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Consider uu_{x}+u_{t}=0\, with the IC u(x,0)=h(x)={\begin{cases}0,&x<0\\u_{0}(x-1),&x>0\end{cases}}\,

Find the weak solution.

When y=0\,, if x>0\, then u=u_{0}(x-1)\, but the characteristic line (see PDEMOC14) is x=ut+x_{0}\,. Therefore u={\frac  {u_{0}(x-1)}{u_{0}t+1}}\,. Now find the line where the shock occurs.

The jump condition is defined as in PDEMOC14.

\xi '(t)={\frac  {1}{2}}{\frac  {u_{0}(x-1)}{u_{0}t+1}}\,

To integrate with respect to t\,, treat everything but u_{0}\, as a constant and you'll see that it's a log:

\xi (t)={\frac  {1}{2}}(x-1)\ln(u_{0}t+1)\,

The weak solution is u={\begin{cases}0,&x<\xi (t)\\{\frac  {u_{0}(x_{0}-1)}{u_{0}t+1}},&x>\xi (t)\end{cases}}\,

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