# Wolfram Language & System 11.0 (2016)|Legacy Documentation

This is documentation for an earlier version of the Wolfram Language.
BUILT-IN WOLFRAM LANGUAGE SYMBOL

# GreenFunction

GreenFunction[{[u[x]],[u[x]]},u,{x,xmin,xmax},y]
gives a Green's function for the linear differential operator with boundary conditions in the range xmin to xmax.

GreenFunction[{[u[x1,x2,]],[u[x1,x2,]]},u,{x1,x2,}Ω,{y1,y2,}]
gives a Green's function for the linear partial differential operator over the region Ω.

GreenFunction[{[u[x,t]],[u[x,t]]},u,{x,xmin,xmax},t,{y,τ}]
gives a Green's function for the linear time-dependent operator in the range xmin to xmax.

GreenFunction[{[u[x1,,t]],[u[x1,,t]]},u,{x1,}Ω,t,{y1,,τ}]]
gives a Green's function for the linear time-dependent operator over the region Ω.

## Details and OptionsDetails and Options

• GreenFunction represents the response of a system to an impulsive DiracDelta driving function.
• GreenFunction for a differential operator is defined to be a solution of that satisfies the given homogeneous boundary conditions .
• A particular solution of with homogeneous boundary conditions can be obtained by performing a convolution integral .
• GreenFunction for a time-dependent differential operator is defined to be a solution of that satisfies the given homogeneous boundary conditions .
• A particular solution of with homogeneous boundary conditions can be obtained by performing a convolution integral .
• The Green's functions for classical PDEs have characteristic geometrical properties:
• is given as an expression in and if the dependent variable is of the form , and as a pure function with formal parameters and if the dependent variable is of the form instead of .  »
• The region Ω can be anything for which RegionQ[Ω] is True.
• All the necessary initial and boundary conditions for ODEs must be specified in .
• Boundary conditions for PDEs must be specified using DirichletCondition or NeumannValue in .
• Assumptions on parameters may be specified using the Assumptions option.

## ExamplesExamplesopen allclose all

### Basic Examples  (2)Basic Examples  (2)

Green's function for a boundary value problem:

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Green's function for the heat operator on the real line:

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