Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 299
... partial differ- ential equation is an initial condition . With an nth order ordinary differential equation , n initial conditions are needed . The number of conditions are correspondingly the same for partial differential equations . Thus ...
... partial differ- ential equation is an initial condition . With an nth order ordinary differential equation , n initial conditions are needed . The number of conditions are correspondingly the same for partial differential equations . Thus ...
Page 303
... partial differential equation governs the perturbed traffic density . However , equation 66.3 is a linear partial differential equation while the exact traffic equation 66.1 is a nonlinear partial differential equation . The coefficient ...
... partial differential equation governs the perturbed traffic density . However , equation 66.3 is a linear partial differential equation while the exact traffic equation 66.1 is a nonlinear partial differential equation . The coefficient ...
Page 306
... partial differential equation 67.1 . Using the chain rule d ( x — ct ) дх дрі дх = dg d ( x — ct ) and d ( x др . = dg d ( x — ct ) -- ct ) δι = dg d ( x — ct ) = -c dg d ( x — ct ) Thus it is verified that equation 67.1 is satisfied by ...
... partial differential equation 67.1 . Using the chain rule d ( x — ct ) дх дрі дх = dg d ( x — ct ) and d ( x др . = dg d ( x — ct ) -- ct ) δι = dg d ( x — ct ) = -c dg d ( x — ct ) Thus it is verified that equation 67.1 is satisfied by ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
Copyright | |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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amplitude applied approximation Assume calculated called cars characteristics conservation Consider constant continuous corresponding curve decreases density wave depends derived described determine differential equation discussed distance dx dt dx/dt energy equal equilibrium population equilibrium position equivalent example exercise expression Figure force formula friction function given growth rate hence highway illustrated increases indicate initial conditions integral intersect 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 Pmax possible problem region result road satisfies shock Show shown in Fig simple sketched sketched in Fig slope solution solve species spring spring-mass system stable straight line Suppose tion traffic density traffic flow trajectories Umax unstable valid variables yields zero