nLab fundamental solution

Context

Differential geometry

differential geometry

synthetic differential geometry

Contents

Idea

Given a linear differential operator (ordinary or partial) $P$ on a domain $M\subset\mathbb{R}^n$ or a manifold $M$, one can consider both the homogeneous equation $P f = 0$ and the nonhomogeneous equation of the form $P f = g$ where $g$ is a given nonhomogeneous term. If $g$ is a delta function and the boundary conditions are given, then the solution of the nonhomogenous equation

$P f = \delta$

is called the fundamental solution for $P$; alternative names like Green function and function of influence are also used. A particular solution of the nonhomogeneous equation for some other $g$ can be obtained by calculating the convolution with the fundamental solution. (Compare the fact that the delta function is the identity element for convolution.)

References

Revised on August 27, 2014 07:33:57 by Urs Schreiber (188.200.54.65)