# Differential equations - Some partial differential equations

1. The Laplace Equation
${\partial^2 u\over \partial x^2}+{\partial^2 u\over \partial y^2}=0$
applies to potential energy function $u(x,y)$ for a conservation force field in the xy-plane, Partial differential equoations of this type are called elliptic.
2. The Heat Equation
${\partial^2 u\over \partial x^2}+{\partial^2 u\over \partial y^2}={\partial u \over \partial t}$
applies to the temperature distribution u(x,y) in the xy-plane when heat is allowed to flow from warm areas to cool ones. The equations of this type are called parabolic.
3. The Wave Equation
${\partial^2 u\over \partial x^2}+{\partial^2 u\over \partial y^2}={\partial^2 u \over \partial t^2}$
applies to the displacement u(x,y) of vibrating membranes and other wave function. The equations of this type are called hyperbolic