4 Laplace’s equation
If you look back at the two-dimensional heat conduction equation (4):
$\frac{\partial u}{\partial t}=k\left(\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}\right)$
it is clear that if the heat flow is steady, i.e. time independent, then $\frac{\partial u}{\partial t}=0$ so the temperature $u\left(x,y\right)$ is a solution of
$\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}=0$ (8)
(8) is the two-dimensional Laplace equation . Both this, and its three-dimensional counterpart
$\frac{{\partial}^{2}u}{\partial {x}^{2}}+\frac{{\partial}^{2}u}{\partial {y}^{2}}+\frac{{\partial}^{2}u}{\partial {z}^{2}}=0$ (9)
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 $R$ for given boundary conditions. Since conditions may involve
- $u$ specified on the boundary curve $C$ (two dimensions) or boundary surface $S$ (three dimensions) of the region $R$ . Such boundary conditions are called Dirichlet conditions .
- The derivative of $u$ normal to the boundary, written $\frac{\partial u}{\partial n},$ specified on $C$ or $S$ . These are referred to as Neumann boundary conditions .
- A mixture of (a) and (b).
Some areas in which Laplace’s equation arises are
- electrostatics ( $u$ being the electrostatic potential in a charge free region)
- gravitation ( $u$ being the gravitational potential in free space)
- steady state flow of inviscid fluids
- steady state heat conduction (as already discussed)