2011 (Day) Coordinator(s) Associate Professor Michael Page

#### 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.

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

#### Contact hours

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

#### Prerequisites

MTH2010 or MTH2015, and MTH2032, or equivalent