GreenFunction
✖
GreenFunction
gives a Green's function for the linear differential operator ℒ with boundary conditions ℬ in the range xmin to xmax.
gives a Green's function for the linear partial differential operator ℒ over the region Ω.
gives a Green's function for the linear time-dependent operator ℒ in the range xmin to xmax.
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
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.

Examples
open allclose allBasic Examples (2)Summary of the most common use cases
Green's function for a boundary value problem:

https://wolfram.com/xid/0en0foyvm-lbh73i


https://wolfram.com/xid/0en0foyvm-c804w3

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

https://wolfram.com/xid/0en0foyvm-brq4pg


https://wolfram.com/xid/0en0foyvm-giizq6

Scope (22)Survey of the scope of standard use cases
Basic Uses (2)
Compute the Green's function for an ordinary differential operator:

https://wolfram.com/xid/0en0foyvm-evdsw1

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

https://wolfram.com/xid/0en0foyvm-ntntw

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

https://wolfram.com/xid/0en0foyvm-0ivno

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

https://wolfram.com/xid/0en0foyvm-bf8co8

Ordinary Differential Equations (4)
Compute the Green's function for an initial value problem:

https://wolfram.com/xid/0en0foyvm-gwrzn


https://wolfram.com/xid/0en0foyvm-0kz1

Compute the Green's function for a Dirichlet problem:

https://wolfram.com/xid/0en0foyvm-d7agwb


https://wolfram.com/xid/0en0foyvm-fdm01a

Compute the Green's function for a Neumann problem:

https://wolfram.com/xid/0en0foyvm-hlvxl4


https://wolfram.com/xid/0en0foyvm-fupax0

Compute the Green's function for a Robin problem:

https://wolfram.com/xid/0en0foyvm-29n5v


https://wolfram.com/xid/0en0foyvm-bbbb1u

Wave Equation (4)
Green's function for the wave operator on the real line:

https://wolfram.com/xid/0en0foyvm-eykqlr


https://wolfram.com/xid/0en0foyvm-gpf3e4

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

https://wolfram.com/xid/0en0foyvm-fethe1


https://wolfram.com/xid/0en0foyvm-oiynwl

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

https://wolfram.com/xid/0en0foyvm-goi70s


https://wolfram.com/xid/0en0foyvm-e0l9mu

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

https://wolfram.com/xid/0en0foyvm-jq8ryi


https://wolfram.com/xid/0en0foyvm-pf0kvr

Heat Equation (5)
Green's function for the heat operator on the real line:

https://wolfram.com/xid/0en0foyvm-cxrkb4


https://wolfram.com/xid/0en0foyvm-x2doh

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

https://wolfram.com/xid/0en0foyvm-cj56hm


https://wolfram.com/xid/0en0foyvm-jlhlwh

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

https://wolfram.com/xid/0en0foyvm-nyzgbs


https://wolfram.com/xid/0en0foyvm-cw5o4p

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

https://wolfram.com/xid/0en0foyvm-if9uvq


https://wolfram.com/xid/0en0foyvm-iptpd0

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

https://wolfram.com/xid/0en0foyvm-ij7l53


https://wolfram.com/xid/0en0foyvm-rhpcm

Laplace Equation (4)
Green's function for the Laplacian in two dimensions:

https://wolfram.com/xid/0en0foyvm-xxz76


https://wolfram.com/xid/0en0foyvm-lrqu6

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

https://wolfram.com/xid/0en0foyvm-5b8rt


https://wolfram.com/xid/0en0foyvm-x9uf3

Dirichlet problem for the Laplacian in a rectangle:

https://wolfram.com/xid/0en0foyvm-cehao


https://wolfram.com/xid/0en0foyvm-jh1zwv

Green's function for the Laplacian in three dimensions:

https://wolfram.com/xid/0en0foyvm-hubtl


https://wolfram.com/xid/0en0foyvm-niz48m

Helmholtz Equation (3)
Green's function for the Helmholtz operator in two dimensions:

https://wolfram.com/xid/0en0foyvm-8b5rg


https://wolfram.com/xid/0en0foyvm-bhay8l

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

https://wolfram.com/xid/0en0foyvm-hpj55u


https://wolfram.com/xid/0en0foyvm-b70rdy

Dirichlet problem for the Helmholtz operator in a rectangle:

