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 34
Page 15
... measured in cycles per second (sometimes known as a Hertz). Since a springmass system normally oscillates with frequency l/ZnN/W, this value is referred to as the natural frequency of a spring-mass system of mass m and spring constant k ...
... measured in cycles per second (sometimes known as a Hertz). Since a springmass system normally oscillates with frequency l/ZnN/W, this value is referred to as the natural frequency of a spring-mass system of mass m and spring constant k ...
Page 16
... measured in radians per unit of time or revolutions per 27: units of time, the circular frequency. 5.8. (a) Show that x = c, cos cot + c; sin cat is the general solution of m(d2x/dt 2) = —kx. What is the value of a)? (b) Show that an ...
... measured in radians per unit of time or revolutions per 27: units of time, the circular frequency. 5.8. (a) Show that x = c, cos cot + c; sin cat is the general solution of m(d2x/dt 2) = —kx. What is the value of a)? (b) Show that an ...
Page 17
... measured in units of feet, inches, miles, meters, or smoots.* In any calculation to eliminate possible confusion only one unit of length should be used. In this text we will use metric units in the m—k—s system, i.e., meters for length ...
... measured in units of feet, inches, miles, meters, or smoots.* In any calculation to eliminate possible confusion only one unit of length should be used. In this text we will use metric units in the m—k—s system, i.e., meters for length ...
Page 41
... measured quantities—there is no possibility that these measurements could be such that c2 : 4mk exactly. Any small deviation from equality will result in either of the previous two cases. For mathematical completeness, the solution in ...
... measured quantities—there is no possibility that these measurements could be such that c2 : 4mk exactly. Any small deviation from equality will result in either of the previous two cases. For mathematical completeness, the solution in ...
Page 53
... measured in a certain way, then we will show that only two parameters are important. Let us scale the time t by any constant time Q, by which we mean let t = Q1. Using the chain rule 31 _ _1_ _d_, dt _ Q d1 yields the differential ...
... measured in a certain way, then we will show that only two parameters are important. Let us scale the time t by any constant time Q, by which we mean let t = Q1. Using the chain rule 31 _ _1_ _d_, dt _ Q d1 yields the differential ...
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