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Computer Modeling: From Sports to Spaceflight... From Order to Chaos

422 Pages, Softbound, 8.5 by 11 inches, 229 Illustrations, Includes CD-ROM, with IBM-PC Software $34.95.

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Calculus is the science of change and differential equations are the engines of calculus. A differential equation is like a machine that is capable of movement and ready for action. It only requires some activation: pressing a button, inserting a coin, lighting a fuse — or running a computer. Then it will describe and give details of something that is continually changing. A chemical compound is formed or broken up (gradually, one hopes), a mortgage is repaid, a population increases, a disease spreads, a miss-hit golf ball curves away from its intended direction or an orbit is described in space: in each of these cases the changes will be modeled by differential equations.

Much of the material in this book has been used for over fifteen years in teaching an interdisciplinary one semester course in computer modeling to undergraduates, principally in engineering, physics and mathematics. The book is structure for those who do their own programming, but is also suitable for individuals using "packages" for solving systems of differential equations. The subjects covered are exceptionally diverse and include chaotic systems; population growth and ecology; sickness and health; competition and economics; sports; travel and recreation; space travel and astronomy; pendulums; springs; chemical and other reacting systems.

Accompanying each book is software on CD-ROM which includes over 50 projects from this book — denoted in the text by a CD-ROM symbol. The programs are menu driven with ample help files and make use of graphic and animation techniques to demonstrate the various phenomena being modeled. The programs are written to be run on MS-DOS platforms. The minimum hardware configuration is an IBM compatible 386-level machine with math coprocessor, mouse and VGA color monitor. The programs require 8.5 Mb of disk space and consist of executable files. Pascal code is also included for the user's information.

About the author:

J.M.A. Danby was born in 1929, and studied mathematics at Oxford and Manchester Universities. He has taught in the U.S. since 1957, before which he was for a time, a professional musician, as first chair oboist in the London Philharmonic Orchestra. He has taught in the Astronomy Departments of the University of Minnesota and Yale University. For more that 30 years he has been a member of the Mathematics Department at North Carolina State University where he was appointed Alumni Distinguished Professor in recognition of teaching. The minor planet Danby was named after him in recognition of his contributions to the study of Celestial Mechanics by the International Astronomical Union. He authored Fundamentals of Celestial Mechanics and co-authored Astrophysics Simulations which included software that received an Annual Educational Software award from Computers in Physics in 1996.

