Laplace Equation

The Laplace equation is

$${\nabla }^{2}u=0$$

It describes the steady-state of the Heat equation (diffusion equation): when $\partial u/\partial t=0$, the heat equation reduces to the Laplace equation.

2D Form

In two spatial dimensions:

$$\frac{{\partial }^{2}u}{\partial x^2}+\frac{{\partial }^{2}u}{\partial y^2}=0$$

Key Properties

• The Laplace equation has no time variable, so there is no initial condition.
• Only Boundary conditions are required to determine the solution uniquely.
• The solution gives the equilibrium temperature distribution inside a domain given fixed boundary temperatures.

The finite difference discretisation of the 2D Laplace equation gives the Laplacian difference equation, which is solved iteratively using Liebmann’s method.

Heat equation | Laplacian difference equation | Liebmann’s method | Boundary conditions