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 ii
... Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F ...
... Methods James M. Ortega, Numerical Analysis: A Second Course Anthony V. Fiacco and Garth P. McCormick, Nonlinear Programming: Sequential Unconstrained Minimization Techniques F. H. Clarke, Optimization and Nonsmooth Analysis George F ...
Page iii
... Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Glkin, and Milton Sobel, Selecting and Ordering ... Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric ...
... Methods in Control: Analysis and Design Jean Dickinson Gibbons, Ingram Glkin, and Milton Sobel, Selecting and Ordering ... Method for Elliptic Problems Jane K. Cullum and Ralph A. Willoughby, Lanczos Algorithms for Large Symmetric ...
Page ix
... Method of Elimination 192 B. Systems Method (using Matrix Theory) ........................................................ ..193 Stability of Two-Species Equilibrium Populations 199 Phase Plane of Linear Systems 203 A. General Remarks ...
... Method of Elimination 192 B. Systems Method (using Matrix Theory) ........................................................ ..193 Stability of Two-Species Equilibrium Populations 199 Phase Plane of Linear Systems 203 A. General Remarks ...
Page x
... Method of Characteristics ................................ .. 319 72. After a Traffic Light Turns Green 7 323 73. A Linear-Velocity Density Relationship 331 74. An Example 339 * 75. Wave Propagation of Automobile Brake Lights ...
... Method of Characteristics ................................ .. 319 72. After a Traffic Light Turns Green 7 323 73. A Linear-Velocity Density Relationship 331 74. An Example 339 * 75. Wave Propagation of Automobile Brake Lights ...
Page xii
... method of characteristics is carefully developed for these nonlinear partial differential equations and applied to various typical traffic situations. In the future, it will be more common for highways to have sensors irnbedded in the ...
... method of characteristics is carefully developed for these nonlinear partial differential equations and applied to various typical traffic situations. In the future, it will be more common for highways to have sensors irnbedded in the ...
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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