Editing Green's function

{{short description|Impulse response of an inhomogeneous linear differential operator}}
{{about|the classical approach to Green's functions|a modern discussion|fundamental solution}}
{{short description|Non-linear second-order differential equation}}
{{Multiple issues|
{{more inline citations needed|date=March 2025}}
{{technical|date=May 2025}}
}}

[[File:Green's function animation.gif|alt=An animation that shows how Green's functions can be superposed to solve a differential equation subject to an arbitrary source.|thumb|360x360px|If one knows the solution <math display="inline">G(x,x')</math> to a differential equation subject to a point source <math display="inline">\hat{L}(x) G(x,x') = \delta(x-x')</math> and the differential operator <math display="inline">\hat{L}(x)</math> is linear, then one can superpose them to build the solution <math display="inline">u(x) = \int f(x') G(x,x') \, dx'</math> for a general source <math display="inline">\hat{L}(x) u(x) = f(x)</math>.]]
In [[mathematics]], a '''Green's function''' (or '''Green function'''<ref>{{Cite journal |last=Wright |first=M. C. M. |date=2006-10-01 |title=Green function or Green's function? |url=https://www.nature.com/articles/nphys411 |journal=Nature Physics |language=en |volume=2 |issue=10 |pages=646–646 |doi=10.1038/nphys411 |issn=1745-2473}}</ref>) is the [[impulse response]] of an [[inhomogeneous ordinary differential equation|inhomogeneous]] linear [[differential operator]] defined on a domain with specified initial conditions or boundary conditions.

This means that if <math>L</math> is a linear differential operator, then

* the Green's function <math>G</math> is the solution of the equation {{nowrap|<math>L G = \delta</math>,}} where <math>\delta</math> is [[Dirac delta function|Dirac's delta function]];
* the solution of the initial-value problem <math>L y = f</math> is the [[convolution]] {{nowrap|(<math>G \ast f</math>).}}

Through the [[superposition principle]], given a [[linear differential equation|linear ordinary differential equation]] (ODE), {{nowrap|<math>L y = f</math>,}} one can first solve {{nowrap|<math>L G = \delta_s</math>,}} for each {{mvar|s}}, and realizing that, since the source is a sum of [[delta function]]s, the solution is a sum of Green's functions as well, by linearity of {{mvar|L}}.

Green's functions are named after the British [[mathematician]] [[George Green (mathematician)|George Green]], who first developed the concept in the 1820s. In the modern study of linear [[partial differential equation]]s, Green's functions are studied largely from the point of view of [[fundamental solution]]s instead.

Under [[Green's function (many-body theory)|many-body theory]], the term is also used in [[physics]], specifically in [[quantum field theory]], [[aerodynamics]], [[aeroacoustics]], [[electrodynamics]], [[seismology]] and [[statistical field theory]], to refer to various types of [[correlation function (quantum field theory)|correlation functions]], even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of [[propagator]]s.

==Definition and uses==
A Green's function, {{math|''G''(''x'',''s'')}}, of a linear [[differential operator]] {{math|1=''L'' = ''L''(''x'')}} acting on [[distribution (mathematics)|distributions]] over a subset of the [[Euclidean space]] {{nowrap|<math>\R^n</math>,}} at a point {{mvar|s}}, is any solution of
{{NumBlk|1=|2=<math display="block">L\,G(x,s) = \delta(s-x) \, ,</math>|3={{EquationRef|1}}}}
where {{mvar|δ}} is the [[Dirac delta function]]. This property of a Green's function can be exploited to solve differential equations of the form
{{NumBlk|1=|2=<math display="block">L\,u(x) = f(x) \,.</math>|3={{EquationRef|2}}}}

If the [[kernel (linear operator)|kernel]] of {{math|''L''}} is non-trivial, then the Green's function is not unique. However, in practice, some combination of [[symmetry]], [[boundary condition]]s and/or other externally imposed criteria will give a unique Green's function. Green's functions may be categorized, by the type of boundary conditions satisfied, by a [[Green's function number]]. Also, Green's functions in general are [[Distribution (mathematics)|distributions]], not necessarily [[Function (mathematics)|functions]] of a real variable.

Green's functions are also useful tools in solving [[wave equation]]s and [[diffusion equation]]s. In [[quantum mechanics]], Green's function of the [[Hamiltonian mechanics|Hamiltonian]] is a key concept with important links to the concept of [[density of states]].

