$$
\newcommand{\uex}{{u_{\small\mbox{e}}}}
\newcommand{\half}{\frac{1}{2}}
\newcommand{\halfi}{{1/2}}
\newcommand{\xpoint}{\boldsymbol{x}}
\newcommand{\normalvec}{\boldsymbol{n}}
\newcommand{\Oof}[1]{\mathcal{O}(#1)}
\newcommand{\Ix}{\mathcal{I}_x}
\newcommand{\Iy}{\mathcal{I}_y}
\newcommand{\It}{\mathcal{I}_t}
\newcommand{\setb}[1]{#1^0} % set begin
\newcommand{\sete}[1]{#1^{-1}} % set end
\newcommand{\setl}[1]{#1^-}
\newcommand{\setr}[1]{#1^+}
\newcommand{\seti}[1]{#1^i}
\newcommand{\Real}{\mathbb{R}}
$$
Analytical work with the PDE problem
Here, choose \( \uex \) such that \( \uex(x,0)=\uex(L,0)=0 \):
$$ \uex (x,t) = x(L-x)(1+{\half}t), $$
Insert in the PDE and find \( f \):
$$ f(x,t)=2(1+t)c^2$$
Initial conditions:
$$ I(x) = x(L-x),\quad V(x)={\half}x(L-x) $$
