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 xiv
... exercises. Many more problems are included than are reasonable for the average reader to do. The exercises have been designed such that their difficulty varies. Almost all readers will probably find some too easy, while others are quite ...
... exercises. Many more problems are included than are reasonable for the average reader to do. The exercises have been designed such that their difficulty varies. Almost all readers will probably find some too easy, while others are quite ...
Page 5
... EXERCISES 2.1. Consider Fig. 2-2, which shows two masses (m1 and m2) attached to the opposite ends of a rigid (and massless) bar: Figure '2-2. *Newton's second law can ... exercise 2.1). k) be (I) N t u m1 is located at. 5 Sec. 2 Newton's ...
... EXERCISES 2.1. Consider Fig. 2-2, which shows two masses (m1 and m2) attached to the opposite ends of a rigid (and massless) bar: Figure '2-2. *Newton's second law can ... exercise 2.1). k) be (I) N t u m1 is located at. 5 Sec. 2 Newton's ...
Page 6
... exercise 2.1 to a rigid body consisting of N masses. Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is the total ...
... exercise 2.1 to a rigid body consisting of N masses. Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is the total ...
Page 7
... exercise 3.1.) Thus the observed variability of the velocity must be due to forces probably exerted by the spring. To develop an appropriate model of the spring force, one should study the motions of spring-mass systems under different ...
... exercise 3.1.) Thus the observed variability of the velocity must be due to forces probably exerted by the spring. To develop an appropriate model of the spring force, one should study the motions of spring-mass systems under different ...
Page 13
... exercise 5.6 using the Taylor series of sines, cosines, and exponentials. A similar expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the evenness of ...
... exercise 5.6 using the Taylor series of sines, cosines, and exponentials. A similar expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the evenness of ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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