The Green's function as used in physics is usually defined with the opposite sign, instead. That is,
<math display="block">L G(x,s) = \delta(x-s)\,.</math>
This definition does not significantly change any of the properties of Green's function due to the evenness of the Dirac delta function.

If the operator is [[translation invariant]], that is, when <math>L</math> has [[constant coefficients]] with respect to {{mvar|x}}, then the Green's function can be taken to be a [[convolution kernel]], that is,
<math display="block">G(x,s) = G(x-s)\,.</math>
In this case, Green's function is the same as the impulse response of [[LTI system theory|linear time-invariant system theory]].

==Motivation==
{{See also |Spectral theory|Volterra integral equation}}
Loosely speaking, if such a function {{mvar|G}} can be found for the operator {{math|''L''}}, then, if we multiply {{EquationNote|1|equation&nbsp;1}} for the Green's function by {{math|''f''(''s'')}}, and then integrate with respect to {{mvar|s}}, we obtain,
<math display="block">\int LG(x,s)\,f(s) \, ds = \int \delta(x-s) \, f(s) \, ds = f(x)\,.</math>
Because the operator <math>L = L(x)</math> is linear and acts only on the variable {{mvar|x}} (and ''not'' on the variable of integration {{mvar|s}}), one may take the operator <math>L</math> outside of the integration, yielding
<math display="block">L\left(\int G(x,s)\,f(s) \,ds \right) = f(x)\,.</math>
This means that
{{NumBlk|1=|2=<math display="block">u(x) = \int G(x,s)\,f(s) \,ds</math>|3={{EquationRef|3}}}}
is a solution to the equation <math>L u(x) = f(x)\,.</math>

Thus, one may obtain the function {{math|''u''(''x'')}} through knowledge of the Green's function in {{EquationNote|1|equation&nbsp;1}} and the source term on the right-hand side in {{EquationNote|2|equation&nbsp;2}}. This process relies upon the linearity of the operator {{math|''L''}}.

In other words, the solution of {{EquationNote|2|equation&nbsp;2}}, {{math| ''u''(''x'')}}, can be determined by the integration given in {{EquationNote|3|equation&nbsp;3}}. Although {{math|''f''(''x'')}} is known, this integration cannot be performed unless {{mvar|G}} is also known. The problem now lies in finding the Green's function {{mvar|G}} that satisfies {{EquationNote|1|equation&nbsp;1}}. For this reason, the Green's function is also sometimes called the [[fundamental solution]] associated to the operator {{math|''L''}}.

Not every operator <math>L</math> admits a Green's function. A Green's function can also be thought of as a [[Inverse function#Left and right inverses|right inverse]] of {{math|''L''}}. Aside from the difficulties of finding a Green's function for a particular operator, the integral in {{EquationNote|3|equation&nbsp;3}} may be quite difficult to evaluate. However the method gives a theoretically exact result.

This can be thought of as an expansion of {{mvar|f}} according to a [[Dirac delta function]] basis (projecting {{mvar|f}} over {{nowrap|<math>\delta(x - s)</math>;}} and a superposition of the solution on each [[Projection (mathematics)|projection]]. Such an integral equation is known as a [[Fredholm integral equation]], the study of which constitutes [[Fredholm theory]].

==Green's functions for solving non-homogeneous boundary value problems==
The primary use of Green's functions in mathematics is to solve non-homogeneous [[boundary value problem]]s. In modern [[theoretical physics]], Green's functions are also usually used as [[propagator]]s in [[Feynman diagram]]s; the term ''Green's function'' is often further used for any [[correlation function (quantum field theory)|correlation function]].

