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.
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 all close allBasic Examples (2)
Green's function for a boundary value problem:
Green's function for the heat operator on the real line:
Scope (22)
Basic Uses (2)
Compute the Green's function for an ordinary differential operator:
Obtain a pure function in the result by using u instead of u[x] in the second argument:
Compute the Green's function for a partial differential operator:
Obtain a pure function in the result by using u instead of u[x,t] in the second argument:
Ordinary Differential Equations (4)
Compute the Green's function for an initial value problem:
Compute the Green's function for a Dirichlet problem:
Compute the Green's function for a Neumann problem:
Compute the Green's function for a Robin problem:
Wave Equation (4)
Green's function for the wave operator on the real line:
Green's function for the wave operator with a Dirichlet condition on a half-line:
Green's function for the wave operator with a Neumann condition on a half-line:
Green's function for the wave operator with a Dirichlet condition on an interval:
Heat Equation (5)
Green's function for the heat operator on the real line:
Green's function for the heat operator with a Dirichlet condition on a half-line:
Green's function for the heat operator with a Dirichlet condition on an interval:
Green's function for the heat operator with a Neumann condition on an interval:
Green's function for the heat operator in the plane:
Laplace Equation (4)
Helmholtz Equation (3)
Green's function for the Helmholtz operator in two dimensions:
Dirichlet problem for the Helmholtz operator in the upper half-plane:
Dirichlet problem for the Helmholtz operator in a rectangle:
Options (1)
Assumptions (1)
Specify Assumptions on parameters in GreenFunction :
Obtain a simpler result under the assumption that t>s:
Applications (9)
Ordinary Differential Equations (4)
Solve an initial value problem for an inhomogeneous differential equation using GreenFunction :
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue :
Solve a Dirichlet problem for an inhomogeneous differential equation using GreenFunction :
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue :
Solve a Neumann problem for an inhomogeneous differential equation using GreenFunction :
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue :
Solve a Robin problem for an inhomogeneous differential equation using GreenFunction :
Define a forcing function:
Perform a convolution of the Green's function with the forcing function:
Compare with the result given by DSolveValue :
Partial Differential Equations (2)
Solve the inhomogeneous wave equation using GreenFunction :
Define the inhomogeneous term:
Solve the inhomogeneous equation using :
Compare with the solution given by DSolveValue :
Solve an initial value problem for the heat equation using GreenFunction :
Specify an initial value:
Solve the initial value problem using :
Compare with the solution given by DSolveValue :
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:
Diagram for the circuit:
Compute the Green's function:
Find the current for a given voltage source:
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:
Force diagram:
Compute the Green's function:
Find the displacement for a given force:
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:
Green's function for the system with homogeneous initial conditions:
Obtain the impulse response by setting s=0:
Plot the impulse response:
Properties & Relations (2)
Compute a Green's function for a differential equation:
Obtain the same result using DSolve :
GreenFunction is related to OutputResponse and TransferFunctionModel :
Tech Notes
Related Guides
History
Text
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.
APA
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: 19-November-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: 19-November-2025]}