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
... correspond to as much as a single lecture. Usually more than one (and occasionally, depending on the background of the reader, many) of the sections can be covered in an amount of time equal to that of a single lecture. In this way the ...
... correspond to as much as a single lecture. Usually more than one (and occasionally, depending on the background of the reader, many) of the sections can be covered in an amount of time equal to that of a single lecture. In this way the ...
Page 8
... (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: approximation of experimental spring force. Thus F=—M pm is a good approximation for the spring ...
... (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: approximation of experimental spring force. Thus F=—M pm is a good approximation for the spring ...
Page 19
... qualitatively. If our intuition about a problem does not correspond to what a mathematical formula predicts, then further investigations of the problem are necessary. 19 Sec. 7 Qualitative and Quantitative Behavior of a Spring-Mass System.
... qualitatively. If our intuition about a problem does not correspond to what a mathematical formula predicts, then further investigations of the problem are necessary. 19 Sec. 7 Qualitative and Quantitative Behavior of a Spring-Mass System.
Page 22
... corresponding to the mass initially at rest at x = x0 and the other corresponding to the mass initially at the equilibrium position with velocity 110. What is the amplitude of the total oscillation? 8.3. Consider a spring-mass system ...
... corresponding to the mass initially at rest at x = x0 and the other corresponding to the mass initially at the equilibrium position with velocity 110. What is the amplitude of the total oscillation? 8.3. Consider a spring-mass system ...
Page 29
... correspond to our experience? If we displaced the mass to the right, as shown in Fig. 10-1, then we would probably expect the mass to oscillate in the manner sketched in Fig. 10-2. We suspect that the mass oscillates around its ...
... correspond to our experience? If we displaced the mass to the right, as shown in Fig. 10-1, then we would probably expect the mass to oscillate in the manner sketched in Fig. 10-2. We suspect that the mass oscillates around its ...
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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