Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 278
... limit as △ a → 0 : - [ **** p ( x , t ) dx = lim 9 ( a , t ) a Ca + 1 lim Aa - 0 Ət -Aa a Aa → 0 - q ( a + △ a , t ) -Δα . ( 60.4b ) The right - hand side of equation 60.4b is exactly the definition of the deriva- tive of q ( a , t ) ...
... limit as △ a → 0 : - [ **** p ( x , t ) dx = lim 9 ( a , t ) a Ca + 1 lim Aa - 0 Ət -Aa a Aa → 0 - q ( a + △ a , t ) -Δα . ( 60.4b ) The right - hand side of equation 60.4b is exactly the definition of the deriva- tive of q ( a , t ) ...
Page 289
... limit , and so on ) in order to maximize the flow on a given roadway . The largest flow q = pu would occur if cars were bumper to bumper ( p = Pmax ) moving at the speed limit ( u = Umax ) . Clearly this is not safe ( is that clear ...
... limit , and so on ) in order to maximize the flow on a given roadway . The largest flow q = pu would occur if cars were bumper to bumper ( p = Pmax ) moving at the speed limit ( u = Umax ) . Clearly this is not safe ( is that clear ...
Page 349
... limit from the left , f ( x , ̄ ) . However , both limits must exist . For example , see Fig . 77-2 . f ( x , - ) f ( x + ) Figure 77-2 Jump discontinuity . Xs † The terminology , shock wave , is introduced because of the analogous ...
... limit from the left , f ( x , ̄ ) . However , both limits must exist . For example , see Fig . 77-2 . f ( x , - ) f ( x + ) Figure 77-2 Jump discontinuity . Xs † The terminology , shock wave , is introduced because of the analogous ...
Contents
Mechanical Vibrations 1 | 3 |
NEWTONS LAW | 4 |
NEWTONS LAW 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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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