===Framework===
Let <math>L</math> be the [[Sturm–Liouville theory|Sturm–Liouville]] operator, a linear differential operator of the form
<math display="block">L = \dfrac{d}{dx} \left[p(x) \dfrac{d}{dx}\right] + q(x)</math>
and let <math>\mathbf{D}</math> be the vector-valued [[boundary condition]]s operator
<math display="block">\mathbf{D} u = \begin{bmatrix}
\alpha_1 u'(0)    + \beta_1 u(0) \\
\alpha_2 u'(\ell) + \beta_2 u(\ell)
\end{bmatrix} \,.</math>

Let <math>f(x)</math> be a [[continuous function]] in {{nowrap|<math>[0,\ell]\,</math>.}} Further suppose that the problem
<math display="block">\begin{align}
 Lu &= f \\
 \mathbf{D}u &= \mathbf{0}
\end{align}</math>
is "regular", i.e., the only solution for <math>f(x) = 0</math> for all {{mvar|x}} is{{nowrap| <math>u(x) = 0</math>.}}{{efn|In technical jargon "regular" means that only the [[Trivial (mathematics)|trivial]] solution {{nowrap|(<math>u(x) = 0</math>)}} exists for the [[homogenization (mathematics)|homogeneous]] problem {{nowrap|(<math>f(x) = 0</math>).}}}}

===Theorem===
There is one and only one solution <math>u(x)</math> that satisfies
<math display="block"> \begin{align}
 Lu & = f \\
 \mathbf{D}u & = \mathbf{0}
\end{align}</math>
and it is given by
<math display="block">u(x) = \int_0^\ell f(s) \, G(x,s) \, ds\,,</math>
where <math>G(x,s)</math> is a Green's function satisfying the following conditions:
# <math>G(x,s)</math> is continuous in <math>x</math> and <math>s</math>.
# For {{nowrap|<math>x \ne s\,</math>,}} {{pad|4em}} {{nowrap|<math> L G(x,s) = 0</math>.}}
# For {{nowrap|<math>s \ne 0\,</math>,}} {{pad|4em}} {{nowrap|<math> \mathbf{D}G(x,s) = \mathbf{0}</math>.}}
# [[Derivative]] "jump": {{pad|0.5em}} {{nowrap|<math> G'(s_{0+},s) - G'(s_{0-},s) = 1 / p(s) \, </math>.}}
# Symmetry: {{pad|4em}} {{nowrap|<math> G(x,s) = G(s,x) \,</math>.}}

===Advanced and retarded Green's functions===
{{See also|Green's function (many-body theory)|propagator}}
Green's function is not necessarily unique since the addition of any solution of the homogeneous equation to one Green's function results in another Green's function. Therefore if the homogeneous equation has nontrivial solutions, multiple Green's functions exist. In some cases, it is possible to find one Green's function that is nonvanishing only for <math>s \leq x</math>, which is called a retarded Green's function, and another Green's function that is nonvanishing only for <math>s \geq x </math>, which is called an advanced Green's function. In such cases, any linear combination of the two Green's functions is also a valid Green's function. The terminology advanced and retarded is especially useful when the variable x corresponds to time. In such cases, the solution provided by the use of the retarded Green's function depends only on the past sources and is [[causal]] whereas the solution provided by the use of the advanced Green's function depends only on the future sources and is acausal. In these problems, it is often the case that the causal solution is the physically important one. The use of advanced and retarded Green's function is especially common for the analysis of solutions of the [[inhomogeneous electromagnetic wave equation]].

==Finding Green's functions==

===Units===
While it does not uniquely fix the form the Green's function will take, performing a [[dimensional analysis]] to find the units a Green's function must have is an important sanity check on any Green's function found through other means. A quick examination of the defining equation,
<math display="block"> L G(x, s) = \delta(x - s), </math>
shows that the units of <math>G</math> depend not only on the units of <math>L</math> but also on the number and units of the space of which the position vectors <math>x</math> and <math>s</math> are elements. This leads to the relationship:
<math display="block"> [[G]] = [[L]]^{-1} [[d x]]^{-1}, </math>
where <math>[[G]]</math> is defined as, "the physical units of {{nowrap|<math>G</math>"}}{{Explain | reason=What does this mean? This is ungoogleable and without a description this section is meaningless. | date = October 2024}}, and <math>dx</math> is the [[volume element]] of the space (or [[spacetime]]).

