Modelling with Ordinary Differential Equations

T.P. Dreyer

• Publisher: CRC Press
• Year: 2017
• ISBN:
  9781351430692
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

Modelling with Ordinary Differential Equations integrates standard material from an elementary course on ordinary differential equations with the skills of mathematical modeling in a number of diverse real-world situations. Each situation highlights a different aspect of the theory or modeling. Carefully selected exercises and projects present excellent opportunities for tutorial sessions and self-study.This text/reference addresses common types of first order ordinary differential equations and the basic theory of linear second order equations with constant coefficients. It also explores the elementary theory of systems of differential equations, Laplace transforms, and numerical solutions. Theorems on the existence and uniqueness of solutions are a central feature. Topics such as curve fitting, time-delay equations, and phase plane diagrams are introduced. The book includes algorithms for computer programs as an integral part of the answer-finding process. Professionals and students in the social and biological sciences, as well as those in physics and mathematics will find this text/reference indispensable for self-study.

1

Introduction

1.1 Mathematical Modelling

Although mathematics had already been applied to real-life problems by the Egyptians and other ancient civilizations, the term “mathematical model” is a fairly recent addition to the mathematical vocabulary. The term signifies an attempt to describe the interplay between the physical world on the one hand and abstract mathematics on the other hand. It is customary to refer to a collection of equations, inequalities, and assumptions as the “model”, but the term “mathematical modelling” means more than that: it is an orderly structured manner in which a real-life problem might be tackled.

The process of mathematical modelling can be broken up into seven different stages, as is shown in the flow chart in Figure 1.1.1.

(1) Identification: The questions to be answered must be clarified. The underlying mechanism at work in the physical situation must be identified as accurately as possible. Formulate the problem in words, and document the relevant data.

(2) Assumptions: The problem must be analysed to decide which factors are important and which factors are to be ignored so that realistic assumptions can be made.

(3) Construction: This is the translation of the problem into mathematical language which normally results in a collection of equations and/or inequalities after the variables had been identified. The “word” problem is transformed into an abstract mathematical problem.

(4) Analysis: The mathematical problem is solved so that the unknown variables are expressed in terms of known quantities, and/or it is analysed to obtain information about parameters.

(5) Interpretation: The solution to the abstract mathematical problem must be compared to the original “word” problem to see if it makes sense in the real-world situation. If not, go back to formulate more realistic assumptions, and construct a new model.

(6) Validation: Check whether the solution agrees with the data of the real-world problem. If the correlation is unsatisfactory, return to the “word” problem for a re-appraisal of the data and the assumptions. Modify or add assumptions and construct a new model.

(7) Implementation: If the solution agrees with the data, then the model can be used to predict what will happen in the future, or conclusions can be drawn to help in future planning, etc. In the case of predictions care should be taken to determine the time interval in which the predictions are valid.

Figure 1.1.1: Flow chart of the modelling process

In a specific problem we may not use all seven stages, or some stages may be trivial. However, even though ...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Preface
• 1 Introduction
• 2 First Order Differential Equations
• 3 Numerical Methods
• 4 Laplace Transforms
• 5 Simultaneous Linear First Order Differential Equations
• 6 Second Order Linear Differential Equations
• 7 Second Order Nonlinear Differential Equations
• 8 Table of Integrals
• 9 Answers
• 10 References
• Index