If you look back at the two-dimensional heat conduction equation (4):
it is clear that if the heat flow is steady, i.e. time independent, then so the temperature is a solution of
(8) is the two-dimensional Laplace equation . Both this, and its three-dimensional counterpart
arise in a wide variety of applications, quite apart from steady state heat conduction theory. Since time does not arise in (8) or (9) it is evident that Laplace’s equation is always a model for equilibrium situations. In any problem involving Laplace’s equation we are interested in solving it in a specific region for given boundary conditions. Since conditions may involve
- specified on the boundary curve (two dimensions) or boundary surface (three dimensions) of the region . Such boundary conditions are called Dirichlet conditions .
- The derivative of normal to the boundary, written specified on or . These are referred to as Neumann boundary conditions .
- A mixture of (a) and (b).
Some areas in which Laplace’s equation arises are
- electrostatics ( being the electrostatic potential in a charge free region)
- gravitation ( being the gravitational potential in free space)
- steady state flow of inviscid fluids
- steady state heat conduction (as already discussed)