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 iii
... Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their ... Variable: Theory and Technique Friedrich Pukelsheim, Optimal Design of Experiments Israel Gohberg, Peter Lancaster, and Leiba ...
... Variables David Kinderlehrer and Guido Stampacchia, An Introduction to Variational Inequalities and Their ... Variable: Theory and Technique Friedrich Pukelsheim, Optimal Design of Experiments Israel Gohberg, Peter Lancaster, and Leiba ...
Page viii
... Variable 53 * Nonlinear Frictionless Systems 54 Linearized Stability Analysis of an Equilibrium Solution ................................ .. 56 Conservation of Energy 61 Energy Curves 67 Phase Plane of a Linear Oscillator 70 Phase Plane ...
... Variable 53 * Nonlinear Frictionless Systems 54 Linearized Stability Analysis of an Equilibrium Solution ................................ .. 56 Conservation of Energy 61 Energy Curves 67 Phase Plane of a Linear Oscillator 70 Phase Plane ...
Page 1
... variables of these partial differential equations. This text is a reflection of my own philosophy of applied mathematics. However, anyone's philosophy is strongly influenced by one's exposure. For my own education, the applied ...
... variables of these partial differential equations. This text is a reflection of my own philosophy of applied mathematics. However, anyone's philosophy is strongly influenced by one's exposure. For my own education, the applied ...
Page 17
... variable which we are differentiating with respect to. This is shown in general directly from the definition of a derivative, d_y _ - ye + AZ) —y(z>_ dz _ hm Az Az—v0 The dimensions of the two sides must agree. The right-hand side is ...
... variable which we are differentiating with respect to. This is shown in general directly from the definition of a derivative, d_y _ - ye + AZ) —y(z>_ dz _ hm Az Az—v0 The dimensions of the two sides must agree. The right-hand side is ...
Page 52
... occurs when sin 0 is approximated by 0 for 6 = 30°? (b) For 0 = 30°, what is the actual percent error? (c) Compare part (a) to part (b). 16. A Dimensionless Time Variable Returning to the equation of. 52 Mechanical Vibrations.
... occurs when sin 0 is approximated by 0 for 6 = 30°? (b) For 0 = 30°, what is the actual percent error? (c) Compare part (a) to part (b). 16. A Dimensionless Time Variable Returning to the equation of. 52 Mechanical Vibrations.
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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