Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 321
... characteristics . V βγ α ' ( x . , t . ) Figure 71-4 Using characteristics to deter- mine the future traffic density . X The density wave velocity , dq / dp , is extremely important . At this velocity the traffic density stays the same ...
... characteristics . V βγ α ' ( x . , t . ) Figure 71-4 Using characteristics to deter- mine the future traffic density . X The density wave velocity , dq / dp , is extremely important . At this velocity the traffic density stays the same ...
Page 325
... characteristics are straight lines . In the x - t plane X dq ( p ) t + k , ( 72.3 ) where each characteristic may have a different integration constant k . Let us analyze all characteristics that intersect the initial data at x > 0 ...
... characteristics are straight lines . In the x - t plane X dq ( p ) t + k , ( 72.3 ) where each characteristic may have a different integration constant k . Let us analyze all characteristics that intersect the initial data at x > 0 ...
Page 365
... characteristics initially a distance Ax ( not necessarily small ) apart , one emanating from x1 , the other from x1 + Ax ; see Fig . 80-3 . If this is the first intersection , then no characteristics could have crossed at an earlier ...
... characteristics initially a distance Ax ( not necessarily small ) apart , one emanating from x1 , the other from x1 + Ax ; see Fig . 80-3 . If this is the first intersection , then no characteristics could have crossed at an earlier ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
NEWTONS LAW AS APPLIED TO A SPRINGMASS SYSTEM | 6 |
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Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic ... Richard Haberman No preview available - 1998 |
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analyze approximation Assume c₁ c₂ calculated conservation of cars Consider corresponding d2x dt2 de/dt density wave velocity depends derived determine dq/dp dx dt dx/dt equilibrium population equilibrium position equilibrium solution example exercise exponential Figure flow-density force formula friction function growth rate highway increases initial conditions initial density initial traffic density initial value problem integral intersect isoclines logistic equation mass mathematical model maximum method of characteristics motion moving Newton's nonlinear pendulum number of cars observer occurs ordinary differential equation oscillation P/Pmax P₁ partial differential equation period phase plane Pmax potential energy problem qualitative region result sharks shock velocity Show shown in Fig simple harmonic motion Sketch the solution sketched in Fig slope solve species spring spring-mass system stable straight line Suppose Taylor series tion traffic flow traffic light trajectories Umax Umaxt unstable equilibrium variables velocity-density x₁ yields zero др