Mathematical Modelling with Case Studies: A Differential Equations Approach using Maple and MATLAB, Second Edition

by Barnes; B.

• ISBN13: 9781420083484
• ISBN10: 1420083481
• Edition: 2nd
• Format: Hardcover
• Copyright: 2008-12-18
• Publisher: Chapman & Hall/

Summary

Focusing on growth and decay processes, interacting populations, and heating/cooling problems, Mathematical Modelling with Case Studies: A Differential Equations Approach using Maple '„¢ and MATLAB ® , Second Editionpresents mathematical techniques applicable to models involving differential equations that describe rates of change. Although the authors concentrate on models involving differential equations, the ideas used can be applied to many other areas.The book carefully details the process of constructing a model, including the conversion of a seemingly complex problem into a much simpler one. It uses flow diagrams and word equations to aid in the model building process and to develop the mathematical equations. Employing theoretical, graphical, and computational tools, the authors analyze the behavior of the models under changing conditions. They discuss the validation of the models and suggest extensions to the models with an emphasis onrecognizing the strengths and limitations of each model.Through applications and the tools of Maple '„¢ and MATLAB ® , this textbook provides hands-on model building skills. It develops, extends, and improves simple models as well as interprets the results.

Table of Contents

Preface	p. v
Acknowledgements	p. viii
Introduction to mathematical modelling	p. 1
Mathematical models	p. 2
An overview of the book	p. 3
Some modelling approaches	p. 5
Modelling for decision-making	p. 7
Compartmental models	p. 9
Introduction	p. 10
Exponential decay and radioactivity	p. 10
Case Study: Detecting art forgeries	p. 17
Case Study: Pacific rats colonise New Zealand	p. 19
Lake pollution models	p. 20
Case Study: Lake Burley Griffin	p. 26
Drug assimilation into the blood	p. 28
Case Study: Dull, dizzy or dead?	p. 33
Cascades of compartments	p. 37
First-order linear DEs	p. 38
Equilibrium points and stability	p. 40
Case Study: Money, money, money makes the world go around	p. 41
Exercises for Chapter 2	p. 44
Models of single populations	p. 51
Exponential growth	p. 52
Density dependent growth	p. 56
Limited growth with harvesting	p. 63
Case Study: Anchovy wipe-out	p. 65
Case Study: How can 2 x 10[superscript 6] birds mean rare?	p. 66
Discrete population growth and chaos	p. 67
Time-delayed regulation	p. 74
Case Study: Australian blowflies	p. 76
Exercises for Chapter 3	p. 77
Numerical solution of differential equations	p. 83
Introduction	p. 84
Basic numerical schemes	p. 84
Computer implementation using Maple and MATLAB	p. 87
Instability	p. 89
Discussion	p. 90
Exercises for Chapter 4	p. 91
Interacting population models	p. 95
Introduction	p. 96
An epidemic model for influenza	p. 97
Predators and prey	p. 106
Case Study: Nile Perch catastrophe	p. 113
Competing species	p. 114
Case Study: Aggressive protection of lerps and nymphs	p. 119
Model of a battle	p. 120
Case Study: Rise and fall of civilisations	p. 125
Exercises for Chapter 5	p. 128
Phase-plane analysis	p. 135
Introduction	p. 136
Phase-plane analysis of epidemic model	p. 139
Analysis of a battle model	p. 143
Analysis of a predator-prey model	p. 148
Analysis of competing species models	p. 152
Closed trajectories for the predator-prey	p. 158
Case Study: Bacteria battle in the gut	p. 160
Exercises for Chapter 6	p. 162
Linearisation analysis	p. 169
Introduction	p. 170
Linear theory	p. 170
Applications of linear theory	p. 179
Nonlinear theory	p. 181
Applications of nonlinear theory	p. 184
Exercises for Chapter 7	p. 187
Some extended population models	p. 191
Introduction	p. 192
Case Study: Competition, predation and diversity	p. 192
Extended predator-prey model	p. 193
Case Study: Lemming mass suicides?	p. 198
Case Study: Prickly-pear meets its moth	p. 200
Case Study: Geese defy mathematical convention	p. 202
Case Study: Possums threaten New Zealand cows	p. 206
Exercises for Chapter 8	p. 212
Formulating basic heat models	p. 217
Introduction	p. 218
Some basic physical laws	p. 219
Model for a hot water heater	p. 222
Heat conduction and Fourier's law	p. 226
Heat conduction through a wall	p. 228
Radial heat conduction	p. 232
Heat fins	p. 234
Exercises for Chapter 9	p. 238
Solving time dependent heat problems	p. 241
The cooling coffee problem revisited	p. 242
The hot water heater problem revisited	p. 244
Case Study: It's hot and stuffy in the attic	p. 248
Spontaneous combustion	p. 250
Case Study: Fish and chips explode	p. 257
Exercises for Chapter 10	p. 258
Solving heat conduction problems	p. 263
Boundary value problems	p. 264
Heat loss through a wall	p. 266
Case Study: Double glazing: What's it worth?	p. 271
Insulating a water pipe	p. 274
Cooling a computer chip	p. 279
Exercises for Chapter 11	p. 284
Introduction to partial differential equations	p. 289
The heat conduction equation	p. 290
Oscillating soil temperatures	p. 292
Case Study: Detecting land mines	p. 296
Lake pollution revisited	p. 299
Exercises for Chapter 12	p. 306
Differential equations	p. 309
Properties of differential equations	p. 309
Solution by inspection	p. 310
First-order separable equations	p. 311
First-order linear equations and integrating factors	p. 312
Homogeneous equations	p. 313
Inhomogeneous equations	p. 314
Further mathematics	p. 317
Linear algebra	p. 317
Partial derivatives and Taylor expansions	p. 320
Review of complex numbers	p. 323
Hyperbolic functions	p. 323
Integration using partial fractions	p. 325
Notes on Maple and MATLAB	p. 327
Brief introduction to Maple	p. 327
Solving differential equations with Maple	p. 327
Brief introduction to MATLAB	p. 329
Solving differential equations with MATLAB	p. 330
Units and scaling	p. 335
Scaling differntial equations	p. 335
SI Units	p. 337
References	p. 339
Index	p. 343
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