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 xiv
... equivalent of the usual first two years of college mathematics (calculus and some elementary ordinary differential equations). Many critical aspects of these prerequisites are briefly reviewed. More specifically, a knowledge of calculus ...
... equivalent of the usual first two years of college mathematics (calculus and some elementary ordinary differential equations). Many critical aspects of these prerequisites are briefly reviewed. More specifically, a knowledge of calculus ...
Page 13
... " and e_'°", x aeiwt + be—iwt' is equivalent to an arbitrary linear combination of cos wt and sin cot, x : cl cos cot + c2 sin cot. In the above manner you should now be able to. 13 Sec. 5 Oscillation of a Spring-Mass System.
... " and e_'°", x aeiwt + be—iwt' is equivalent to an arbitrary linear combination of cos wt and sin cot, x : cl cos cot + c2 sin cot. In the above manner you should now be able to. 13 Sec. 5 Oscillation of a Spring-Mass System.
Page 14
... equivalent expression for the solution is x = A sin (cot + (150). (5.5) This is shown by noting sin (cot + (#0) : sin (or cos ¢,, + cos cot sin ¢,, in which case 0, I A sin 45,, c2 : A cos ¢,. If you are given 0, and c2, it is seen that ...
... equivalent expression for the solution is x = A sin (cot + (150). (5.5) This is shown by noting sin (cot + (#0) : sin (or cos ¢,, + cos cot sin ¢,, in which case 0, I A sin 45,, c2 : A cos ¢,. If you are given 0, and c2, it is seen that ...
Page 16
... equivalent to x = c1 cos cor + c; sin wt. Show that if c; and c; are real (that is if x is real), then b is the complex conjugate of a. 5.6. The Taylor series for sin x, cos x, and e" are well known for real x: 3 5 7 sinx=x—%-+J§CT—%+ 2 ...
... equivalent to x = c1 cos cor + c; sin wt. Show that if c; and c; are real (that is if x is real), then b is the complex conjugate of a. 5.6. The Taylor series for sin x, cos x, and e" are well known for real x: 3 5 7 sinx=x—%-+J§CT—%+ 2 ...
Page 17
... equivalent length in feet or miles and the equivalent mass in pounds will appear afterwards in parentheses. What is the dimension of dx/dt, the velocity? Clearly, [ape dt — 1: a length L divided by a time 1. Mathematically we note that ...
... equivalent length in feet or miles and the equivalent mass in pounds will appear afterwards in parentheses. What is the dimension of dx/dt, the velocity? Clearly, [ape dt — 1: a length L divided by a time 1. Mathematically we note that ...
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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