In the following problem, obtain $u(x,t)$ for $(x, t)$ in regions $I$, $II$, $III$, and $IV$. Verify that the initial and boundary conditions are compatible.
\begin{align} u_{tt} &= 4u_{xx}, \hspace{0.2cm} 0<x<1, \hspace{0.2cm} t>0\\ u(x,0) &= x\cos( \pi x)+x, \hspace{0,2cm} u_{t}( x,0) = x^2, \hspace{0,2cm} 0 \leq x \leq1\\ u( 0,t) &= u( 1,t) = 2 \pi t^2, \hspace{0,2cm} t \geq 0 \end{align}
My attempt:
For this problem I draw the characteristic lines that flow from the limit points to a point $(x, t)$, then I consider their projections and those parallel to these lines. So, I get the regions $I$, $II$, $III$ and $IV$ mentioned in the problem:
The region $I$ is the characteristic triangle and, therefore, I can solve using the general equation for this type of problem when the vibrating string is not fixed at the ends. In that case the solution would be given by
$$u(x,t)= \sum_{n=1}^{\infty}\left[ a_{n}\cos\left( \frac{n\pi ct}{L}\right)+b_{n}\sin \left( \frac{n\pi ct}{L} \right) \right]\sin \left( \frac{n\pi x}{L} \right) + \frac{1}{L}[\beta(t)-\alpha(t)]x + \alpha(t)$$ $$a_{n} = \frac{2}{L} \int_{0}^{L}\left[\varphi(\epsilon)- \frac{1}{L}\left[\beta(t)- \alpha(t)\right]\epsilon - \alpha(t)\right]\sin\left(\frac{n \pi \epsilon}{L}\right) d\epsilon$$ $$b_{n} = \frac{2}{n\pi c} \int_{0}^{L} \psi(\epsilon) \sin \left( \frac{n\pi \epsilon}{L}\right)d \epsilon$$ Which solves the general problem:
\begin{align} u_{tt} &= c^{2}u_{xx} \hspace{0,3cm} 0 < x <L, \hspace{0,3cm} t>0\\ u(x,0) &= \varphi( x), \hspace{0,3cm} u_{t}( x,0)= \psi( x), \hspace{0,3cm} 0\leq x \leq L\\ u( 0,t) &= \alpha( t), \hspace{0,3cm} u( L,t) = \beta( t), \hspace{0,3cm} t\geq 0 \end{align}
Now, let $A$ be a point in region $II$, then I must find the value of $u$ at that point. The identity of the parallelogram says, according to the figure, that $u (A) + u (C) = u (B) + u (D)$. How can I use this identity to solve this problem? I can obviously notice that $B$ is on the boundary and $C$ and $D$ on the characteristic triangle. On the other hand, how can I proceed for example in region $IV$? I don't have good ideas for this part. Any good help is appreciated.