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