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}