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 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 L(G(x;y))=TemplateBox[{{x, -, y}}, DiracDeltaSeq] 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 L(G(x,t;y,tau))=TemplateBox[{{x, -, y}}, DiracDeltaSeq]TemplateBox[{{t, -, tau}}, DiracDeltaSeq] 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.

Examples

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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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Scope  (22)

Options  (1)

Applications  (9)

Properties & Relations  (2)

See Also

DSolve  DSolveValue  DiracDelta  Convolve  Wronskian  DEigensystem  TransferFunctionModel  OutputResponse

Introduced in 2016
(10.4)