Mathematical Models : Mechanical Vibrations, Population Dynamics, and Traffic Flow

by Haberman, Richard

• ISBN13: 9780898714081
• ISBN10: 0898714087
• Format: Paperback
• Copyright: 1998-02-01
• Publisher: SIAM

Summary

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.

Table of Contents

Foreword

Preface to the classics edition

Preface

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

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

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

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