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 vi
... manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA l9104e2688 USA. Library of Congress Catalog Card ...
... manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA l9104e2688 USA. Library of Congress Catalog Card ...
Page 3
... manner mathematics becomes an integral part of the scientific method. Applied mathematics can be said to involve three steps": 1. the formulation of a problem—the approximations and assumptions, based on experiments or observations ...
... manner mathematics becomes an integral part of the scientific method. Applied mathematics can be said to involve three steps": 1. the formulation of a problem—the approximations and assumptions, based on experiments or observations ...
Page 4
... manner: 1. 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 ...
... manner: 1. 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 ...
Page 8
... manner, on the stretching. However, for stretching of the spring which is not too large (corresponding to at most a moderate force), Fig. 3-3 shows that this curve can be approximated by a straight line: Figure 3-3 Hooke's Law ...
... manner, on the stretching. However, for stretching of the spring which is not too large (corresponding to at most a moderate force), Fig. 3-3 shows that this curve can be approximated by a straight line: Figure 3-3 Hooke's Law ...
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... manner as the mass would move horizontally around its horizontal equilibrium position. For this reason we may continue to study the horizontal spring-mass system even though vertical systems are more commonplace. EXERCISES 4.1. A mass m ...
... manner as the mass would move horizontally around its horizontal equilibrium position. For this reason we may continue to study the horizontal spring-mass system even though vertical systems are more commonplace. EXERCISES 4.1. A mass m ...
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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