Green's function

A Green's function is the response of a specified linear differential operator $L$ to a point source, together with specified boundary conditions or causality conditions. In distributional notation, $L_{x}G(x,y)=\delta (x-y)$, where $\delta$ is the Dirac delta function. Integrating this response against a source gives $u(x)=\int G(x,y)f(y)dy$ when the representation is valid.

For $L=-d^2/d{x}^2$ on $0<x<1$ with zero Dirichlet boundary conditions, $G(x,y)=\min (x,y)(1-\max (x,y))$. It is linear on either side of $y$, continuous there, and its first derivative jumps by $-1$, so $-G^{\prime \prime }={\delta }_y$. Changing either the operator sign or the boundary conditions changes the kernel. The example shows how a difficult source is assembled from elementary impulse responses.

Green's function is a powerful mathematical tool used primarily in the fields of differential equations and mathematical physics. It serves a variety of purposes, but its main role is to solve inhomogeneous linear differential equations subject to specific boundary conditions.