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GreenFunction
✖
GreenFunction

✖
GreenFunction[{ℒ[u[x]],ℬ[u[x]]},u,{x,xmin,xmax},y]

gives a Green's function for the linear differential operator ℒ with boundary conditions ℬ in the range xmin to xmax.

✖
GreenFunction[{ℒ[u[x1,x2,…]],ℬ[u[x1,x2,…]]},u,{x1,x2,…}∈Ω,{y1,y2,…}]

gives a Green's function for the linear partial differential operator ℒ over the region Ω.

✖
GreenFunction[{ℒ[u[x,t]],ℬ[u[x,t]]},u,{x,xmin,xmax},t,{y,τ}]

gives a Green's function for the linear time-dependent operator ℒ in the range xmin to xmax.

✖
GreenFunction[{ℒ[u[x1,…,t]],ℬ[u[x1,…,t]]},u,{x1,…}∈Ω,t,{y1,…,τ}]

gives a Green's function for the linear time-dependent operator ℒ 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

open allclose all

Basic Examples  (2)Summary of the most common use cases

Green's function for a boundary value problem:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-lbh73i
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-c804w3
Out[2]=2

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

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-brq4pg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-giizq6
Out[2]=2

Scope  (22)Survey of the scope of standard use cases

Basic Uses  (2)

Compute the Green's function for an ordinary differential operator:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-evdsw1
Out[1]=1

Obtain a pure function in the result by using u instead of u[x] in the second argument:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-ntntw
Out[2]=2

Compute the Green's function for a partial differential operator:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-0ivno
Out[1]=1

Obtain a pure function in the result by using u instead of u[x,t] in the second argument:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-bf8co8
Out[2]=2

Ordinary Differential Equations  (4)

Compute the Green's function for an initial value problem:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-gwrzn
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-0kz1
Out[2]=2

Compute the Green's function for a Dirichlet problem:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-d7agwb
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-fdm01a
Out[2]=2

Compute the Green's function for a Neumann problem:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-hlvxl4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-fupax0
Out[2]=2

Compute the Green's function for a Robin problem:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-29n5v
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-bbbb1u
Out[2]=2

Wave Equation  (4)

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

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-eykqlr
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-gpf3e4
Out[2]=2

Green's function for the wave operator with a Dirichlet condition on a half-line:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-fethe1
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-oiynwl
Out[2]=2

Green's function for the wave operator with a Neumann condition on a half-line:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-goi70s
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-e0l9mu
Out[2]=2

Green's function for the wave operator with a Dirichlet condition on an interval:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-jq8ryi
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-pf0kvr
Out[2]=2

Heat Equation  (5)

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

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-cxrkb4
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-x2doh
Out[2]=2

Green's function for the heat operator with a Dirichlet condition on a half-line:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-cj56hm
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-jlhlwh
Out[2]=2

Green's function for the heat operator with a Dirichlet condition on an interval:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-nyzgbs
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-cw5o4p
Out[2]=2

Green's function for the heat operator with a Neumann condition on an interval:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-if9uvq
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-iptpd0
Out[2]=2

Green's function for the heat operator in the plane:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-ij7l53
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-rhpcm
Out[2]=2

Laplace Equation  (4)

Green's function for the Laplacian in two dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-xxz76
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-lrqu6
Out[2]=2

Dirichlet problem for the Laplacian in a quadrant of the plane:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-5b8rt
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-x9uf3
Out[2]=2

Dirichlet problem for the Laplacian in a rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-cehao
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-jh1zwv
Out[2]=2

Green's function for the Laplacian in three dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-hubtl
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-niz48m
Out[2]=2

Helmholtz Equation  (3)

Green's function for the Helmholtz operator in two dimensions:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-8b5rg
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-bhay8l
Out[2]=2

Dirichlet problem for the Helmholtz operator in the upper half-plane:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-hpj55u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-b70rdy
Out[2]=2

Dirichlet problem for the Helmholtz operator in a rectangle:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-bb8m4u
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-naspd3
Out[2]=2

Options  (1)Common values & functionality for each option

Assumptions  (1)

Specify Assumptions on parameters in GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-ee42yq
Out[1]=1

Obtain a simpler result under the assumption that t>s:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-laq4wl
Out[2]=2

Applications  (9)Sample problems that can be solved with this function

Ordinary Differential Equations  (4)

Solve an initial value problem for an inhomogeneous differential equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-6kk6s
Out[1]=1

Define a forcing function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-c2y0ea

Perform a convolution of the Green's function with the forcing function:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-b5r517
Out[3]=3

Compare with the result given by DSolveValue:

In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-k5f2bm
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-lgosh8
Out[5]=5

Solve a Dirichlet problem for an inhomogeneous differential equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-3dhlb
Out[1]=1

Define a forcing function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-cymrui

