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 vii
... ............... .. 18 8. Initial Value Problem 20 9. A Two-Mass Oscillator 23 * IO. Friction Z9 11. Oscillations of a Damped System 33 12. Underdamped Oscillations 34 13. Overdamped and Critically Damped Oscillations ...
... ............... .. 18 8. Initial Value Problem 20 9. A Two-Mass Oscillator 23 * IO. Friction Z9 11. Oscillations of a Damped System 33 12. Underdamped Oscillations 34 13. Overdamped and Critically Damped Oscillations ...
Page 11
... initial speed 220 at an angle 0 with respect to the horizon. Where does the mass land ? What trajectory did the mass take? For what angle does the mass land the farthest away from where it was thrown (assuming the same initial speed)? ...
... initial speed 220 at an angle 0 with respect to the horizon. Where does the mass land ? What trajectory did the mass take? For what angle does the mass land the farthest away from where it was thrown (assuming the same initial speed)? ...
Page 20
... Initial Value Problem In the previous sections, we have shown that x : cl cos cot + c2 sin cot (8.1) is the general solution of the differential equation describing a spring-mass system, 2 m '5% : ——kx, (8.2) where c1 and c2 are ...
... Initial Value Problem In the previous sections, we have shown that x : cl cos cot + c2 sin cot (8.1) is the general solution of the differential equation describing a spring-mass system, 2 m '5% : ——kx, (8.2) where c1 and c2 are ...
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 22
... initial velocity 1:0. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at x = x0 and the other corresponding to the mass initially at the equilibrium position with velocity ...
... initial velocity 1:0. Show that the resulting motion is the sum of two oscillations, one corresponding to the mass initially at rest at x = x0 and the other corresponding to the mass initially at the equilibrium position with velocity ...
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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