https://wolfram.com/xid/0en0foyvm-bb8m4u


https://wolfram.com/xid/0en0foyvm-naspd3

Options (1)Common values & functionality for each option
Assumptions (1)
Specify Assumptions on parameters in GreenFunction:

https://wolfram.com/xid/0en0foyvm-ee42yq

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

https://wolfram.com/xid/0en0foyvm-laq4wl

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:

https://wolfram.com/xid/0en0foyvm-6kk6s


https://wolfram.com/xid/0en0foyvm-c2y0ea
Perform a convolution of the Green's function with the forcing function:

https://wolfram.com/xid/0en0foyvm-b5r517

Compare with the result given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-k5f2bm


https://wolfram.com/xid/0en0foyvm-lgosh8

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

https://wolfram.com/xid/0en0foyvm-3dhlb


https://wolfram.com/xid/0en0foyvm-cymrui
Perform a convolution of the Green's function with the forcing function:

https://wolfram.com/xid/0en0foyvm-cwluap

Compare with the result given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-cvdhyc


https://wolfram.com/xid/0en0foyvm-dvjee7

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

https://wolfram.com/xid/0en0foyvm-bbx52u


https://wolfram.com/xid/0en0foyvm-u4zm8
Perform a convolution of the Green's function with the forcing function:

https://wolfram.com/xid/0en0foyvm-jjxf0d

Compare with the result given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-46vnj


https://wolfram.com/xid/0en0foyvm-ejzhvb

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

https://wolfram.com/xid/0en0foyvm-bmshm


https://wolfram.com/xid/0en0foyvm-m1owgc
Perform a convolution of the Green's function with the forcing function:

https://wolfram.com/xid/0en0foyvm-eryc8i

Compare with the result given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-b63wx0


https://wolfram.com/xid/0en0foyvm-bf5eko

Partial Differential Equations (2)
Solve the inhomogeneous wave equation using GreenFunction:

https://wolfram.com/xid/0en0foyvm-blm0qr

Define the inhomogeneous term:

https://wolfram.com/xid/0en0foyvm-o4imn
Solve the inhomogeneous equation using :

https://wolfram.com/xid/0en0foyvm-l9wik6


https://wolfram.com/xid/0en0foyvm-fwngy

Compare with the solution given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-i2776

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

https://wolfram.com/xid/0en0foyvm-ewt1wa


https://wolfram.com/xid/0en0foyvm-chtfeb
Solve the initial value problem using :

https://wolfram.com/xid/0en0foyvm-g7wuj4


https://wolfram.com/xid/0en0foyvm-eh8rhj

Compare with the solution given by DSolveValue:

https://wolfram.com/xid/0en0foyvm-cg4wuz

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:

https://wolfram.com/xid/0en0foyvm-pxw81


https://wolfram.com/xid/0en0foyvm-bgd0w

Find the current for a given voltage source:

https://wolfram.com/xid/0en0foyvm-cf2psp

https://wolfram.com/xid/0en0foyvm-b4tjyp


https://wolfram.com/xid/0en0foyvm-ei2uj4

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:

https://wolfram.com/xid/0en0foyvm-c9lgm6


https://wolfram.com/xid/0en0foyvm-elg55v

Find the displacement for a given force:

https://wolfram.com/xid/0en0foyvm-hai3f8

https://wolfram.com/xid/0en0foyvm-c903mt


https://wolfram.com/xid/0en0foyvm-cg7t1z

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:

https://wolfram.com/xid/0en0foyvm-badslv
Green's function for the system with homogeneous initial conditions:

https://wolfram.com/xid/0en0foyvm-d3vl9

Obtain the impulse response by setting s=0:

https://wolfram.com/xid/0en0foyvm-nhpn97


https://wolfram.com/xid/0en0foyvm-0tnq4

Properties & Relations (2)Properties of the function, and connections to other functions
Compute a Green's function for a differential equation:

https://wolfram.com/xid/0en0foyvm-juxiwg

Obtain the same result using DSolve:

https://wolfram.com/xid/0en0foyvm-gtobhx

GreenFunction is related to OutputResponse and TransferFunctionModel:

https://wolfram.com/xid/0en0foyvm-ekbrbj


https://wolfram.com/xid/0en0foyvm-nhs57

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
]}
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
]}