The partial differential equation: ∇2ψ = 0. It is a special case of the Helmholtz equation or Poisson's equation.
A solution to Laplace's equation is called a harmonic function.
A unique solution to Laplace's equation can be determined if either:
- The value of the function is specified on all boundaries (Dirichlet boundary conditions), or
- The normal derivative of the function is specified on all boundaries (Neumann boundary conditions)
The most common method for solving the equation is by separation of variables in a coordinate system suitable for the symmetries present in the problem and subsequent Fourier expansion of the boundary conditions.
This equation is common in applied mathematics, and mathematical physics, including problems in thermodynamics, electromagnetics, potential theory, and many others.