For example, if <math>L = \partial_t^2</math> and time is the only variable then:
<math display="block">\begin{align}[]
[[L]] &= [[\text{time}]]^{-2}, \\[1ex]
[[dx]] &= [[\text{time}]],\ \text{and} \\[1ex]
[[G]] &= [[\text{time}]].
\end{align}</math>
If {{nowrap|<math>L = \square = \tfrac{1}{c^2}\partial_t^2 - \nabla^2</math>,}} the [[d'Alembert operator]], and space has 3 dimensions then:
<math display="block">\begin{align}[]
[[L]] &= [[\text{length}]]^{-2}, \\[1ex]
[[dx]] &= [[\text{time}]] [[\text{length}]]^3,\ \text{and} \\[1ex]
[[G]] &= [[\text{time}]]^{-1} [[\text{length}]]^{-1}.
\end{align}</math>

===Eigenvalue expansions===
If a [[differential operator]] {{math|''L''}} admits a set of [[eigenvectors]] {{math|Ψ<sub>''n''</sub>(''x'')}} (i.e., a set of functions {{math|Ψ<sub>''n''</sub>}} and scalars {{math|''λ''<sub>''n''</sub>}} such that {{math|1=''L''Ψ<sub>''n''</sub> = ''λ''<sub>''n''</sub> Ψ<sub>''n''</sub>}}&thinsp;) that is complete, then it is possible to construct a Green's function from these eigenvectors and [[eigenvalues]].

"Complete" means that the set of functions {{math|{{mset|Ψ<sub>''n''</sub>}}}} satisfies the following [[completeness relation]],
<math display="block">\delta(x-x') = \sum_{n=0}^\infty \Psi_n^\dagger(x') \Psi_n(x).</math>

Then the following holds, 
{{Equation box 1
|indent =:
|equation = <math>G(x, x') = \sum_{n=0}^\infty \dfrac{\Psi_n^\dagger(x') \Psi_n(x)}{\lambda_n},</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}
where <math>\dagger</math> represents complex conjugation.

Applying the operator {{math|''L''}} to each side of this equation results in the completeness relation, which was assumed.

The general study of Green's function written in the above form, and its relationship to the [[function space]]s formed by the eigenvectors, is known as [[Fredholm theory]].

There are several other methods for finding Green's functions, including the [[method of images]], [[separation of variables]], and [[Laplace transform]]s.<ref>{{cite book |first1=K.D. |last1=Cole |first2=J.V. |last2=Beck |first3=A. |last3=Haji-Sheikh |first4=B. |last4=Litkouhi |chapter=Methods for obtaining Green's functions |title=Heat Conduction Using Green's Functions |publisher=Taylor and Francis |year=2011 |pages=101–148 |isbn=978-1-4398-1354-6}}</ref>

===Combining Green's functions===
If the differential operator <math>L</math> can be factored as <math>L = L_1 L_2</math> then the Green's function of <math>L</math> can be constructed from the Green's functions for <math>L_1</math> and {{nowrap|<math>L_2</math>:}}
<math display="block"> G(x, s) = \int G_2(x, s_1) \, G_1(s_1, s) \, ds_1. </math>
The above identity follows immediately from taking <math>G(x, s)</math> to be the representation of the right operator inverse of {{nowrap|<math>L</math>,}} analogous to how for the [[Invertible matrix#Other properties|invertible linear operator]] {{nowrap|<math>C</math>,}} defined by {{nowrap|<math>C = (AB)^{-1} = B^{-1} A^{-1}</math>,}} is represented by its matrix elements {{nowrap|<math>C_{i,j}</math>.}}

A further identity follows for differential operators that are scalar polynomials of the derivative, {{nowrap|<math>L = P_N(\partial_x)</math>.}} The [[fundamental theorem of algebra]], combined with the fact that <math>\partial_x</math> [[Commutative property|commutes with itself]], guarantees that the polynomial can be factored, putting <math>L</math> in the form:
<math display="block"> L = \prod_{i=1}^N \left(\partial_x - z_i\right),</math>
where <math>z_i</math> are the zeros of {{nowrap|<math>P_N(z)</math>.}} Taking the [[Fourier transform]] of <math>L G(x, s) = \delta(x - s)</math> with respect to both <math>x</math> and <math>s</math> gives:
<math display="block"> \widehat{G}(k_x, k_s) = \frac{\delta(k_x - k_s)}{\prod_{i=1}^N (ik_x - z_i)}. </math>
The fraction can then be split into a sum using a [[partial fraction decomposition]] before Fourier transforming back to <math>x</math> and <math>s</math> space. This process yields identities that relate integrals of Green's functions and sums of the same. For example, if <math>L = \left(\partial_x + \gamma\right) \left(\partial_x + \alpha\right)^2</math> then one form for its Green's function is:
<math display="block"> \begin{align}
G(x, s) & = \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\gamma(x-s)} - \frac{1}{\left(\gamma - \alpha\right)^2}\Theta(x-s) e^{-\alpha(x-s)} + \frac{1}{\gamma-\alpha} \Theta(x - s) \left(x - s\right) e^{-\alpha(x-s)} \\[1ex]
& = \int \Theta(x - s_1) \left(x - s_1\right) e^{-\alpha(x-s_1)} \Theta(s_1 - s) e^{-\gamma (s_1 - s)} \, ds_1.
\end{align} </math>
While the example presented is tractable analytically, it illustrates a process that works when the integral is not trivial (for example, when <math>\nabla^2</math> is the operator in the polynomial).

