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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 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.
Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fan-shaped characteristics describe the traffic situation that occurs when a traffic light turns green and shock waves describe the effects of a red light or traffic accident.
Although it was written over 20 years ago, this book is still relevant. It is intended as an introduction to applied mathematics, but can be used for undergraduate courses in mathematical modeling or nonlinear dynamical systems or to supplement courses in ordinary or partial differential equations.
Author Bio
Haberman, Richard : Southern Methodist University
Richard Haberman is Professor of Mathematics at Southern Methodist University. He has published two other texts and numerous journal articles.
Foreword
Preface to the Classics Edition
Preface
Part 1: Mechanical Vibrations.
Introduction to Mathematical Models in the Physical Sciences
Newton's Law
Newton's Law as Applied to a Spring-Mass System
Gravity
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
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 2: 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 3: 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
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 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
Index.
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.
Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fan-shaped characteristics describe the traffic situation that occurs when a traffic light turns green and shock waves describe the effects of a red light or traffic accident.
Although it was written over 20 years ago, this book is still relevant. It is intended as an introduction to applied mathematics, but can be used for undergraduate courses in mathematical modeling or nonlinear dynamical systems or to supplement courses in ordinary or partial differential equations.
Author Bio
Haberman, Richard : Southern Methodist University
Richard Haberman is Professor of Mathematics at Southern Methodist University. He has published two other texts and numerous journal articles.
Table of Contents
Foreword
Preface to the Classics Edition
Preface
Part 1: Mechanical Vibrations.
Introduction to Mathematical Models in the Physical Sciences
Newton's Law
Newton's Law as Applied to a Spring-Mass System
Gravity
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
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 2: 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 3: 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
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 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
Index.