Richard Haberman
Classics in Applied Mathematics 21
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, 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 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.
Haberman develops the method of characteristics to analyze the nonlinear partial differential equations that describe traffic flow. Fanshaped 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.
Audience
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.
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 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 2: 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 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 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.
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.
1998 / xvii + 402 pages / Softcover / ISBN13: 9780898714081 / ISBN10: 0898714087 / List Price $69.00 / SIAM Member Price $48.30 / Order Code CL21