===Table of Green's functions===
{{Disputed section|Dimensional inconsistencies and wrong scaling|date=April 2025}}

The following table gives an overview of Green's functions of frequently appearing differential operators, where {{nowrap|<math display="inline"> r = \sqrt{x^2 + y^2 + z^2}</math>,}} {{nowrap|<math display="inline"> \rho = \sqrt{x^2 + y^2}</math>,}} <math display="inline"> \Theta(t)</math> is the [[Heaviside step function]], <math display="inline"> J_\nu(z)</math> is a [[Bessel function]], <math display="inline"> I_\nu(z)</math> is a [[modified Bessel function of the first kind]], and <math display="inline"> K_\nu(z)</math> is a [[modified Bessel function of the second kind]].<ref>some examples taken from {{cite book | last = Schulz | first = Hermann | title = Physik mit Bleistift: das analytische Handwerkszeug des Naturwissenschaftlers | date = 2001 | publisher = Deutsch | isbn = 978-3-8171-1661-4 | edition = 4. Aufl | location = Frankfurt am Main}}</ref> Where time ({{mvar|t}}) appears in the first column, the retarded (causal) Green's function is listed.

{| class="wikitable"
|-
! Differential operator {{math|''L''}} !! Green's function {{mvar|G}} !! Example of application
|-
| <math>\partial_t^{n+1}</math> || <math>\frac{t^n}{n!} \Theta(t)</math> ||
|-
| <math>\partial_t + \gamma </math> || <math>\Theta(t) e^{-\gamma t}</math> ||
|-
| <math>\left(\partial_t + \gamma \right)^2</math> || <math>\Theta(t)t e^{-\gamma t}</math> ||
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma < \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} \, \frac{\sin(\omega t)}{\omega}</math> &nbsp; with &nbsp; <math>\omega=\sqrt{\omega_0^2-\gamma^2}</math>|| [[Harmonic oscillator#Damped harmonic oscillator|1D underdamped harmonic oscillator]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma > \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} \, \frac{\sinh(\omega t)}{\omega}</math> &nbsp; with &nbsp; <math>\omega = \sqrt{\gamma^2-\omega_0^2}</math>|| 1D overdamped harmonic oscillator
|-
| <math>\partial_t^2 + 2\gamma\partial_t + \omega_0^2</math> where <math> \gamma = \omega_0 </math> || <math>\Theta(t) e^{-\gamma t} t</math> || 1D critically damped harmonic oscillator
|-
| 1D Laplace operator <math> \frac {d^2}{dx^2} </math>
| <math> \left(x - s\right) \Theta(x-s) + x\alpha (s) + \beta(s) </math>
| 1D Poisson equation
|-
| [[Laplace operator#Two dimensions|2D Laplace operator]] <math>\nabla^2_{\text{2D}} = \partial_x^2 + \partial_y^2</math> || <math>\frac{1}{2 \pi}\ln \rho </math> &nbsp; with &nbsp; <math>\rho=\sqrt{x^2+y^2}</math>|| 2D Poisson equation
|-
| [[Laplace operator#Three dimensions| 3D Laplace operator]] <math>\nabla^2_{\text{3D}} = \partial_x^2 + \partial_y^2 + \partial_z^2</math> || <math>-\frac{1}{4 \pi r}</math> &nbsp; with &nbsp; <math> r = \sqrt{x^2 + y^2 + z^2} </math> || [[Poisson equation]]
|-
| [[Helmholtz equation|Helmholtz operator]] <math>\nabla^2_{\text{3D}} + k^2</math> || <math>\frac{-e^{-ikr}}{4 \pi r} = i \sqrt{\frac{k}{32 \pi r}} H^{(2)}_{1/2}(kr) = i \frac{k}{4\pi} \, h^{(2)}_{0}(kr)</math> {{br}} where <math>H_\alpha^{(2)}</math> is the [[Hankel function of the second kind]], and <math>h_0^{(2)}</math> is the [[spherical Hankel function of the second kind]] || stationary 3D [[Schrödinger equation]] for [[free particle]]
|-
|Divergence operator <math>\nabla \cdot \mathbf{v}</math>
|<math>\frac{1}{4 \pi} \frac{\mathbf{x} - \mathbf{x}_0}{\left\|\mathbf{x} - \mathbf{x}_0\right\|^3} </math>
