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 x
... 69. An Interpretation of Traffic Waves 314 70. A Nearly Uniform Traffic Flow Example ......................................................... .. 315 * 71. Nonuniform Traffic - The Method of Characteristics .............................
... 69. An Interpretation of Traffic Waves 314 70. A Nearly Uniform Traffic Flow Example ......................................................... .. 315 * 71. Nonuniform Traffic - The Method of Characteristics .............................
Page xiv
... example, the divergence theorem) is never used (nor is it needed). 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 ...
... example, the divergence theorem) is never used (nor is it needed). 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 ...
Page 4
... Examples of nonlinear frictionless oscillators are worked out in detail (Secs. 21-25). Nonlinear systems which are damped ... example, the oscillations of a spring-mass system resemble the motions of clock-like mechanisms and, in a sense ...
... Examples of nonlinear frictionless oscillators are worked out in detail (Secs. 21-25). Nonlinear systems which are damped ... example, the oscillations of a spring-mass system resemble the motions of clock-like mechanisms and, in a sense ...
Page 7
... example, the force is assumed to be the same no matter what speed the mass is moving at. *Througho ut this text, we assume that the width of the mass is negligible. A careful examination of the experimental data shows that the. 7 Sec. 3 ...
... example, the force is assumed to be the same no matter what speed the mass is moving at. *Througho ut this text, we assume that the width of the mass is negligible. A careful examination of the experimental data shows that the. 7 Sec. 3 ...
Page 16
... errors. In dimensional analysis, brackets indicate the dimension of a quantity. For example, the notation [x] designates the dimension of x, which is a length L, i.e., [x] : L, measured in units of feet, 16 Mechanical Vibrations.
... errors. In dimensional analysis, brackets indicate the dimension of a quantity. For example, the notation [x] designates the dimension of x, which is a length L, i.e., [x] : L, measured in units of feet, 16 Mechanical Vibrations.
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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