nLab
fundamental solution

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Contents

Idea

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

Pf=δ P f = \delta

is called the fundamental solution for PP; alternative names like Green function and function of influence are also used. A particular solution of the nonhomogeneous equation for some other gg 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 September 9, 2013 21:10:56 by Urs Schreiber (89.204.155.147)