Perform a convolution of the Green's function with the forcing function:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-cwluap
Out[3]=3

Compare with the result given by DSolveValue:

In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-cvdhyc
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-dvjee7
Out[5]=5

Solve a Neumann problem for an inhomogeneous differential equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-bbx52u
Out[1]=1

Define a forcing function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-u4zm8

Perform a convolution of the Green's function with the forcing function:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-jjxf0d
Out[3]=3

Compare with the result given by DSolveValue:

In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-46vnj
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-ejzhvb
Out[5]=5

Solve a Robin problem for an inhomogeneous differential equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-bmshm
Out[1]=1

Define a forcing function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-m1owgc

Perform a convolution of the Green's function with the forcing function:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-eryc8i
Out[3]=3

Compare with the result given by DSolveValue:

In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-b63wx0
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-bf5eko
Out[5]=5

Partial Differential Equations  (2)

Solve the inhomogeneous wave equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-blm0qr
Out[1]=1

Define the inhomogeneous term:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-o4imn

Solve the inhomogeneous equation using :

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-l9wik6
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-fwngy
Out[4]=4

Compare with the solution given by DSolveValue:

In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-i2776
Out[5]=5

Solve an initial value problem for the heat equation using GreenFunction:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-ewt1wa
Out[1]=1

Specify an initial value:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-chtfeb

Solve the initial value problem using :

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-g7wuj4
Out[3]=3
In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-eh8rhj
Out[4]=4

Compare with the solution given by DSolveValue:

In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-cg4wuz
Out[5]=5

Physics and Engineering  (3)

Compute the current i[t] in a circuit with a voltage source v[t] that is connected to a resistor R and an inductor L. The operator for this circuit is given by:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-pxw81

Diagram for the circuit:

Compute the Green's function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-bgd0w
Out[2]=2

Find the current for a given voltage source:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-cf2psp
In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-b4tjyp
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-ei2uj4
Out[5]=5

Compute the displacement u[x] for a string of length p and tension T that is fixed at the two ends and is subjected to a force per unit length of f[x]. The operator for the displacement is given by:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-c9lgm6

Force diagram:

Compute the Green's function:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-elg55v
Out[2]=2

Find the displacement for a given force:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-hai3f8
In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-c903mt
Out[4]=4
In[5]:=5
✖
https://wolfram.com/xid/0en0foyvm-cg7t1z
Out[5]=5

The impulse response of a continuous linear time-invariant system can be found by using the Green's function for the system with homogeneous initial conditions. Compute the impulse response for the system defined by:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-badslv

Green's function for the system with homogeneous initial conditions:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-d3vl9
Out[2]=2

Obtain the impulse response by setting s=0:

In[3]:=3
✖
https://wolfram.com/xid/0en0foyvm-nhpn97
Out[3]=3

Plot the impulse response:

In[4]:=4
✖
https://wolfram.com/xid/0en0foyvm-0tnq4
Out[4]=4

Properties & Relations  (2)Properties of the function, and connections to other functions

Compute a Green's function for a differential equation:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-juxiwg
Out[1]=1

Obtain the same result using DSolve:

In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-gtobhx
Out[2]=2

GreenFunction is related to OutputResponse and TransferFunctionModel:

In[1]:=1
✖
https://wolfram.com/xid/0en0foyvm-ekbrbj
Out[1]=1
In[2]:=2
✖
https://wolfram.com/xid/0en0foyvm-nhs57
Out[2]=2
Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.
✖
Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.

Text

Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.

✖
Wolfram Research (2016), GreenFunction, Wolfram Language function, https://reference.wolfram.com/language/ref/GreenFunction.html.

CMS

Wolfram Language. 2016. "GreenFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GreenFunction.html.

✖
Wolfram Language. 2016. "GreenFunction." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/GreenFunction.html.

APA

Wolfram Language. (2016). GreenFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GreenFunction.html

✖
Wolfram Language. (2016). GreenFunction. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/GreenFunction.html

BibTeX

@misc{reference.wolfram_2025_greenfunction, author="Wolfram Research", title="{GreenFunction}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GreenFunction.html}", note=[Accessed: 06-May-2025 ]}

✖
@misc{reference.wolfram_2025_greenfunction, author="Wolfram Research", title="{GreenFunction}", year="2016", howpublished="\url{https://reference.wolfram.com/language/ref/GreenFunction.html}", note=[Accessed: 06-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_greenfunction, organization={Wolfram Research}, title={GreenFunction}, year={2016}, url={https://reference.wolfram.com/language/ref/GreenFunction.html}, note=[Accessed: 06-May-2025 ]}

✖
@online{reference.wolfram_2025_greenfunction, organization={Wolfram Research}, title={GreenFunction}, year={2016}, url={https://reference.wolfram.com/language/ref/GreenFunction.html}, note=[Accessed: 06-May-2025 ]}