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 xii
... hence suitable for selfstudy, reading, or group projects. The mathematical models and techniques developed continue to be of fundamental interest and hence provide excellent background and motivation to the reader for further studies ...
... hence suitable for selfstudy, reading, or group projects. The mathematical models and techniques developed continue to be of fundamental interest and hence provide excellent background and motivation to the reader for further studies ...
Page xiv
... hence are relatively independent of the previous material. The method of characteristics is slowly and carefully explained, resulting in the concept of traffic density wave propagation. Throughout, mathematical techniques are developed ...
... hence are relatively independent of the previous material. The method of characteristics is slowly and carefully explained, resulting in the concept of traffic density wave propagation. Throughout, mathematical techniques are developed ...
Page 10
... hence Newton's law becomes m F = —ky — mg, (4.2) where y is the vertical coordinate. y = 0 is the position at which the spring exerts no force. Is there a position at which we could place the mass and it would not move, what we have ...
... hence Newton's law becomes m F = —ky — mg, (4.2) where y is the vertical coordinate. y = 0 is the position at which the spring exerts no force. Is there a position at which we could place the mass and it would not move, what we have ...
Page 17
... (hence having the dimension of y) divided by a small value of 2 (having the dimension of 2). Thus, j—fl = lyl/IzlThis result can be used to determine the dimensions of an acceleration: dzx _[x]_L_ WJ—[tTZ—F dzx L2 [W] i 'F This is ...
... (hence having the dimension of y) divided by a small value of 2 (having the dimension of 2). Thus, j—fl = lyl/IzlThis result can be used to determine the dimensions of an acceleration: dzx _[x]_L_ WJ—[tTZ—F dzx L2 [W] i 'F This is ...
Page 19
... hence it returns more quickly to its equilibrium position. Thus we suspect that the larger k is, the shorter the period- Equation 7.1 also predicts this qualitative feature. On the other hand, if the mass is increased using the same ...
... hence it returns more quickly to its equilibrium position. Thus we suspect that the larger k is, the shorter the period- Equation 7.1 also predicts this qualitative feature. On the other hand, if the mass is increased using the same ...
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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