3rd Edition

# Mathematical Modelling with Case Studies Using Maple and MATLAB, Third Edition

By B. Barnes, G..R. Fulford Copyright 2015
390 Pages 143 B/W Illustrations
by Chapman & Hall

388 Pages
by Chapman & Hall

Also available as eBook on:

Mathematical Modelling with Case Studies: Using Maple™ and MATLAB®, Third Edition provides students with hands-on modelling skills for a wide variety of problems involving differential equations that describe rates of change. While the book focuses on growth and decay processes, interacting populations, and heating/cooling problems, the mathematical techniques presented can be applied to many other areas.

The text 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. The authors often examine a model numerically before solving it analytically. They also discuss the validation of the models and suggest extensions to the models with an emphasis on recognizing the strengths and limitations of each model.

The highly recommended second edition was praised for its lucid writing style and numerous real-world examples. With updated Maple™ and MATLAB® code as well as new case studies and exercises, this third edition continues to give students a clear, practical understanding of the development and interpretation of mathematical models.

Introduction to Mathematical Modeling
Mathematical models
An overview of the book
Some modeling approaches
Modeling for decision making

Compartmental Models
Introduction
Case study: detecting art forgeries
Case study: Pacific rats colonize New Zealand
Lake pollution models
Case study: Lake Burley Griffin
Drug assimilation into the blood
Case study: dull, dizzy, or dead?
First-order linear DEs
Equilibrium points and stability
Case study: money, money, money makes the world go around

Models of Single Populations
Exponential growth
Density-dependent growth
Limited growth with harvesting
Case study: anchovy wipe-out
Case study: how can 2 × 106 birds mean rare?
Discrete population growth and chaos
Time-delayed regulation
Case study: Australian blowflies

Numerical Solution of Differential Equations
Introduction
Basic numerical schemes
Computer implementation using Maple and MATLAB
Instability
Discussion

Interacting Population Models
Introduction
An epidemic model for influenza
Predators and prey
Case study: Nile Perch catastrophe
Competing species
Case study: aggressive protection of lerps and nymphs
Model of a battle
Case study: rise and fall of civilizations

Phase-Plane Analysis
Introduction
Phase-plane analysis of epidemic model
Analysis of a battle model
Analysis of a predator-prey model
Analysis of competing species models
The predator-prey model revisited
Case study: bacteria battle in the gut

Linearization Analysis
Introduction
Linear theory
Applications of linear theory
Nonlinear theory
Applications of nonlinear theory

Some Extended Population Models
Introduction
Case study: competition, predation, and diversity
Extended predator-prey model
Case study: lemming mass suicides?
Case study: prickly pear meets its moth
Case study: geese defy mathematical convention
Case study: possums threaten New Zealand cows

Formulating Heat and Mass Transport Models
Introduction
Some basic physical laws
Model for a hot water heater
Heat conduction and Fourier’s law
Heat conduction through a wall
Heat fins
Diffusion

Solving Time-Dependent Heat Problems
The cooling coffee problem revisited
The water heater problem revisited
Case study: it’s hot and stuffy in the attic
Spontaneous combustion
Case study: fish and chips explode

Solving Heat Conduction and Diffusion Problems
Boundary condition problems
Heat loss through a wall
Case study: double glazing: what’s it worth?
Insulating a water pipe
Cooling a computer chip
Case Study: Tumor growth

Introduction to Partial Differential Equations
The heat conduction equation
Oscillating soil temperatures
Case study: detecting land mines
Lake pollution revisited

Appendix A: Differential Equations
Appendix B: Further Mathematics
Appendix C: Notes on Maple and MATLAB
Appendix D: Units and Scaling
Appendix E: Parameters

References

Index

Exercises appear at the end of each chapter.

### Biography

B. Barnes is a director in the Australian Government Research Bureau and a visiting fellow at the National Centre for Epidemiology and Population Health at the Australian National University, Canberra. She has published work in a number of applied areas, such as bifurcation theory, population dynamics, carbon sequestration, biological processes, and disease transmission.

G.R. Fulford was recently a research associate and senior lecturer in applicable mathematics at the Queensland University of Technology. He has published several textbooks on mathematical modeling and industrial mathematics as well as other work in areas, such as mucus transport, spermatozoa propulsion, infectious disease modeling, tuberculosis in possums, tear-flow dynamics in the eye, and population genetics.

Praise for the Second Edition:
"The book is written in a very lucid manner, with numerous case studies and examples thoroughly discussed. The material is very well organized, generously illustrated, and delightfully presented. All chapters, except the first one, conclude with scores of nicely designed exercises that can be used for independent study. The book contains enough material to organize a new well-structured one-semester course or to complement the existing one with additional examples and problems and is highly recommended for either purpose"
Zentralblatt MATH, 1168

"The book can be useful for students of mathematical modeling. They will find many skills for modeling and solving real problems. Useful sheets for Maple and MATLAB are included for numerical solution. The most important feature of the book is that it contains many real-life examples. … The main examples are solved in detail and the others are left for the reader. This is the best treasury of real case problems seen in a single book."