Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic FlowThe 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. |
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Page viii
... Linear Oscillator 70 Phase Plane of a Nonlinear Pendulum 76 Can a Pendulum Stop? 82 What Happens if a Pendulum is Pushed Too Hard? ....................................... .. 84 Period of a Nonlinear Pendulum 87 * Nonlinear Oscillations ...
... Linear Oscillator 70 Phase Plane of a Nonlinear Pendulum 76 Can a Pendulum Stop? 82 What Happens if a Pendulum is Pushed Too Hard? ....................................... .. 84 Period of a Nonlinear Pendulum 87 * Nonlinear Oscillations ...
Page ix
... Linear Systems 203 A. General Remarks 203 B. Saddle Points 205 C. Nodes 212 D. Spirals 216 E. Summary 223 Predator-Prey Models 224 Derivation of the Lotka-Volterra Equations ....................................................... .. 225 ...
... Linear Systems 203 A. General Remarks 203 B. Saddle Points 205 C. Nodes 212 D. Spirals 216 E. Summary 223 Predator-Prey Models 224 Derivation of the Lotka-Volterra Equations ....................................................... .. 225 ...
Page x
... Linear Partial Differential Equation 303 68. Traffic Density Waves 309 69. An Interpretation of Traffic Waves 314 70 ... Linear-Velocity Density Relationship 331 74. An Example 339 * 75. Wave Propagation of Automobile Brake Lights ...
... Linear Partial Differential Equation 303 68. Traffic Density Waves 309 69. An Interpretation of Traffic Waves 314 70 ... Linear-Velocity Density Relationship 331 74. An Example 339 * 75. Wave Propagation of Automobile Brake Lights ...
Page xiv
... Linear algebra and probability are also not required (although they are briefly utilized in a few sections which the reader may skip). Although some knowledge of differential equations is required, it is mostly restricted to first- and ...
... Linear algebra and probability are also not required (although they are briefly utilized in a few sections which the reader may skip). Although some knowledge of differential equations is required, it is mostly restricted to first- and ...
Page 4
... linear systems (frictionless). 2. linear systems with friction. 3. nonlinear systems (frictionless). 4. nonlinear systems with friction. 2. Newton's Law To begin our investigations of mathematical models, a problem with which most of ...
... linear systems (frictionless). 2. linear systems with friction. 3. nonlinear systems (frictionless). 4. nonlinear systems with friction. 2. Newton's Law To begin our investigations of mathematical models, a problem with which most of ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
Common terms and phrases
amplitude analysis applied approximately Assume birth calculated called cars characteristics Consider constant continuous corresponding curve decreases delay depends derived described determine differential equation discussed distance energy equal equilibrium population equilibrium position equivalent example exercise experiments expression Figure first fish flow force formulate friction function given growth rate hence highway illustrated increases initial initial conditions integral isoclines known length light limit linear manner mass mathematical model maximum measured method motion moving nonlinear number of cars observer obtained occurs oscillation partial differential equation pendulum period phase plane possible probability problem region result roots sharks shock Show shown in Fig simple sketched sketched in Fig solution solve species spring spring-mass system stable straight line Suppose tion traffic density traflic trajectories unstable variables velocity yields zero