traffic flow module

142 days ago by swinarsk

2. a

p(x,t) = 1

var('x,t') f(x) = 1 plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=1

characteristics=sum(plot(x/c + E, (x,0,20)) for E in [-5..-1,1..5]) characteristics

characteristics+point([6,2],color='red', pointsize=50)

ρ(6,2) is the same value as ρ(4,0)

f(4).n()

1.00000000000000

rho(x,t)=f(x-c*t) rho(x,t)

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

2.b

forget()

var('x,t') f(x) = x+1 plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=1

characteristics=sum(plot(x/c + E, (x,0,20)) for E in [-5..-1,1..5]) characteristics

f(4).n()

5.00000000000000

rho(x,t)=f(x-c*t) rho(x,t)

-t + x + 1

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

2.c

forget() var('x,t') f(x) = 1 plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=x+1

characteristics=sum(plot(x/c + E, (x,0,20)) for E in [-5..-1,1..5]) characteristics

f(4).n()

1.00000000000000

rho(x,t)=f(x-c*t) rho(x,t)

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

prob 3

forget() var('x,t') f(x)=e^(-(x-2)^2) plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=2

characteristics=sum(plot(x/c+E, (x,0,20)) for E in [-5..5]) characteristics

characteristics+point([12,3],color='red', pointsize=50)

f(5).n()

0.000123409804086680

rho(x,t)=f(x-c*t) rho(x,t)

e^(-(2*t - x + 2)^2)

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

problem 4

initial condition

ρ(x,t) = 0

forget() var('x,t') f(x)=0 plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=2

characteristics=sum(plot(x/c+E, (x,0,20)) for E in [-5..5]) characteristics

characteristics+point([12,3],color='red', pointsize=50)

f(6).n()

0.000000000000000

rho(x,t)=f(x-c*t) rho(x,t)

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

ρ(x,t)=x

forget() var('x,t') f(x)=x plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=2

characteristics=sum(plot(x/c+E, (x,0,20)) for E in [-5..5]) characteristics

characteristics+point([12,3],color='red', pointsize=50)

f(6).n()

6.00000000000000

rho(x,t)=f(x-c*t) rho(x,t)

-2*t + x

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

ρ(x,t)=x^2

forget() var('x,t') f(x)=x^2 plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=2

characteristics=sum(plot(x/c+E, (x,0,20)) for E in [-5..5]) characteristics

characteristics+point([12,3],color='red', pointsize=50)

f(6).n()

36.0000000000000

rho(x,t)=f(x-c*t) rho(x,t)

(2*t - x)^2

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()

ρ(x,t)=e^x

forget() var('x,t') f(x)=e^x plot(f(x), (x,0,20), axes_labels=['$x$',r'$\rho(x,0)$'])

c=2

characteristics=sum(plot(x/c+E, (x,0,20)) for E in [-5..5]) characteristics

characteristics+point([12,3],color='red', pointsize=50)

f(6).n()

403.428793492735

rho(x,t)=f(x-c*t) rho(x,t)

e^(-2*t + x)

plot3d(rho(x,t), (x,0,20), (t,0,12),plot_points=100)

animate([plot(rho(x,t), (x,0,20))+text("time=%d"%t,(10,.5)) for t in [0..5]]).show()