# Unit information: Delay and stochastic equations in engineering and biology in 2015/16

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Unit name Delay and stochastic equations in engineering and biology EMATM0024 10 M/7 Teaching Block 1 (weeks 1 - 12) Professor. John Hogan Not open or an alternative (including Laplace transforms) none Department of Engineering Mathematics Faculty of Engineering

## Description including Unit Aims

This unit covers the theory and application of two important types of equations that occur widely in engineering and biology. Delay equations occur when the dynamics of a system depends on the state of the system at a past time. For example, almost any control system has delays in it due the time taken for measurement and processing. Delay equations are infinite dimensional systems that require careful solution and interpretation. Stochastic equations occur whenever there is noise in a system. Brownian motion is the most widely known example, and all of modern financial mathematics is based on these equations.

## Intended Learning Outcomes

By the end of this unit, you will be able to use appropriate tools to analyse systems with either delays or stochastic effects. Specifically, you will know how to:

1. Solve simple linear delay equations using Laplace transforms and see if the solution is stable or not.

2. Solve simple stochastic equations and understand concepts such as the first passage time and the time evolution of the probability density function.

3. Numerically solve nonlinear delay or nonlinear stochastic equations using appropriate methods.

Lectures

## Assessment Information

2 hr written exam

## Reading and References

Delay Differential Equations With Applications in Population Dynamics.
Yang Kuang

Semi-Discretization for Time-Delay Systems. Stability and Engineering Applications
Tamás Insperger & Gabor Stépán

Stochastic methods: A Handbook for the Natural and Social Sciences
Crispin Gardiner.