Famous PDEs, Laplace's equation
Another familiar equation from electrostatics is Laplace's equation, which looks similar
to the wave equation in Eq.
(1) except that we have set \( A=0 \)
$$
\begin{equation}
\tag{3}
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=0,
\end{equation}
$$
or if we have a finite electric charge represented by a charge density
\( \rho(\mathbf{x}) \) we have the familiar Poisson equation
$$
\begin{equation}
\tag{4}
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=-4\pi \rho(\mathbf{x}).
\end{equation}
$$