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Mathematical Modelling with Case Studies

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Mathematical Modelling with Case Studies: Using Maple (TM) and MATLAB (R), 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 (TM) and MATLAB (R) 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.
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Table of Contents

Introduction to Mathematical Modeling Mathematical models An overview of the book Some modeling approaches Modeling for decision making Compartmental Models Introduction 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 Exponential growth Density-dependent growth Limited growth with harvesting Case study: anchovy wipe-out Case study: how can 2 x 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 Radial heat conduction 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 Appendix F: Answers and Hints References Index Exercises appear at the end of each chapter.

About the Author

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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