|
|-
| <math>\nabla^2 - k^2</math> in <math>n</math> dimensions || <math>- \left(2\pi\right)^{-n/2} \left(\frac{k}{r}\right)^{n/2-1} K_{n/2-1}(kr)</math>|| [[Yukawa potential]], [[Propagator#Feynman propagator|Feynman propagator]], [[Screened Poisson equation]]
|-
| <math>\partial_t^2 - c^2\partial_x^2</math>|| <math>\frac{1}{2c} \Theta(ct - x)</math>|| 1D [[wave equation]]
|-
| <math>\partial_t^2 - c^2\,\nabla^2_{\text{2D}}</math>|| <math>\frac{\Theta(ct - \rho)}{2\pi c\sqrt{c^2t^2 - \rho^2}}</math>|| 2D [[wave equation]]
|-
| [[D'Alembert operator]] <math>\square = \frac{1}{c^2}\partial_t^2 - \nabla^2_{\text{3D}}</math>|| <math>\frac{1}{4 \pi r} \delta\left(t-\frac{r}{c}\right)</math>|| 3D [[wave equation]]
|-
| <math>\partial_t - k\partial_x^2</math>|| <math>\left(\frac{1}{4\pi kt}\right)^{1/2} \Theta(t) e^{-x^2/4kt}</math>|| 1D [[diffusion]]
|-
| <math>\partial_t - k\,\nabla^2_{\text{2D}}</math>|| <math>\left(\frac{1}{4\pi kt}\right) \Theta(t) e^{-\rho^2/4kt}</math>|| 2D [[diffusion]]
|-
| <math>\partial_t - k\,\nabla^2_{\text{3D}}</math>|| <math>\left(\frac{1}{4\pi kt}\right)^{3/2} \Theta(t) e^{-r^2/4kt}</math>|| 3D [[diffusion]]
|-
| <math>\frac{1}{c^2}\partial_t^2 - \partial_x^2+\mu^2</math>|| <math>\begin{align}
&\tfrac{1}{2} \left(1-\sin{\mu ct}\right) \left[\delta(ct-x) + \delta(ct+x)\right] \\[0.5ex]
&+\tfrac{1}{2} \mu \Theta(ct - |x|) J_0(\mu u)
\end{align} </math> {{br}} with <math> u = \sqrt{c^2 t^2 - x^2}</math>|| 1D [[Klein–Gordon equation]]
|-
| <math>\frac{1}{c^2}\partial_t^2 - \nabla^2_{\text{2D}}+\mu^2</math>|| <math>\begin{align}
&\frac{\delta(ct-\rho)}{4\pi\rho} \left(1 + \cos(\mu ct)\right) \\[0.5ex]
&+ \frac{\mu^2\Theta(ct - \rho)}{4\pi} \operatorname{sinc}(\mu u)
\end{align}</math> {{br}} with <math> u=\sqrt{c^2t^2-\rho^2} </math>|| 2D [[Klein–Gordon equation]]
|-
| <math>\square + \mu^2</math>|| <math>\frac{1}{4\pi r} \delta{\left(t - \frac{r}{c}\right)} + \frac{\mu c}{4\pi u} \Theta(ct - r) J_1{\left(\mu u\right)}</math> &nbsp; with &nbsp; <math> u = \sqrt{c^2t^2-r^2}</math>|| 3D [[Klein–Gordon equation]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\partial_x^2</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{2} \left[ \delta(ct - x) + \delta(ct + x) \right] \\[0.5ex]
&+ \frac{e^{-\gamma t}}{2} \Theta(ct - |x|) \left(k I_0(k u) + \frac{\gamma t}{u} I_1(k u)\right)
\end{align}</math> {{br}} with &nbsp; <math> u=\sqrt{c^2t^2-x^2}</math> and <math>k = \gamma / c </math>|| [[telegrapher's equation]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\,\nabla^2_{\text{2D}}</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{4\pi\rho} \delta(ct-\rho) \left(1 + e^{-\gamma t} + 3\gamma t\right) \\
&+ \frac{e^{-\gamma t}}{4\pi u^2} \Theta(ct - \rho) \left(\frac{k u^2 - 3 c t}{c u} \sinh\left(k u\right) + 3\gamma t \cosh\left(k u\right)\right)
\end{align}</math> {{br}} with &nbsp; <math> u = \sqrt{c^2 t^2 - \rho^2}</math> and <math>k = \gamma / c</math>|| 2D [[relativistic heat conduction]]
|-
| <math>\partial_t^2 + 2\gamma\partial_t - c^2\,\nabla^2_{\text{3D}}</math>
| <math>\begin{align}
&\frac{e^{-\gamma t}}{20\pi r^2} \delta(ct - r) \left(8 - 3e^{-\gamma t} + 2\gamma t + 4\gamma^2 t^2\right) \\[0.5ex]
&+ \frac{k e^{-\gamma t}}{20 \pi u} \Theta(ct - r) \left(k I_1(k u) + \frac{4 \gamma t}{u} I_2(k u)\right)
\end{align}</math> {{br}} with &nbsp; <math> u = \sqrt{c^2 t^2 - r^2}</math> and <math>k = \gamma / c</math>|| 3D [[relativistic heat conduction]]
|}

