Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow
SIAM, Dec 1, 1998 - Mathematics - 419 pages
Mathematics is a grand subject in the way it can be applied to various problems in science and engineering. To use mathematics, one needs to understand the physical context. The 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.
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amplitude analysis applied approximately Assume behavior birth calculated called cars characteristics Consider constant continuous corresponding curve decreases delay density wave depends derived described determine differential equation discussed distance dx/dt energy equal equilibrium population equilibrium position equivalent example exercise experiments expression Figure fish force formulate friction function given growth rate hence highway illustrated increases initial initial conditions integral isoclines known length light limit linear 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 satisfies 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 traffic flow trajectories unstable variables velocity yields zero