What are the characteristic lines for the given PDE?

[mediocre]

Homework Statement

For the given PDE:

$\partial_t u+\frac{1}{2}\partial_x u=0$

Where:

$u(0,t)=-1/2t\;;\;t>0$

$u(x,0)=x\;;\;0\leq x\leq2$

$u(x,0)=4-x\;;\;2<x\leq4$

$u(x,0)=0\;;\;4<x$

The Attempt at a Solution

Characteristic equations give :

$x=\frac{1}{2}t+x_0$

$u(x,t)=f(x-\frac{1}{2}t)=f(x_0)=const.$

Which,on the (x,u) graph would look something like:

Is this correct?

The next thing i have to do is to find the solution to it via Lax-Wendroff scheme,but i won't be bothering anyone with that :)

 

[lippyka]

Yes, your answer is correct. The characteristic lines are straight lines with slope 1/2, as you said. In the (x,t) plane, these lines have the equation x = 1/2 t + x_0. If we plot these lines starting from the initial conditions, we get the following sketch: The solution of the PDE is indeed a constant along each of these lines, so the solution is the union of all these line segments. You can see that the solution is continuous along the x-axis, and it interpolates between the given initial condition (for t=0).