Fundamental solution

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 (although unlike Green's functions, fundamental solutions do not address boundary conditions).

In terms of the Dirac delta "function" $δ (x)$ , a fundamental solution $F$ is a solution of the inhomogeneous equation

LF = δ (x)

.

Here $F$ is a priori only assumed to be a distribution.

This concept has long been utilized 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 right hand side — was shown by Bernard Malgrange and Leon Ehrenpreis, and a proof is available on Joel Smoller (1994).^[1] In the context of functional analysis, fundamental solutions are usually developed via the Fredholm alternative and explored in Fredholm theory.

^ Smoller, Joel (1994). "7. Distribution Theory". Shock Waves and Reaction—Diffusion Equations (2 ed.). Springer New York, NY. ISBN 978-0-387-94259-9.

[1] Smoller, Joel (1994). "7. Distribution Theory". Shock Waves and Reaction—Diffusion Equations (2 ed.). Springer New York, NY. ISBN 978-0-387-94259-9.

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