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 20
... Initial Value Problem In the previous sections, we have shown that x : cl ... conditions of the spring-mass system. One way to initiate motion in a spring ... initial conditions that the mass is at x0 at t = 0, x(0) = x0. and at t : O ...
... Initial Value Problem In the previous sections, we have shown that x : cl ... conditions of the spring-mass system. One way to initiate motion in a spring ... initial conditions that the mass is at x0 at t = 0, x(0) = x0. and at t : O ...
Page 21
... initial value problem the mass is initially at rest. Two initial conditions are necessary since the differential equation involves the second derivative in time. To solve this initial value problem, the arbitrary constants c1 and c2 are ...
... initial value problem the mass is initially at rest. Two initial conditions are necessary since the differential equation involves the second derivative in time. To solve this initial value problem, the arbitrary constants c1 and c2 are ...
Page 24
... initial conditions such that initially x2 — x1 at I. Now from Newton's law of motion, it follows that if F , is the force on mass ml and if F2 is the force on mass m,, then 2 d x1 dzxz dz, =FI and m2 d', m1 = F,. To complete the ...
... initial conditions such that initially x2 — x1 at I. Now from Newton's law of motion, it follows that if F , is the force on mass ml and if F2 is the force on mass m,, then 2 d x1 dzxz dz, =FI and m2 d', m1 = F,. To complete the ...
Page 25
... initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not accelerate. By saying there is no ...
... initial conditions). If this system of two masses attached to a spring is viewed as a single entity, then since there are no external forces on it, the system obeys Newton's first law and will not accelerate. By saying there is no ...
Page 38
... value? Does this time depend in a reasonable way on k, m, and c? Show that the ratio of two consecutive local maximum ... initial conditions. If c2 < 4mk, show that for large times the oscillation approximately obeys the following ...
... value? Does this time depend in a reasonable way on k, m, and c? Show that the ratio of two consecutive local maximum ... initial conditions. If c2 < 4mk, show that for large times the oscillation approximately obeys the following ...
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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