Mathematical Models
Mechanical Vibrations, Population Dynamics, and Traffic Flow
£53.00
Part of Classics in Applied Mathematics
 Author: Richard Haberman
 Date Published: February 1998
 availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
 format: Paperback
 isbn: 9780898714081
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Mathematics is a grand subject in the way it can be applied to various problems in science and engineering. To use mathematics, one needs to understand the physical context. The author uses mathematical techniques along with observations and experiments to give an indepth look at models for mechanical vibrations, population dynamics, and traffic flow. Equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. In the sections on mechanical vibrations and population dynamics, the author emphasizes the nonlinear aspects of ordinary differential equations and develops the concepts of equilibrium solutions and their stability. He introduces phase plane methods for the nonlinear pendulum and for predatorprey and competing species models.
Reviews & endorsements
'Before courses in math modeling became de rigueur, Richard Haberman had already demonstrated that mathematical techniques could be unusually effective in understanding elementary mechanical vibrations, population dynamics, and traffic flow, as well as how such intriguing applications could motivate the further study of nonlinear ordinary and partial differential equations. My students and I can attest that this carefully crafted book is perfect for both selfstudy and classroom use.' Robert E. O'Malley, Jr, University of Washington
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×Product details
 Date Published: February 1998
 format: Paperback
 isbn: 9780898714081
 length: 422 pages
 dimensions: 230 x 155 x 24 mm
 weight: 0.57kg
 availability: This item is not supplied by Cambridge University Press in your region. Please contact Soc for Industrial & Applied Mathematics for availability.
Table of Contents
Foreword
Preface to the classics edition
Preface
Part I. Mechanical Vibrations: Introduction to Mathematical Models in the Physical Sciences
Newton's Law
Newton's Law as Applied to a SpringMass System
Gravity
Oscillation of a SpringMass System
Dimensions and Units
Qualitative and Quantitative Behavior of a SpringMass System
Initial Value Problem
A TwoMass Oscillator
Friction
Oscillations of a Damped System
Underdamped Oscillations
Overdamped and Critically Damped Oscillations
A Pendulum
How Small is Small?
A Dimensionless Time Variable
Nonlinear Frictionless Systems
Linearized Stability Analysis of an Equilibrium Solution
Conservation of Energy
Energy Curves
Phase Plane of a Linear Oscillator
Phase Plane of a Nonlinear Pendulum
Can a Pendulum Stop?
What Happens if a Pendulum is Pushed Too Hard?
Period of a Nonlinear Pendulum
Nonlinear Oscillations with Damping
Equilibrium Positions and Linearized Stability
Nonlinear Pendulum with Damping
Further Readings in Mechanical Vibrations
Part II. Population DynamicsMathematical Ecology. Introduction to Mathematical Models in Biology
Population Models
A Discrete OneSpecies Model
Constant Coefficient FirstOrder Difference Equations
Exponential Growth
Discrete OnceSpecies Models with an Age Distribution
Stochastic Birth Processes
DensityDependent Growth
Phase Plane Solution of the Logistic Equation
Explicit Solution of the Logistic Equation
Growth Models with Time Delays
Linear Constant Coefficient Difference Equations
Destabilizing Influence of Delays
Introduction to TwoSpecies Models
Phase Plane, Equilibrium, and linearization
System of Two Constant Coefficient FirstOrder Differential Equations, Stability of TwoSpecies Equilibrium Populations
Phase Plane of Linear Systems
PredatorPrey Models
Derivation of the LotkaVolterra Equations
Qualitative Solution of the Lotka Volterra Equations
Average Populations of Predators and Preys
Man's Influence on PredatorPrey Ecosystems
Limitations of the LotkaVolterra Equation
Two Competing Species
Further Reading in Mathematical Ecology
Part III. Traffic Flow. Introduction to Traffic Flow
Automobile Velocities and a Velocity Field
Traffic Flow and Traffic Density
Flow Equals Density Times Velocity
Conservation of the Number of Cars
A VelocityDensity Relationship
Experimental Observations
Traffic Flow
SteadyState CarFollowing Models
Partial Differential Equations
Linearization
A Linear Partial Differential Equation
Traffic Density Waves
An Interpretation of Traffic Waves
A Nearly Uniform Traffic Flow Example
Nonuniform Traffic  The Method of Characteristics
After a Traffic Light Turns Green
A Linear VelocityDensity Relationship
An Example
Wave Propagation of Automobile Brake Lights
Congestion Ahead
Discontinuous Traffic
Uniform Traffic Stopped by a Red Light
A Stationary Shock Wave
The Earliest Shock
Validity of Linearization
Effect of a Red Light or an Accident
Exits and Entrances
Constantly Entering Cars
A Highway Entrance
Further reading in traffic flow
Index.
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