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 4
... phase plane analysis are used (Secs. 19-20). Examples of nonlinear frictionless oscillators are worked out in detail (Secs. 21-25). Nonlinear systems which are damped are then discussed (Secs. 26-28). Mathematical models of increasing ...
... phase plane analysis are used (Secs. 19-20). Examples of nonlinear frictionless oscillators are worked out in detail (Secs. 21-25). Nonlinear systems which are damped are then discussed (Secs. 26-28). Mathematical models of increasing ...
Page 68
... phase plane, since we have expressed the equation in terms of the two variables x and dx/dt, referred to as the two phases of the system (position and velocity). The curve sketching the path of the solution is called the trajectory in the ...
... phase plane, since we have expressed the equation in terms of the two variables x and dx/dt, referred to as the two phases of the system (position and velocity). The curve sketching the path of the solution is called the trajectory in the ...
Page 69
... phase plane relating x and dx/dt is known, and looks as sketched in Fig. 20-3. Although x and dx/dl are as yet unknown functions of t, they satisfy a relation indicated by the curve in \ Q5 dr Figure 20-3. the phase plane. This curve is ...
... phase plane relating x and dx/dt is known, and looks as sketched in Fig. 20-3. Although x and dx/dl are as yet unknown functions of t, they satisfy a relation indicated by the curve in \ Q5 dr Figure 20-3. the phase plane. This curve is ...
Page 70
Mechanical Vibrations, Population Dynamics, and Traffic Flow Richard Haberman. 21. Phase Plane of a Linear Oscillator As a simple example of the analysis of a problem using a phase plane, consider the linear spring-mass system 2 mist—f ...
Mechanical Vibrations, Population Dynamics, and Traffic Flow Richard Haberman. 21. Phase Plane of a Linear Oscillator As a simple example of the analysis of a problem using a phase plane, consider the linear spring-mass system 2 mist—f ...
Page 71
... phase plane. ./2EO//< X How does this solution behave in time? Recall, in the upper half plane x increases, while in the lower half plane x decreases. Thus we have Fig. 21-3. The solution goes around and around (clockwise) in the phase ...
... phase plane. ./2EO//< X How does this solution behave in time? Recall, in the upper half plane x increases, while in the lower half plane x decreases. Thus we have Fig. 21-3. The solution goes around and around (clockwise) in the phase ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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