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 xiv
... consider the relationships between the mathematics and the models. Each major part is divided into many subsections. However, these sections are not of equal length. Few correspond to as much as a single lecture. Usually more than one ...
... consider the relationships between the mathematics and the models. Each major part is divided into many subsections. However, these sections are not of equal length. Few correspond to as much as a single lecture. Usually more than one ...
Page 16
... Consider a particle moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt is constant, d0/dt = a). Show that the x component of the particle's position executes simple harmonic ...
... Consider a particle moving around a circle, with its position designated by the polar angle 0. Assume its angular velocity dQ/dt is constant, d0/dt = a). Show that the x component of the particle's position executes simple harmonic ...
Page 23
... consider a more complicated spring-mass system. Suppose instead of attaching a spring and a mass to a rigid wall, we attach a spring and a mass to another mass which is also free to move. Let us assume that the two masses are m1 and m2 ...
... consider a more complicated spring-mass system. Suppose instead of attaching a spring and a mass to a rigid wall, we attach a spring and a mass to another mass which is also free to move. Let us assume that the two masses are m1 and m2 ...
Page 26
... Consider two masses (of mass m; and m2) attached to a spring (of unstretched length l and spring constant k) in a manner similar to that discussed in this section. However, suppose the system is aligned vertically (rather than ...
... Consider two masses (of mass m; and m2) attached to a spring (of unstretched length l and spring constant k) in a manner similar to that discussed in this section. However, suppose the system is aligned vertically (rather than ...
Page 28
... Consider two masses each of mass m attached between two walls a distance d apart (refer to Figs. 9-3 and 9-7): N. m. tQQSlQQSlQL. m. $1? '1. 'm. Figure 9-7. Assume that all three springs have the same spring constant and unstretched length ...
... Consider two masses each of mass m attached between two walls a distance d apart (refer to Figs. 9-3 and 9-7): N. m. tQQSlQQSlQL. m. $1? '1. 'm. Figure 9-7. Assume that all three springs have the same spring constant and unstretched length ...
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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