Table of Contents
Chapter 1 Getting to Know a Differential Equation
1.1 What a Differential Equation Might Mean
1.2 What is a Differential Equation Telling Us To Do?
1.3 Looking at and Interpreting a Solution
1.4 When is there a solution? Or when is there a solution? Or when is there a solution?
1.5 Assignments
Chapter 2 Some Fundamental Concepts in the Solution of Differential Equations
2.1 Euler's Method
2.2 Truncation Error
2.3 The Order of a Method
2.4 Numerical Confirmation of the Order of Euler's Method
Chapter 3 One Approach to Solving a System of Ordinary Differential Equations
3.1 Generalities
3.2 A Look at the Improved Euler's Method
3.3 Runge-Kutta Formulas
3.4 Stepsize Control—Some Tactics to Avoid
3.5 Fehlberg's RKF4(5) Method
3.6 The Implementation of RKF4(5). One Equation
3.7 The Implementation of RKF4(5). A System of Equations
3.8 Projects for Debugging, Testing and Understanding the Program
3.9 Running the Program
Chapter 4 Introduction to Modeling
4.1 Generalities
4.2 A Model for Discussion: Richardson's "Arms Race"
4.3 Linear Systems, Linear Stability and Nonlinear Stability
Chapter 5 An Introduction to Chaotic Systems
5.1 Introduction
5.2 Chaos in Dynamical Systems
5.3 The Lorenz Equations
5.4 A Periodically Forced Pendulum
5.5 Chaos in Difference Equations
Chapter 6 Population Growth and Ecology
6.1 The Predator-Prey Model of Volterra
6.2 Volterra's Model with Periodic Birth-rate
6.3 The Predator-Prey Model with Fishing
6.4 The Predator-Prey Model with Logistic Growth for the Prey
6.5 The Predator-Prey Model with Logistic Growth for Both Species
6.6 An Alternative Law for Predation
6.7 A Predator-Prey Model Due to R.M.May: Limit Cycles
6.8 The Predator-Prey Model with Internal Competition
6.9 Cooperation Between Two Species
6.10 Competition Between Two Species
6.11 The Predator-Prey Model with Child Care
6.12 Predator-Prey Models with More than Two Species
6.13 Carnivores, Vegetarians and Plants
6.14 Violets, Ants and Rodents
6.15 A Model for the Population Growth of a Parasite
6.16 A Model for Cannibalism
6.17 Population Growth in a Changing Environment
6.18 A Model for Lake Pollution
6.19 A Model in Ecology: The Spruce Budworm versus the Balsam Fir
Chapter 7 Sickness and Health
7.1 A Model for the Spread of Disease
7.2 Possible Effects of Vaccination
7.3 Possible Effects of Migration
7.4 The Spread of Disease in a Population with Birth and Death Included in the Model: 1
7.5 The Spread of Disease in a Population with Birth and Death Included in the Model: 2
7.6 Cross-Infection Between Two Species
7.7 The Spread of Disease with Incubation Included
7.8 Seasonal Changes in Infectiousness
7.9 The Epidemiology of Malaria
7.10 The Spread of Gonorrhea
7.11 A Model for the Initial Spread of the HIV Virus
7.12 A Model for Weight Change
7.13 Zeeman's Model for the Heartbeat
Chapter 8 Competition and Economics
8.1 Lanchester's Combat Models
8.2 Production and Exchange
8.3 The Economics of Fishing: One Species of Fish
8.4 The Economics of Fishing: Two Species of Fish
8.5 Goodwin's Growth Cycle
8.6 A One-sector, Two-capital Model of Economic Growth
Chapter 9 Sports
9.1 The Dynamics of a Spinning Ball
9.2 A Model for a Baseball Pitch
9.3 Pitching a Knuckle Ball
9.4 The Flight of a Fly Ball
9.5 Pitching a Softball
9.6 Driving a Golf Ball
9.7 Serving in Tennis
9.8 Kicking a Ball in Soccer
9.9 Bowling a Cricket Ball
9.10 The Swing of a Cricket Ball
9.11 Shuttlecock Trajectories in Badminton
9.12 Table Tennis
9.13 Shooting in Basketball
9.14 A Badly Kicked Football
9.15 The Path of a Discus
9.16 The Motion of a Javelin
9.17 A Model for the Ski Jump
9.18 Running
9.19 Diving
9.20 The Pole Vault
Chapter 10 Travel and Recreation
10.1 The Dynamics of Flight
10.2 The Motion of a Hovercraft
10.3 Pitching and Rolling at Sea
10.4 The Motion of a Balloon and its Payload
10.5 Jogging with a Companion
10.6 A Bunjy Jump
10.7 A Model for the Motion of a Yo-yo
10.8 Chaos in the Amusement Park
10.9 Playing Ball in a Space Station
10.10 Fireworks
