Famous PDEs, two dimension
In two dimension we have
u=u(x,y,t) . We will, unless otherwise stated, simply use
u in our discussion below.
Familiar situations which this equation can model
are waves on a string, pressure waves, waves on the surface of a fjord or a
lake, electromagnetic waves and sound waves to mention a few. For e.g., electromagnetic
waves we have the constant
A=c^2 , with
c the speed of light. It is rather
straightforward to extend this equation to two or three dimension. In two dimensions
we have
\begin{equation*}
\frac{\partial^2 u}{\partial x^2}+\frac{\partial^2 u}{\partial y^2}=A\frac{\partial^2 u}{\partial t^2},
\end{equation*}