GreenFunktion

GreenFunktion, also known as Green's function, is a fundamental concept in mathematics and physics. It is the kernel that solves a linear differential equation L G(x, x') = δ(x - x') on a domain Ω, subject to specified boundary conditions. The Green function acts as the impulse response of the system described by L.

In practical use, the solution to L u = f is expressed as u(x) = ∫Ω G(x, x') f(x') dx' plus terms determined by boundary data. For self-adjoint operators with appropriate boundary conditions, the Green function is often symmetric: G(x, x') = G(x', x). It encodes how local sources influence the field at other points.

Examples and notable cases include:

- Poisson equation ∇^2 u = f with Dirichlet or Neumann boundaries. The Green function in free space satisfies ∇^2 G = -δ, yielding G(r, r') = -1/(4π|r - r'|) in three dimensions.

- In two dimensions, the free-space Green function for the Laplacian behaves like G ~ -(1/2π) log|r - r'|.

- Time-dependent problems use retarded Green's functions, incorporating causality: G(x, t; x', t') = 0 for t < t'. For the heat equation, the kernel is the heat kernel; for the wave equation, one obtains propagating signals with finite speed.

Construction methods include eigenfunction expansions, Fourier transform techniques, and the method of images, often tailored to boundary conditions and geometry. GreenFunktionen provide a unifying framework to solve linear boundary value problems across physics and engineering, linking differential operators to integral representations and enabling the computation of solutions from source distributions.