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 (which normally further addresses boundary conditions).

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

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.

Consider the following differential equation Lf = sin(x) with

The fundamental solutions can be obtained by solving LF = δ(x), explicitly,

Since for the Heaviside function H we have

there is a solution

Here C is an arbitrary constant introduced by the integration. For convenience, set C = − 1/2.

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