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 81
Page x
... Differential Equations 298 66. Linearization 301 67. A Linear Partial Differential Equation 303 68. Traffic Density Waves 309 69. An Interpretation of Traffic Waves 314 70. A Nearly Uniform Traffic Flow Example ...
... Differential Equations 298 66. Linearization 301 67. A Linear Partial Differential Equation 303 68. Traffic Density Waves 309 69. An Interpretation of Traffic Waves 314 70. A Nearly Uniform Traffic Flow Example ...
Page 12
... Equation 5.1 is a second-order linear differential equation with constant coefficients. As you should recall from a course in differential equations, the general solution of this differential equation is x = 0, cos cot + 02 sin cot ...
... Equation 5.1 is a second-order linear differential equation with constant coefficients. As you should recall from a course in differential equations, the general solution of this differential equation is x = 0, cos cot + 02 sin cot ...
Page 25
... Equation 9.3 can be re-expressed as d2 Wonlxi + mzxz) : 0- (9'4) Thus the center of mass of the system, (m,xl + m2xz)/( ... differential equation for the stretching of the spring. We see this can be accomplished by dividing each equation by ...
... Equation 9.3 can be re-expressed as d2 Wonlxi + mzxz) : 0- (9'4) Thus the center of mass of the system, (m,xl + m2xz)/( ... differential equation for the stretching of the spring. We see this can be accomplished by dividing each equation by ...
Page 32
... differential equation is nonlinear. From the differential equation for a spring-mass system with linear damping, show that x = 0 is the only equilibrium position of the mass. What is the dimension of the constant 0 defined for linear ...
... differential equation is nonlinear. From the differential equation for a spring-mass system with linear damping, show that x = 0 is the only equilibrium position of the mass. What is the dimension of the constant 0 defined for linear ...
Page 33
... equation 10.3? (b) Suppose that the frictional force instead is —y - c Z-f if 2%' > 0 F' = d d y - c i if g < 0 ... differential equation recall that the two linearly independent solutions “almost always” can be written in the form of ...
... equation 10.3? (b) Suppose that the frictional force instead is —y - c Z-f if 2%' > 0 F' = d d y - c i if g < 0 ... differential equation recall that the two linearly independent solutions “almost always” can be written in the form of ...
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