Famous PDEs, general equation in two dimensions
A general partial differential equation with two given dimensions
reads
$$
\begin{equation*}
A(x,y)\frac{\partial^2 u}{\partial x^2}+B(x,y)\frac{\partial^2 u}{\partial x\partial y}
+C(x,y)\frac{\partial^2 u}{\partial y^2}=F(x,y,u,\frac{\partial u}{\partial x}, \frac{\partial u}{\partial y}),
\end{equation*}
$$
and if we set
$$
\begin{equation*}
B=C=0,
\end{equation*}
$$
we recover the \( 1+1 \)-dimensional diffusion equation which is an example
of a so-called parabolic partial differential equation.
With
$$
\begin{equation*}
B=0, \hspace{1cm} AC < 0
\end{equation*}
$$
we get the \( 2+1 \)-dim wave equation which is an example of a so-called
elliptic PDE, where more generally we have
\( B^2 > AC \).
For \( B^2 < AC \)
we obtain a so-called hyperbolic PDE, with the Laplace equation in
Eq.
(3) as one of the
classical examples.
These equations can all be easily extended to non-linear partial differential
equations and \( 3+1 \) dimensional cases.