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
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Page 4
... simple spring-mass system exhibits behavior of more complex systems. For example, the oscillations of a spring-mass system resemble the motions of clock-like mechanisms and, in a sense, also aid in the understanding of the up-and-down ...
... simple spring-mass system exhibits behavior of more complex systems. For example, the oscillations of a spring-mass system resemble the motions of clock-like mechanisms and, in a sense, also aid in the understanding of the up-and-down ...
Page 12
... simple exponentials, e". The specific exponential(s) are obtained by directly substituting the assumed form e" into the differential equation. If e" is substituted into equation 5.1, then a quadratic equation for r results, mrZ : —k ...
... simple exponentials, e". The specific exponential(s) are obtained by directly substituting the assumed form e" into the differential equation. If e" is substituted into equation 5.1, then a quadratic equation for r results, mrZ : —k ...
Page 15
... simple harmonic motion. The mass oscillates sinusoidally around the equilibrium position x : O. The solution is periodic in time. As illustrated in Fig. 5-1, the mass after reaching its maximum' displacement (x largest), again returns ...
... simple harmonic motion. The mass oscillates sinusoidally around the equilibrium position x : O. The solution is periodic in time. As illustrated in Fig. 5-1, the mass after reaching its maximum' displacement (x largest), again returns ...
Page 16
... simple harmonic motion (and also the y component). (bis measured in radians per unit of time or revolutions per 27: units of time, the circular frequency. 5.8. (a) Show that x = c, cos cot + c; sin cat is the general solution of m(d2x ...
... simple harmonic motion (and also the y component). (bis measured in radians per unit of time or revolutions per 27: units of time, the circular frequency. 5.8. (a) Show that x = c, cos cot + c; sin cat is the general solution of m(d2x ...
Page 21
... simple harmonic motion, as shown in Fig. 8-1. The spring is initially stretched and hence the spring pulls the mass to the left. This process continues indefinitely. Other initial value problems are posed in the exercises. It should be ...
... simple harmonic motion, as shown in Fig. 8-1. The spring is initially stretched and hence the spring pulls the mass to the left. This process continues indefinitely. Other initial value problems are posed in the exercises. It should be ...
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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