Green's function

In mathematics, if L is a linear operator acting upon distributions over a manifold, M, then any solution of (Lf)(x) = δ(x − x₀), where δ is the Dirac delta function, is called a Green's function of L at x₀. If the kernel of L is nontrivial, then the Green's function is not unique. However, in practice, some combination of symmetry, boundary conditions and/or other externally imposed criteria would give us a unique Green's function. Also, please note Green's functions are distributions in general, not functions.

Not every operator L admits a Green's function, though. A Green's function can also be thought of as a one-sided inverse of L.

Motivation

Convolving with a Green's function gives solutions to inhomogeneous differentio-integral equations. If g is the Green's function of an operator L, then the solution for f of the equation Lf = h is the convolution of g with h.

Examples

• Let the manifold be R and L be d/dx. Then, the Heaviside function H(x − x₀) is a Green's function of L at x₀.
• Let the manifold be the quarter-plane { (x, y) : x, y ≥ 0 } and L be the Laplacian. Also, assume a Dirichlet boundary condition is imposed at x=0 and a Neumann boundary condition is imposed at y=0. Then the Green's function is G(x, y ; x₀, y₀)

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