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 viii
... Constant Coefficient First-Order Difference Equations ............................... .. 129 Exponential Growth 1 3 1 Discrete One-Species Models with an Age Distribution ................................ .. 138 * Stochastic Birth ...
... Constant Coefficient First-Order Difference Equations ............................... .. 129 Exponential Growth 1 3 1 Discrete One-Species Models with an Age Distribution ................................ .. 138 * Stochastic Birth ...
Page 8
... constant. It depends on the elasticity of the spring. This linear relationship between the force and the position of the mass was discovered by the seventeenth century physicist Hooke and is thus known as Hooke's law. Doubling the ...
... constant. It depends on the elasticity of the spring. This linear relationship between the force and the position of the mass was discovered by the seventeenth century physicist Hooke and is thus known as Hooke's law. Doubling the ...
Page 10
... constant* —mg, the mass m times the acceleration due to gravity —g. The two forces add vectorially and hence Newton's law becomes m F = —ky — mg, (4.2) where y is the vertical coordinate. y = 0 is the position at which the spring exerts ...
... constant* —mg, the mass m times the acceleration due to gravity —g. The two forces add vectorially and hence Newton's law becomes m F = —ky — mg, (4.2) where y is the vertical coordinate. y = 0 is the position at which the spring exerts ...
Page 11
... constant. How high does the mass go before it begins to fall? Does this height depend in a reasonable way on m, 00 ... constant. Thus the universal constant G is related to g by GM g = WThe rotation of the earth only causes very small ...
... constant. How high does the mass go before it begins to fall? Does this height depend in a reasonable way on m, 00 ... constant. Thus the universal constant G is related to g by GM g = WThe rotation of the earth only causes very small ...
Page 12
... constant coefficient linear differential equations is given. The general solution of a second-order linear homogeneous differential equation is a linear combination of two homogeneous solutions. For constant coefficient differential ...
... constant coefficient linear differential equations is given. The general solution of a second-order linear homogeneous differential equation is a linear combination of two homogeneous solutions. For constant coefficient differential ...
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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