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. In terms of the Dirac delta function δ(x), a fundamental solution f is the solution of the inhomogeneous equation

Related Topics:
Mathematics - Partial differential operator - Distribution theory - Green's function - Dirac delta function - Inhomogeneous equation

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:Lf = δ(x).

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Here f is a priori only assumed to be a Schwartz distribution.

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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.

Related Topics:
Laplacian - Marcel Riesz - Convolution - RHS

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