Solve the following PDE numerically using the pseudo-transient approach. Vary initial conditions to show that your final solution is independent of the initial condition.

\[
\begin{equation*}
\frac{\partial^{2} u}{\partial x^{2}}+\frac{\partial^{2} u}{\partial y^{2}}=0 \tag{11.107}
\end{equation*}
\]

The boundary conditions are given by

\[
\begin{equation*}
u(x=0)=1, u\left(x=L_{x}ight)=0, u(y=0)=1, u\left(y=L_{y}ight)=0 \tag{11.108}
\end{equation*}
\]

a. Solve this PDE numerically without using the pseudo-transient approach.

b. Compare your results with the analytical solution.