==Green's functions for the Laplacian==
Green's functions for linear differential operators involving the [[Laplacian]] may be readily put to use using the second of [[Green's identities]].

To derive Green's theorem, begin with the [[divergence theorem]] (otherwise known as [[Gauss's theorem]]),
<math display="block">\int_V \nabla \cdot \mathbf A\, dV = \int_S \mathbf A \cdot d\hat\boldsymbol\sigma \,.</math>

Let <math>\mathbf A = \varphi\,\nabla\psi - \psi\,\nabla\varphi</math> and substitute into Gauss' law.

Compute <math>\nabla\cdot\mathbf A</math> and apply the product rule for the ∇ operator,
<math display="block">\begin{align}
 \nabla\cdot\mathbf A &= \nabla\cdot \left(\varphi\,\nabla\psi \;-\; \psi\,\nabla\varphi\right)\\
 &= (\nabla\varphi)\cdot(\nabla\psi) \;+\; \varphi\,\nabla^2\psi \;-\; (\nabla\varphi)\cdot(\nabla\psi) \;-\; \psi\nabla^2\varphi\\
 &= \varphi\,\nabla^2\psi \;-\; \psi\,\nabla^2\varphi.
\end{align}</math>

Plugging this into the divergence theorem produces [[Green's theorem]],
<math display="block">\int_V \left(\varphi\,\nabla^2\psi-\psi\,\nabla^2\varphi\right) dV = \int_S \left(\varphi\,\nabla\psi-\psi\nabla\,\varphi\right) \cdot d\hat\boldsymbol\sigma.</math>

Suppose that the linear differential operator {{mvar|L}} is the [[Laplacian]], ∇<sup>2</sup>, and that there is a Green's function {{mvar|G}} for the Laplacian. The defining property of the Green's function still holds,
<math display="block">L G(\mathbf{x},\mathbf{x}') = \nabla^2 G(\mathbf{x},\mathbf{x}') = \delta(\mathbf{x}-\mathbf{x}').</math>

Let <math>\psi=G</math> in Green's second identity, see [[Green's identities]]. Then,
<math display="block">\int_V \left[ \varphi(\mathbf{x}') \delta(\mathbf{x}-\mathbf{x}') - G(\mathbf{x},\mathbf{x}') \, {\nabla'}^2\,\varphi(\mathbf{x}')\right] d^3\mathbf{x}' = \int_S \left[\varphi(\mathbf{x}')\,{\nabla'} G(\mathbf{x},\mathbf{x}') - G(\mathbf{x},\mathbf{x}') \, {\nabla'}\varphi(\mathbf{x}')\right] \cdot d\hat\boldsymbol\sigma'.</math>

