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 right hand side — was shown by Malgrange and Leon Ehrenpreis.

Read more about Fundamental Solution: Example, Motivation, Signal Processing

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—Ralph Waldo Emerson (1803–1882)

“POLITICIAN, n. An eel in the *fundamental* mud upon which the superstructure of organized society is reared. When he wriggles he mistakes the agitation of his tail for the trembling of the edifice. As compared with the statesman, he suffers the disadvantage of being alive.”

—Ambrose Bierce (1842–1914?)