Edit: The question is about Problem 2 for "Problems to Section 2.5, 2.6" which you can find at:

http://www.math.toronto.edu/ivrii/PDE-textbook/Chapter2/S2.6.P.htmlI looked at the other post about this problem but I'm still confused about how to even begin solving it. The problem says to solve for $0<x<c_1t$ and $−c_2t<x<0$ separately, but we need $u$ or $u_x$ at $x=0$ for $t>0$. We have the limits at $x=0$, and I thought about using the formulas for Dirchlet or Neumann B.C., but I realized I couldn't use them since we have both conditions at $x=0$. So I thought of maybe combining the two to get a Robin B.C. but I'm not sure if that's right either. Any help would be appreciated.