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
... resulting mathematical problem is solved, requiring at times the introduction of new mathematical methods. The ... results. Furthermore these three topics were chosen for inclusion in this text because each serves as an introduction to ...
... resulting mathematical problem is solved, requiring at times the introduction of new mathematical methods. The ... results. Furthermore these three topics were chosen for inclusion in this text because each serves as an introduction to ...
Page xiv
... resulting in the concept of traffic density wave propagation. Throughout, mathematical techniques are developed, but equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. I ...
... resulting in the concept of traffic density wave propagation. Throughout, mathematical techniques are developed, but equal emphasis is placed on the mathematical formulation of the problem and the interpretation of the results. I ...
Page 6
... result of exercise 2.1 to a rigid body consisting of N masses. Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is ...
... result of exercise 2.1 to a rigid body consisting of N masses. Figure 2-4 shows a rigid bar of length L: [:1 <_—L——-——> Figure 2-4. a.) If the mass density p(x) (mass per unit length) depends on the position along the bar, then what is ...
Page 11
... result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k larger), the smaller the sag of the spring (also quite reasonable). It is frequently advantageous to translate coordinate systems ...
... result that should not be surprising. For a larger mass, the spring sags more. The stiffer the spring (k larger), the smaller the sag of the spring (also quite reasonable). It is frequently advantageous to translate coordinate systems ...
Page 12
... results, mrZ : —k. The two roots are imaginary, r = iico where a) = A/k/m. Thus the general solution is a linear combination of eimt and e—t'art' x : ae'm + be"°", (5.3) where a and b are arbitrary constants. However, the above. 12 ...
... results, mrZ : —k. The two roots are imaginary, r = iico where a) = A/k/m. Thus the general solution is a linear combination of eimt and e—t'art' x : ae'm + be"°", (5.3) where a and b are arbitrary constants. However, the above. 12 ...
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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