Mathematical Models: Mechanical Vibrations, Population Dynamics, and Traffic Flow : an Introduction to Applied Mathematics |
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Page 278
... variables . Thus , the full derivative with respect to time in equation 60.3 must be replaced by a partial derivative , a δι S a b p ( x , t ) dx = q ( a , t ) − q ( b , t ) , - − ( 60.4a ) since the derivation of equation 60.3 ...
... variables . Thus , the full derivative with respect to time in equation 60.3 must be replaced by a partial derivative , a δι S a b p ( x , t ) dx = q ( a , t ) − q ( b , t ) , - − ( 60.4a ) since the derivation of equation 60.3 ...
Page 305
... variables . Thus for the moving coordinate system , we use the variables x ' and t ' , where t ' t . Consequently , the change of variables we use is x ' - = x - ct t ' = t . In order to express the partial differential equation in ...
... variables . Thus for the moving coordinate system , we use the variables x ' and t ' , where t ' t . Consequently , the change of variables we use is x ' - = x - ct t ' = t . In order to express the partial differential equation in ...
Page 306
... variables , P1 = g ( x — ct ) . - ( 67.2 ) To again verify that this really is the solution , we substitute it back into the partial differential equation 67.1 . Using the chain rule d ( x — ct ) дх дрі дх = dg d ( x — ct ) and d ( x др ...
... variables , P1 = g ( x — ct ) . - ( 67.2 ) To again verify that this really is the solution , we substitute it back into the partial differential equation 67.1 . Using the chain rule d ( x — ct ) дх дрі дх = dg d ( x — ct ) and d ( x др ...
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