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

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 SolutionExample, Motivation, Signal Processing

Other articles related to "fundamental solution, solution":

Fundamental Solution - Signal Processing
... In signal processing, the analog of the fundamental solution of a differential equation is called the impulse response of a filter ...
... equations (PDEs), a parametrix is an approximation to a fundamental solution of a PDE, and is essentially an approximate inverse to a differential operator ... for a differential operator is often easier to construct than a fundamental solution, and for many purposes is almost as good ... It is sometimes possible to construct a fundamental solution from a parametrix by iteratively improving it ...
Pell's Equation - Solutions - Fundamental Solution Via Continued Fractions
... This pair is called the fundamental solution ... Thus, the fundamental solution may be found by performing the continued fraction expansion and testing each successive convergent until a solution to Pell's equation is found ... As Lenstra (2002) describes, the time for finding the fundamental solution using the continued fraction method, with the aid of the Schönhage–Strassen algorithm for fast integer multiplication, is within a ...
Construction of A Fundamental Solution From A Parametrix
... A sufficiently good parametrix can often be used to construct an exact fundamental solution by a convergent iterative procedure as follows (Berger, Gauduchon Mazet 1971) ... and P is a parametrix, this gives a right inverse to L, in other words a fundamental solution, provided that R is "small enough" which in practice ... functions, so the terms of the infinite sum giving the fundamental solution of L involve convolution of P with copies of R ...

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