Chapter 11 Space Travel and Astronomy
11.1 The Motion of Three Bodies
11.2 A Trip to the Moon
11.3 The Motion of a Space Station Around L₄ or L₅: 1
11.4 The Motion of a Space Station Around L₄ or L₅: 2
11.5 The Descent of Skylab
11.6 The Range of an ICBM
11.7 The Accuracy of an ICBM
11.8 Aero-Braking the Orbit of a Spacecraft
11.9 The Motion of a Rocket. 1: Introduction
11.10 The Motion of a Rocket. 2: Multi-Stage Rockets
11.11 The Motion of a Rocket. 3: Two-Dimensional Motion
11.12 The Motion of a Rocket. 4: Low-Thrust Orbits
11.13 Reentering the Earth's Atmosphere
11.14 De-spinning a Satellite 1: A Space Yo-yo
11.15 De-spinning a Satellite 2: The Stretch Yo-yo
11.16 If You Were Jupiter, Could You Catch a Comet?
11.17 Using Jupiter to Boost the Orbit of a Spacecraft
11.18 A Grand Tour of the Solar System
11.19 The Motion of the Perihelion of a Planetary Orbit under General Relativity: An Application of the Method of Variation of Parameters
11.20 The Poynting-Robertson Effect: or "How would you like to be a grain of dust in the Solar System?"
11.21 Gravitational Interaction Between Two Galaxies
11.22 If You Were on Venus, Could You See the Back of Your Head?
11.23 Helium Burning in a Hot Star
11.24 An Approximate Model for a Star
11.25 The Limiting Mass of a White Dwarf
Chapter 12 Pendulums
12.1 The Simple Pendulum
12.2 The Period of a Simple Pendulum
12.3 The Pendulum with Linear Damping
12.4 The Pendulum with Dry Friction
12.5 The Pendulum of a Clock
12.6 The Simple Pendulum in a Stiff Wind
12.7 The Simple Pendulum With Added Constant Load
12.8 Pull-out Torques of Synchronous Motors
12.9 A Gravity Pendulum
12.10 A Magnetic Pendulum in Two Dimensions
12.11 A Child on a Swing: 1
12.12 A Child on a Swing: 2
12.13 A Pendulum with Varying Length
12.14 A Swinging Censer
12.15 A Pendulum with Moving Pivot
12.16 A Pendulum Wrapped Around a Peg
12.17 The Motion of a Sliding Pendulum: 1
12.18 The Motion of a Sliding Pendulum: 2
12.19 The Swinging Atwood Machine
12.20 The Motion of a Pendulum Connected to a Mass Moving on a Table
12.21 A Double Pendulum
12.22 The Motion of a Pendulum Attached to a Freely Spinning Wheel
12.23 A Magnetic Pendulum in Three Dimensions
12.24 A Dumbbell Satellite
12.25 The Rotation of the Moon
12.26 The Rotation of Hyperion
12.27 The Rotation of Mercury
12.28 The Rotation of the Moon and Mercury, Including Effects of Tidal Friction
Chapter 13 Springs
13.1 Non-Linear Springs
13.2 Duffing's Equation
13.3 Van der Pol's Equation, and Friends
13.4 Variation of Parameters and Van der Pol's Equation
13.5 A Dynamic Model for the Buckling of a Column
13.6 Zeeman's Catastrophe Machine
13.7 Two Attracting Wires
13.8 The Action Between a Violin Bow and a String
13.9 Landing an Airplane on an Aircraft Carrier
13.10 A Pendulum with Sprung Pivot
13.11 A Spring Pendulum
13.12 A Chaotic Driven Wheel
Chapter 14 Chemical and other Reacting Systems
14.1 The Decomposition of a Molecule
14.2 Enzyme Kinetics
14.3 An Application of Enzyme Kinetics
14.4 More Enzyme Kinetics
14.5 Still More Enzyme Kinetics
14.6 Limit Cycles in Chemical Reactions: The "Brusselator"
14.7 Limit Cycles in Chemical Reactions: The "Oregonator"
14.8 Chemical-Tank-Reactor Stability
14.9 Temperature and Volume Control in a Tank
14.10 The Dynamics of a Reservoir System
Chapter 15 Bits and Pieces
15.1 Fireflies
15.2 Curves of Pursuit
15.3 Low-level Bombing
15.4 A Carbon Microphone Circuit
15.5 The Motion of a Ball in a Rotating Circular Ring: 1
15.6 The Motion of a Ball in a Rotating Circular Ring: 2
15.7 A Compass Needle in an Oscillating Magnetic Field
15.8 The Motion of a Piston and Flywheel
15.9 Watt's Governor
15.10 A Dynamo System with Magnetic Reversal
15.11 A Two-Magnet Toy
15.12 Bernoulli's Problem
Appendix A Lagrange's Equations
A.1 Introduction
A.2 The Simple Pendulum
Appendix B Software
B.1 Introduction
B.2 Chaotic Systems
B.3 Predator-Prey Models
B.4 Sickness and Health
B.5 Sports
B.6 Spaceflight and Astronomy
B.7 Pendulums
B.8 Bits and Pieces
Bibliography
Index