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
Results 1-5 of 85
Page 3
... occurs because the real world consists of many interacting processes. It may be impossible in an experiment to entirely eliminate certain undesirable effects. Furthermore one is never sure which effects may be negligible in nature. A ...
... occurs because the real world consists of many interacting processes. It may be impossible in an experiment to entirely eliminate certain undesirable effects. Furthermore one is never sure which effects may be negligible in nature. A ...
Page 19
... occur? Let us compare two different springs represented in Fig. 7-1: (k1 > k2) Figure 7-1. The one which is firmer has a larger restoring force and hence it returns more quickly to its equilibrium position. Thus we suspect that the ...
... occur? Let us compare two different springs represented in Fig. 7-1: (k1 > k2) Figure 7-1. The one which is firmer has a larger restoring force and hence it returns more quickly to its equilibrium position. Thus we suspect that the ...
Page 20
... occur that the intuition is correct and consequently that either there was a mathematical error in the derivation of the formula or the model upon which the analysis is based may need improvement. EXERCISES 7.1. A .227 kilogram Q pound ...
... occur that the intuition is correct and consequently that either there was a mathematical error in the derivation of the formula or the model upon which the analysis is based may need improvement. EXERCISES 7.1. A .227 kilogram Q pound ...
Page 23
... occur halfway between times at which the mass passes its equilibrium position. 9. A Two-Mass Oscillator In the previous sections we carefully developed a mathematical model of a spring-mass system. We analyzed the resulting oscillations ...
... occur halfway between times at which the mass passes its equilibrium position. 9. A Two-Mass Oscillator In the previous sections we carefully developed a mathematical model of a spring-mass system. We analyzed the resulting oscillations ...
Page 25
... occurs in applied mathematics, the solution of the mathematical equation of interest simplifies if we use insight gained from the physical problem. We also note that this is the same type of. 25 Sec. 9 A Two-Mass Oscillator.
... occurs in applied mathematics, the solution of the mathematical equation of interest simplifies if we use insight gained from the physical problem. We also note that this is the same type of. 25 Sec. 9 A Two-Mass Oscillator.
Other editions - View all
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