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. |
From inside the book
Results 1-5 of 63
Page ix
... Equals Density Times Velocity 273 Conservation of the Number of Cars 275 A Velocity-Density Relationship 282 Experimental Observations 286 Traffic Flow 289 ix 2: Contents 64. Steady-State Car-Following Models 293 * 65. Partial.
... Equals Density Times Velocity 273 Conservation of the Number of Cars 275 A Velocity-Density Relationship 282 Experimental Observations 286 Traffic Flow 289 ix 2: Contents 64. Steady-State Car-Following Models 293 * 65. Partial.
Page xiv
... equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. I ... equal length. Few correspond to as much as a single lecture. Usually more than one (and occasionally, depending on the ...
... equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. I ... equal length. Few correspond to as much as a single lecture. Usually more than one (and occasionally, depending on the ...
Page 3
... equal emphasis to all three aspects. One cannot underestimate the importance of good experiments in developing mathematical models. However, mathematical models are important in their own right, aside from an attempt to mimic nature ...
... equal emphasis to all three aspects. One cannot underestimate the importance of good experiments in developing mathematical models. However, mathematical models are important in their own right, aside from an attempt to mimic nature ...
Page 5
... equal the rate of change of the momentum me, where v is the velocity of the mass and x its position: _\ —\ _ dx_ 1 ... equals its mass times its acceleration, easily remembered as “F equals ma.” The resulting acceleration of a point mass ...
... equal the rate of change of the momentum me, where v is the velocity of the mass and x its position: _\ —\ _ dx_ 1 ... equals its mass times its acceleration, easily remembered as “F equals ma.” The resulting acceleration of a point mass ...
Page 6
... equal and opposite, implies that F; = —F,. (a) Suppose that an external force a, is applied to m], and 62 to m2. By applying Newton's second law to each mass, show the law can be applied to the rigid body consisting of both masses, if ...
... equal and opposite, implies that F; = —F,. (a) Suppose that an external force a, is applied to m], and 62 to m2. By applying Newton's second law to each mass, show the law can be applied to the rigid body consisting of both masses, if ...
Other editions - View all
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