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 7
... increase the stretching of the spring, the force exerted by the spring would increase. Thus we might obtain the results shown in Fig. 3-2, where a curve is smoothly drawn connecting the experimental data points marked with an a n, X . F ...
... increase the stretching of the spring, the force exerted by the spring would increase. Thus we might obtain the results shown in Fig. 3-2, where a curve is smoothly drawn connecting the experimental data points marked with an a n, X . F ...
Page 15
... increases to t + T, from equation 5.5 wt + (#0 must change by 21:: a>(t+T)+¢0——wt—¢o—_~21z. Consequently the period T is T = 20- : zia/fl- (5.6) (0, called the circular frequency (as is explained in exercise 5.7), is the number of ...
... increases to t + T, from equation 5.5 wt + (#0 must change by 21:: a>(t+T)+¢0——wt—¢o—_~21z. Consequently the period T is T = 20- : zia/fl- (5.6) (0, called the circular frequency (as is explained in exercise 5.7), is the number of ...
Page 19
... increases the period, but it is doubtful that we could have known that quadrupling the weight results in an increase in the period by a factor of two! In mathematical models, usually the qualitative effects are at least partially ...
... increases the period, but it is doubtful that we could have known that quadrupling the weight results in an increase in the period by a factor of two! In mathematical models, usually the qualitative effects are at least partially ...
Page 39
... increase the velocity by I/m. This explains a method by which a nonzero initial velocity occurs. Reconsider exercise 12.8 for an alternate derivation. (a) Show that for 0 g t g At, (d) I mj—f+cx—cxo+kf xdt'=f(,t. O (b) If At is small ...
... increase the velocity by I/m. This explains a method by which a nonzero initial velocity occurs. Reconsider exercise 12.8 for an alternate derivation. (a) Show that for 0 g t g At, (d) I mj—f+cx—cxo+kf xdt'=f(,t. O (b) If At is small ...
Page 69
... increases as 1 increases. Arrows are added to the phase plane diagram to indicate the direction the solution changes with time. In the phase plane shown in Fig. 20-4, since x increases, the solution x(t) moves to the right as time increases ...
... increases as 1 increases. Arrows are added to the phase plane diagram to indicate the direction the solution changes with time. In the phase plane shown in Fig. 20-4, since x increases, the solution x(t) moves to the right as time increases ...
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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