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 14
... sketched in Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + ...
... sketched in Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the amplitude of the oscillation; it is easily computed from the above equation if c, and c2 are known. The phase of oscillation is cot + ...
Page 29
... sketched in Fig. 10-2. We suspect that the mass oscillates around its equilibrium position with smaller and smaller magnitude until it stops. 7 x=0 x=xo Figure 10-1. x = O x = x0 Figure 10-2. Must we reject a mathematical model that ...
... sketched in Fig. 10-2. We suspect that the mass oscillates around its equilibrium position with smaller and smaller magnitude until it stops. 7 x=0 x=xo Figure 10-1. x = O x = x0 Figure 10-2. Must we reject a mathematical model that ...
Page 31
... sketched in Fig. 10-6 (see exercise 10.4). )2 depends on the roughness of the surface (and the weight of the mass). In later sections exercises will discuss the mathematical solution of problems involving this type of friction, called ...
... sketched in Fig. 10-6 (see exercise 10.4). )2 depends on the roughness of the surface (and the weight of the mass). In later sections exercises will discuss the mathematical solution of problems involving this type of friction, called ...
Page 35
... sketched in Fig. 12-1). At the maximum value of the sinusoidal function, sin (out + $0) Figure 12-1. x equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first ...
... sketched in Fig. 12-1). At the maximum value of the sinusoidal function, sin (out + $0) Figure 12-1. x equals the exponential alone, while at the minimum value of the sinusoidal function, x equals minus the exponential. Thus we first ...
Page 64
... sketched, yielding (for example) Fig. 19-1. The derivative of the potential energy is f (x), dF (x) __ 7; 4°')The applied force is — f (x). Thus the derivative of the potential energy is minus the force. Since f (x,.,) = 0, potential ...
... sketched, yielding (for example) Fig. 19-1. The derivative of the potential energy is f (x), dF (x) __ 7; 4°')The applied force is — f (x). Thus the derivative of the potential energy is minus the force. Since f (x,.,) = 0, potential ...
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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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