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

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By Richard Haberman


1998 / xvii + 402 pages / Softcover / ISBN: 978-0-898714-08-1 / List Price $69.00 / SIAM Member Price $48.30 / Order Code CL21

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, from the foreword.

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.

This book is intended for undergraduate engineering, science, or mathematics students. It may also be useful for graduate students in various disciplines including ecology and transportation engineering.

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.

A partial solutions manual is available for instructors through email. If you would like to receive a PDF of the solutions manual please email textbooks@siam.org.

ISBN: 9780898714081

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