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 xii
... derive the nonlinear partial differential equations for the traffic density. The method of characteristics is carefully developed for these nonlinear partial differential equations and applied to various typical traffic situations. In ...
... derive the nonlinear partial differential equations for the traffic density. The method of characteristics is carefully developed for these nonlinear partial differential equations and applied to various typical traffic situations. In ...
Page 12
... derived using many approximations and assumptions, it is hoped that the understanding of its solution will aid in more exact investigations (some of which we will pursue). Equation 5.1 is a second-order linear differential equation with ...
... derived using many approximations and assumptions, it is hoped that the understanding of its solution will aid in more exact investigations (some of which we will pursue). Equation 5.1 is a second-order linear differential equation with ...
Page 13
... derived in exercise 5.6 using the Taylor series of sines, cosines, and exponentials. A similar expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the ...
... derived in exercise 5.6 using the Taylor series of sines, cosines, and exponentials. A similar expression for e"'°", can be derived from equation 5.4a by replacing to by —co. This results in e"imt = cos cot — i sin cot (5.4b) where the ...
Page 16
... derived, a, : x/£. m As a check on our calculations we claim that the dimensions of both sides of this equation agree. Checking formulas by dimensional analysis is an important general procedure you should follow. Frequently this type ...
... derived, a, : x/£. m As a check on our calculations we claim that the dimensions of both sides of this equation agree. Checking formulas by dimensional analysis is an important general procedure you should follow. Frequently this type ...
Page 18
... a spring-mass system, the eifect of varying the different parameters is investigated. An important formula is the one derived for the period of oscillation, T : 21rx/% - (7.1) Suppose that we use a. 18 Mechanical Vibrations.
... a spring-mass system, the eifect of varying the different parameters is investigated. An important formula is the one derived for the period of oscillation, T : 21rx/% - (7.1) Suppose that we use a. 18 Mechanical Vibrations.
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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