TT1 = Problem 1

[Victor Ivrii]

Consider the first order equation:
\begin{equation}
u_t + x u_x = 0.
\label{eq-1} 
\end{equation}

• (a) Find the characteristic curves and sketch them in the $(x,t)$ plane.
• (b) Write the general solution.
• (c) Solve  equation (\ref{eq-1})  with the initial condition $u(x,0)= \cos(2x)$.

Explain why the solution is fully  determined by the initial condition.

• (d) bonus Describe domain in which solution of
  \begin{equation}
  u_t + x^2 u_x = 0, \qquad x>0
  \label{eq-2} 
  \end{equation}
  is fully determined by the initial condition $u(x,0)=g(x)$ ($x>0$)?

[Djirar]

My solution. Please check there might be mistakes. Sorry for quality of scan and handwriting.

[Aida Razi]

Check the attachment please,

[Aida Razi]

My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you 

[Djirar]

My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you 

I was scribbling my solutions as fast as I could 

Edit: I forgot to write down the axes to part a.  The vertical is x-axis and the horizontal is y-axis. I hope I didn't forget this on the test 

[Zarak Mahmud]

My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you 

I was scribbling my solutions as fast as I could 

Edit: I forgot to write down the axes to part a.  The vertical is x-axis and the horizontal is y-axis. I hope I didn't forget this on the test 

Do you mean the $t$ axis?

[Qitan Cui]

I have a different solution for bonus

[Djirar]

I think Qitan Cui is correct. In my haste I must have messed up the integration of the bonus part.

[Jinchao Lin]

Subqueston (d):

$ \frac{dt}{1} = \frac{dx}{x^2} $
$ t = -x^{-1}+c $
so the general solution is $ u(t,x)=f(t+x^{-1})$.
$u(0,x)=f(x^{-1})=g(x)$
$f(y)=g(y^{-1})$
Since $y=x^{-1}$, so when $x>0$ we have $y>0$ as well.
$u(t,x)=f(t+x^{-1})$
We need $t+x^{-1}>0$
Since $x>0$,
therefore $tx+1>0$
so the domain be defined is $\{(t,x) | tx>-1 \}.

[Djirar]

My solution. Please check there might be mistakes. Sorry for quality of print and hand writing.

Djirar, I posted my solution 10 seconds after you 

I was scribbling my solutions as fast as I could 

Edit: I forgot to write down the axes to part a.  The vertical is x-axis and the horizontal is y-axis. I hope I didn't forget this on the test 

Do you mean the $t$ axis?

Yes $t$ not $y$. Thank you.

[Ian Kivlichan]

Subqueston (d):

$ \frac{dt}{1} = \frac{dx}{x^2} $
$ t = -x^{-1}+c $
so the general solution is $ u(t,x)=f(t+x^{-1})$.
$u(0,x)=f(x^{-1})=g(x)$
$f(y)=g(y^{-1})$
Since $y=x^{-1}$, so when $x>0$ we have $y>0$ as well.
$u(t,x)=f(t+x^{-1})$
We need $t+x^{-1}>0$
Since $x>0$,
therefore $tx+1>0$
so the domain be defined is $\{(t,x) | tx>-1 \}.

I think Jinchao has the most correct solution.

Qitan, is it possible to only have the one discontinuity in your solution - won't your characteristic curves be "blocked" by the discontinuity at tx=-1, and not able to go any further?

[Victor Ivrii]

Djirar solved (a), (c) correctly and made a mistake in (b), Aida solved (a)-(c) correctly. If we consider $x$ which could be negative and positive $\log |x|-t$ is not good enough  as it is the same on two disjoint curves, but $xe^{-t}$ works. If initial function was not even then Djirar would not be able honestly to satisfy it from the "general" solution.

Jinchao (BTW, change your name with proper capitalization and as on BlackBoard) is correct and Ian' remark explain fallacy of Qui solutions.

In two attached pictures one can see vector fields and curves for (1) and (2) respectively; one can see that certain characteristics never intersect initial curve. Condition $x>0$ was given only for a sake of simplicity.