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Mathematical Models

Mathematical Models
Mechanical Vibrations, Population Dynamics, and Traffic Flow


Part of Classics in Applied Mathematics

  • 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

£ 53.00

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About the Authors
  • 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 in-depth 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 predator-prey 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 self-study 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

    Preface to the classics edition
    Part I. Mechanical Vibrations: Introduction to Mathematical Models in the Physical Sciences
    Newton's Law
    Newton's Law as Applied to a Spring-Mass System
    Oscillation of a Spring-Mass System
    Dimensions and Units
    Qualitative and Quantitative Behavior of a Spring-Mass System
    Initial Value Problem
    A Two-Mass Oscillator
    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 Dynamics-Mathematical Ecology. Introduction to Mathematical Models in Biology
    Population Models
    A Discrete One-Species Model
    Constant Coefficient First-Order Difference Equations
    Exponential Growth
    Discrete Once-Species Models with an Age Distribution
    Stochastic Birth Processes
    Density-Dependent 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 Two-Species Models
    Phase Plane, Equilibrium, and linearization
    System of Two Constant Coefficient First-Order Differential Equations, Stability of Two-Species Equilibrium Populations
    Phase Plane of Linear Systems
    Predator-Prey Models
    Derivation of the Lotka-Volterra Equations
    Qualitative Solution of the Lotka- Volterra Equations
    Average Populations of Predators and Preys
    Man's Influence on Predator-Prey Ecosystems
    Limitations of the Lotka-Volterra 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 Velocity-Density Relationship
    Experimental Observations
    Traffic Flow
    Steady-State Car-Following Models
    Partial Differential Equations
    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 Velocity-Density 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

  • Author

    Richard Haberman

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