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 16
... solution of m(d2x/dt 2) = —kx. What is the value of a)? (b) Show that an equivalent expression for the general solution is x = Bcos (tot + 90). How do B and 00 depend on c1 and c2? 6. Dimensions and Units In the previous section, the ...
... solution of m(d2x/dt 2) = —kx. What is the value of a)? (b) Show that an equivalent expression for the general solution is x = Bcos (tot + 90). How do B and 00 depend on c1 and c2? 6. Dimensions and Units In the previous section, the ...
Page 20
... solution has aided in directly improving one's qualitative understanding. On the other hand, it may occur that the intuition is correct and consequently that either there was a mathematical error in the derivation of the formula or the ...
... solution has aided in directly improving one's qualitative understanding. On the other hand, it may occur that the intuition is correct and consequently that either there was a mathematical error in the derivation of the formula or the ...
Page 24
... solution of this problem in a different way; one which in fact yields a solution more easily interpreted. By inspection we note that the equations simplify if they are added together. In that manner 2 2 m,% + m, '17'? = 0. 24 Mechanical ...
... solution of this problem in a different way; one which in fact yields a solution more easily interpreted. By inspection we note that the equations simplify if they are added together. In that manner 2 2 m,% + m, '17'? = 0. 24 Mechanical ...
Page 34
... solution corresponds to each case, since the roots are respectively real and unequal, real and equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic ...
... solution corresponds to each case, since the roots are respectively real and unequal, real and equal, and complex. EXERCISES 11.1. Show that c2 has the same dimension as mk. 11.2. What dimension should the roots of the characteristic ...
Page 35
... solution is most easily accomplished using the latter form. The solution is the product of an exponential and a sinusoidal function (each sketched in Fig. 12-1). At the maximum value of the sinusoidal function, sin (out + $0) Figure 12 ...
... solution is most easily accomplished using the latter form. The solution is the product of an exponential and a sinusoidal function (each sketched in Fig. 12-1). At the maximum value of the sinusoidal function, sin (out + $0) Figure 12 ...
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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