Fundamental solution
From Biocrawler, the free encyclopedia.
In mathematics, a fundamental solution for a linear partial differential operator L is a formulation in the language of distribution theory of the older idea of a Green's function. In terms of the Dirac delta function δ(x), a fundamental solution f is the solution of the inhomogeneous equation
- Lf = δ(x).
Here f is a priori only assumed to be a Schwartz distribution.
This concept was long known for the Laplacian in two and three dimensions. It was investigated for all dimensions for the Laplacian by Marcel Riesz. The existence of a fundamental solution for any operator with constant coefficients — the most important case, directly linked to the possibility of using convolution to solve an arbitrary RHS — was shown by Malgrange and Ehrenpreis.
