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 1.00000000000000 |
|
|
rho(x,t)=f(x-c*t)
rho(x,t)
1 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.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 5.00000000000000 |
rho(x,t)=f(x-c*t)
rho(x,t)
-t + x + 1 -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 1.00000000000000 |
rho(x,t)=f(x-c*t)
rho(x,t)
1 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()
|
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 0.000123409804086680 |
rho(x,t)=f(x-c*t)
rho(x,t)
e^(-(2*t - x + 2)^2) 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 0.000000000000000 |
rho(x,t)=f(x-c*t)
rho(x,t)
0 0 |
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 6.00000000000000 |
rho(x,t)=f(x-c*t)
rho(x,t)
-2*t + x -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 36.0000000000000 |
rho(x,t)=f(x-c*t)
rho(x,t)
(2*t - x)^2 (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 403.428793492735 |
rho(x,t)=f(x-c*t)
rho(x,t)
e^(-2*t + x) 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()
|
|
|