Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 325
... intersect the initial data at x > 0. There p ( x , 0 ) = 0. Thus p = 0 along all lines such that ―― dx dq dt dp 0 = = u ( 0 ) = Umax where this velocity has been evaluated using equation 72.2 . The characteristic velocity for zero ...
... intersect the initial data at x > 0. There p ( x , 0 ) = 0. Thus p = 0 along all lines such that ―― dx dq dt dp 0 = = u ( 0 ) = Umax where this velocity has been evaluated using equation 72.2 . The characteristic velocity for zero ...
Page 346
... intersect . the lighter traffic density wave . Thus we have Fig . 76-2 . Eventually these two families of characteristics intersect as illustrated in the figure . The sketched characteristics are moving forward ; the velocities of both ...
... intersect . the lighter traffic density wave . Thus we have Fig . 76-2 . Eventually these two families of characteristics intersect as illustrated in the figure . The sketched characteristics are moving forward ; the velocities of both ...
Page 369
... intersect ( see equation 80.2a ) . Before t = to , no neighboring characteristics intersect and dp / dx is never infinite . When t = to , dp / dx = ∞ at the position of breaking , the first place where neighboring characteristics ...
... intersect ( see equation 80.2a ) . Before t = to , no neighboring characteristics intersect and dp / dx is never infinite . When t = to , dp / dx = ∞ at the position of breaking , the first place where neighboring characteristics ...
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