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 13
... function of an imaginary argument. The displacement x must be real. To show how equation 5.3 can be expressed in ... function [cos (— y) = cos y] and the oddness of the sine function [sin (—~y) = —sin y] has been used. Equations 5.4a and ...
... function of an imaginary argument. The displacement x must be real. To show how equation 5.3 can be expressed in ... function [cos (— y) = cos y] and the oddness of the sine function [sin (—~y) = —sin y] has been used. Equations 5.4a and ...
Page 14
... functions, a cosine and a sine. An equivalent expression for the solution is x = A sin (cot + (150). (5.5) This is ... function as sketched in Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the ...
... functions, a cosine and a sine. An equivalent expression for the solution is x = A sin (cot + (150). (5.5) This is ... function as sketched in Fig. 5-1: |<—T—>l \/ \/ t Figure 5-1 Period and amplitude of oscillation. A is called the ...
Page 15
... function f (t) is said to be periodic with period Tif ff! + T) =f(t)To determine the periogL T, we recall that the trigonometric functions are periodic with period 21f." {Thus for a complete oscillation, as 1 increases to t + T, from ...
... function f (t) is said to be periodic with period Tif ff! + T) =f(t)To determine the periogL T, we recall that the trigonometric functions are periodic with period 21f." {Thus for a complete oscillation, as 1 increases to t + T, from ...
Page 16
... function. 5.5. It was shown that x = ae'“" + be"“" is 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 ...
... function. 5.5. It was shown that x = ae'“" + be"“" is 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 ...
Page 18
... function of k and m. (a) Give an example of a possible dependence of y on k and m. (b) Can you describe the most general dependence that y can have on k and m ? 7. Qualitative and Quantitative Behavior of a Spring-Mass System To ...
... function of k and m. (a) Give an example of a possible dependence of y on k and m. (b) Can you describe the most general dependence that y can have on k and m ? 7. Qualitative and Quantitative Behavior of a Spring-Mass System To ...
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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