Mathematical Modelling

1. Introduction

Mathematical modelling is concerned with using mathematics to solve real life problems-. These arc often problems arising from science or engineering but needn't be. Mathematical models have been successfully employed in other areas such as economics social science, military tactics and history. Mathematical modelling involves making a mathematical abstraction of the real life problem, usually resulting in a set of mathematical equations, solving the equations, relating the solution to the real life problem and if necessary modifying the model in the light of this relationship.

To some extent the whole of engineering and science is concerned with obtaining mathematical models of real life phenomena. For example statics, dynamics, fluid flow, electromagnetism and quantum mechanics are based on sophisticated mathematical models. Engineering design for buildings, bridges, engines and electrical circuits are all based to a large extent on mathematical models. -In other areas governments use mathematical models to make demographic and economic predictions, businesses use mathematical models to decide on stock levels, staffing requirements and to control plants and machinery.

Mathematical models fall into two main camps. Deterministic in which the behaviour of a physical system can be predicted with certainty e.-g.- the orbit of an artificial earth satellite, and stochastic in which the behaviour is subject to random fluctuations e.g. the economy of Great Britain. In this course we will be concerned almost entirely with deterministic models. As well as studying mathematical models we will also consider the computer implementation of these models, called simulation using such computer based tools as spreadsheets and computer algebra packages.

One of necessary skills required of the mathematical modeller is that of problem solving. We will begin our course by looking at how human beings solve problems. One way is by thinking.