Past exam of the algebra, analysis and differential equations 1 course of the University of Oxford / 2013 / Differential Equations / 8 / b / Solution

In the lower triangular region, the characteristic coordinates obey $-1\le \xi <\eta \le 1$. Integrating the canonical wave equation $u_{\xi \eta }=0$ givesOn the right characteristic edge, $\eta =1$ and $x=(\xi +1)/2$, so its prescribed value gives $F(\xi )+G(1)=-\sin (\pi \xi )$. On the left edge, $\xi =-1$ and $x=(\eta -1)/2$, giving $F(-1)+G(\eta )=\sin (\pi \eta )$. At the common vertex the two values agree and equal zero; hence $F(-1)+G(1)=0$.

Adding the two edge identities and subtracting this vertex identity determines the entire solution:The apparent freedom to add a constant to $F$ and subtract it from $G$ does not change $u$. Every classical lower-region solution has the displayed separated form, so these two characteristic boundary traces determine it uniquely. Substitution into $u_{xx}-u_{yy}=0$ and into both edge values directly verifies existence.