From AMS Glossary
(Also called influence function.) A function that is the known solution of a homogenous differential equation in a specified region and that may be generalized (if the equation is linear) to satisfy given boundary or initial conditions, or a nonhomogeneous differential equation.
It is thus an alternative method to the Fourier transform or Laplace transform, applicable to many of the same problems. The Green's function method takes a fundamental solution and assigns it a weight at each point, say, of the boundary, according to the value of the given boundary condition there; the Fourier method analyzes the entire boundary condition into wave components thus assigning each of these a weight or amplitude. Both methods then obtain the final solution by summation or integration.