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 xiii
... formulate and understand the mathematical models are relatively well known to the average reader. We will not find it necessary to refer to exceedingly technical research results. Furthermore these three topics were chosen for inclusion ...
... formulate and understand the mathematical models are relatively well known to the average reader. We will not find it necessary to refer to exceedingly technical research results. Furthermore these three topics were chosen for inclusion ...
Page 4
... formulating an equation which describes its motion. Fortunately many experimental observations culminated in Newton's second law of motion describing how a particle reacts to a force. Newton discovered that the motion of a point mass is ...
... formulating an equation which describes its motion. Fortunately many experimental observations culminated in Newton's second law of motion describing how a particle reacts to a force. Newton discovered that the motion of a point mass is ...
Page 23
... formulate Newton's law of motion for each mass. The force on each mass equals its mass times its acceleration. In order to obtain expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 ...
... formulate Newton's law of motion for each mass. The force on each mass equals its mass times its acceleration. In order to obtain expressions for the accelerations, we introduce the position of each mass (for example, in Fig. 9-2, x1 ...
Page 66
... formulate conservation of energy. Show that this potential energy (as a function of 0) has a relative minimum at the stable equilibrium position and a relative maximum at the unstable equilibrium position. The equation of a linearized ...
... formulate conservation of energy. Show that this potential energy (as a function of 0) has a relative minimum at the stable equilibrium position and a relative maximum at the unstable equilibrium position. The equation of a linearized ...
Page 87
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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