Green's Identities - Green's Third Identity

Green's Third Identity

Green's third identity derives from the second identity by choosing, where G is a Green's function of the Laplace operator. This means that:

For example in, a solution has the form:

Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then

A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation. Then and the identity simplifies to:

The second term in the integral above can be eliminated if we choose G to be the Green's function that vanishes on the boundary of region U (Dirichlet boundary condition):

This form is used to construct solutions to Dirichlet boundary condition problems. To find solutions for Neumann boundary condition problems, the Green's function with vanishing normal gradient on the boundary is used instead.

It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens Principle.