Modelling with Differential and Difference Equations

Preface ix

Introduction to the student 1

Part one: Simple Models in Mechanics 5
1 Newtonian mechanics 7
1.1 Mechanics before Newton 7
1.2 Kinematics and dynamics 10
1.3 Newton\'s laws 13
1.4 Gravity near the Earth 16
1.5 Units and dimensions 18

2 Kinematics on a line 21
2.1 Displacement and velocity 22
2.2 Acceleration 28
2.3 Derivatives as slopes 33
2.4 Differential equations and antiderivatives 37

3 Ropes and pulleys 41
3.1 Tension in the rope 41
3.2 Solving pulley problems 44
3.3 Further pulley systems 49
3.4 Symmetry 57

4 Friction 60
4.1 Coefficients of friction 60
4.2 Further applications 64
4.3 Why does the wheel work? 68

5 Differential equations: linearity and SHM 71
5.1 Guessing solutions 71
5.2 How many solutions? 74
5.3 Linearity 77
5.4 The SHM equation 81

6 Springs and oscillations 85
6.1 Force in a spring 85
6.2 A basic example 88
6.3 Further spring problems 94

Part two: Models with Difference Equations 103
7 Difference equations 105
7.1 Introductory example 105
7.2 Difference equations - basic ideas 109
7.3 Constant solutions and fixed points 114
7.4 Iteration and cobweb diagrams 118

8 Linear difference equations in finance and economics 126
8.1 Linearity 127
8.2 Interest and loan repayment 133
8.3 The cobweb model of supply and demand 138
8.4 National income: \'acceleration models\' 142

9 Non-linear difference equations and population growth 146
9.1 Linear models for population growth 146
9.2 Restricted growth - non-linear models 152
9.3 A computer experiment 157
9.4 A coupled model of a measles epidemic 164
9.5 Linearizing non-linear equations 170

10 Models for population genetics 177
10.1 Some background genetics 177
10.2 Random mating with equal survival 185
10.3 Lethal recessives, selection and mutation 193

Part three: Models with Differential Equations 201
11 Continuous growth and decay models 203
11.1 First-order differential equations 203
11.2 Exponential growth 212
11.3 Restricted growth 218
11.4 Exponential decay 227

12 Modelling heat flow 232
12.1 Newton\'s model of heating and cooling 232
12.2 More physics in the model 237
12.3 Conduction and insulation 241
12.4 Insulating a pipe 249

13 Compartment models of mixing 257
13.1 A mixing problem 257
13.2 Modelling pollution in a lake 265
13.3 Modelling heat loss from a hot water tank 270


Part four: Further Mechanics 275
14 Motion in a fluid medium 277
14.1 Some basic fluid mechanics 277
14.2 Archimedes\' Principle 282
14.3 Falling sphere with Stokes\' resistance 286
14.4 Falling sphere with velocity-squared drag 290

15 Damped and forced oscillations 295
15.1 Constant-coefficient differential equations 295
15.2 Damped oscillations 302
15.3 Forced harmonic motion 311

16 Motion in a plane 318
16.1 Kinematics in a plane 318
16.2 Motion down an inclined plane 326
16.3 Projectiles 331

17 Motion on a circle 336
17.1 Kinematics on a circle 336
17.2 Uniform circular motion 343
17.3 The pendulum and linearization 348

Part five: Coupled Models 353
18 Models with linear interactions 355
18.1 Two-compartment mixing 355
18.2 Solving constant-coefficient equations 360
18.3 A model for detecting diabetes 366
18.4 Nutrient exchange in the placenta 373

19 Non-linear coupled models 379
19.1 Predator-prey interactions 379
19.2 Phase-plane analysis 384
19.3 Models of combat 389
19.4 Epidemics 394

References 399

Index 403