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.
Table of Contents
Introduction to Mathematical Modeling
An overview of the book
Some modeling approaches
Modeling for decision making
Exponential decay and radioactivity
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?
Cascades of compartments
First-order linear DEs
Equilibrium points and stability
Case study: money, money, money makes the world go around
Models of Single Populations
Limited growth with harvesting
Case study: anchovy wipe-out
Case study: how can 2 × 106 birds mean rare?
Discrete population growth and chaos
Case study: Australian blowflies
Numerical Solution of Differential Equations
Basic numerical schemes
Computer implementation using Maple and MATLAB
Interacting Population Models
An epidemic model for influenza
Predators and prey
Case study: Nile Perch catastrophe
Case study: aggressive protection of lerps and nymphs
Model of a battle
Case study: rise and fall of civilizations
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
Applications of linear theory
Applications of nonlinear theory
Some Extended Population Models
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
Some basic physical laws
Model for a hot water heater
Heat conduction and Fourier’s law
Heat conduction through a wall
Radial heat conduction
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
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
Appendix F: Answers and Hints
Exercises appear at the end of each chapter.
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."
—EMS Newsletter, September 2009
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