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 1
... given to me by Dr. Eugene Speer and Dr. Richard Falk who have co-taught the material with me. Also I would like to express my appreciation to Dr. Mark Ablowitz for his many thoughtful and useful suggestions. For the opportunity and ...
... given to me by Dr. Eugene Speer and Dr. Richard Falk who have co-taught the material with me. Also I would like to express my appreciation to Dr. Mark Ablowitz for his many thoughtful and useful suggestions. For the opportunity and ...
Page 3
... given that agree qualitatively with experiments or observations. However, when theory and experiment quantitatively agree, then we can usually be more confident in the validity of the theory. In this manner mathematics becomes an ...
... given that agree qualitatively with experiments or observations. However, when theory and experiment quantitatively agree, then we can usually be more confident in the validity of the theory. In this manner mathematics becomes an ...
Page 12
... given. The general solution of a second-order linear homogeneous differential equation is a linear combination of two homogeneous solutions. For constant coefficient differential equations, the homogeneous solutions are usually in the ...
... given. The general solution of a second-order linear homogeneous differential equation is a linear combination of two homogeneous solutions. For constant coefficient differential equations, the homogeneous solutions are usually in the ...
Page 13
... given any value of 01 and c2, there exists values of a and b, namely Since the algebra is a bit involved, it is useful to memorize the result we have just derived: An arbitrary linear combination of e"'" and e_'°", x aeiwt + be—iwt' is ...
... given any value of 01 and c2, there exists values of a and b, namely Since the algebra is a bit involved, it is useful to memorize the result we have just derived: An arbitrary linear combination of e"'" and e_'°", x aeiwt + be—iwt' is ...
Page 14
... given 0, and c2, it is seen that both A and (1),, can be determined. Dividing the two equations yields an expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The ...
... given 0, and c2, it is seen that both A and (1),, can be determined. Dividing the two equations yields an expression for tan (#0, and using sinZ on + cos2 (1),, = 1 results in an equation for A2: A I or + cal/2 _ c ¢0:tan1—l6?. The ...
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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