Using this expression, it is possible to solve [[Laplace's equation]] {{math|1=∇<sup>2</sup>''φ''('''x''') = 0}} or [[Poisson's equation]] {{math|1=∇<sup>2</sup>''φ''('''x''') = −''ρ''('''x''')}}, subject to either [[Neumann boundary condition|Neumann]] or [[Dirichlet boundary condition|Dirichlet]] boundary conditions. In other words, we can solve for {{math|''φ''('''x''')}} everywhere inside a volume where either (1) the value of {{math|''φ''('''x''')}} is specified on the bounding surface of the volume (Dirichlet boundary conditions), or (2) the normal derivative of {{math|''φ''('''x''')}} is specified on the bounding surface (Neumann boundary conditions).

Suppose the problem is to solve for {{math|''φ''('''x''')}} inside the region. Then the integral
<math display="block">\int_V \varphi(\mathbf{x}') \, \delta(\mathbf{x}-\mathbf{x}') \, d^3\mathbf{x}'</math>
reduces to simply {{math|''φ''('''x''')}} due to the defining property of the [[Dirac delta function]] and we have
<math display="block">\varphi(\mathbf{x})
= -\int_V G(\mathbf{x},\mathbf{x}') \, \rho(\mathbf{x}')\, d^3\mathbf{x}'
+ \int_S \left[\varphi(\mathbf{x}') \, \nabla' G(\mathbf{x},\mathbf{x}') - G(\mathbf{x},\mathbf{x}') \, \nabla'\varphi(\mathbf{x}')\right] \cdot d\hat\boldsymbol\sigma'.</math>

This form expresses the well-known property of [[harmonic function]]s, that ''if the value or normal derivative is known on a bounding surface, then the value of the function inside the volume is known everywhere''.

In [[electrostatics]], {{math|''φ''('''x''')}} is interpreted as the [[electric potential]], {{math|''ρ''('''x''')}} as [[electric charge]] [[density]], and the normal derivative <math>\nabla\varphi(\mathbf{x}')\cdot d\hat\boldsymbol\sigma'</math> as the normal component of the electric field.

If the problem is to solve a Dirichlet boundary value problem, the Green's function should be chosen such that {{math|''G''(''x'',''x''&prime;)}} vanishes when either {{mvar|x}} or {{mvar|x′}} is on the bounding surface. Thus only one of the two terms in the [[surface integral]] remains. If the problem is to solve a Neumann boundary value problem, it might seem logical to choose Green's function so that its normal derivative vanishes on the bounding surface. However, application of Gauss's theorem to the differential equation defining the Green's function yields
<math display="block">\int_S \nabla' G(\mathbf{x},\mathbf{x}') \cdot d\hat\boldsymbol\sigma'
= \int_V \nabla'^2 G(\mathbf{x},\mathbf{x}') \, d^3\mathbf{x}'
= \int_V \delta (\mathbf{x}-\mathbf{x}')\, d^3\mathbf{x}'
= 1 \,,</math>
meaning the normal derivative of ''G''('''x''','''x'''&prime;) cannot vanish on the surface, because it must integrate to 1 on the surface.<ref>{{cite book |last1=Jackson |first1=John David |title=Classical Electrodynamics |date=1998-08-14 |publisher=John Wiley & Sons |pages=39}}</ref>

The simplest form the normal derivative can take is that of a constant, namely {{math|1/''S''}}, where {{math|''S''}} is the surface area of the surface. The surface term in the solution becomes
<math display="block">\int_S \varphi(\mathbf{x}') \, \nabla' G(\mathbf{x},\mathbf{x}') \cdot d\hat\boldsymbol\sigma' = \langle\varphi\rangle_S </math>
where <math>\langle\varphi\rangle_S </math> is the average value of the potential on the surface. This number is not known in general, but is often unimportant, as the goal is often to obtain the electric field given by the gradient of the potential, rather than the potential itself.

With no boundary conditions, the Green's function for the Laplacian ([[Green's function for the three-variable Laplace equation]]) is
<math display="block">G(\mathbf{x},\mathbf{x}') = -\frac{1}{4 \pi \left|\mathbf{x}-\mathbf{x}'\right|}.</math>

Supposing that the bounding surface goes out to infinity and plugging in this expression for the Green's function finally yields the standard expression for electric potential in terms of electric charge density as

{{Equation box 1
|indent =:
|equation = <math>\varphi(\mathbf{x}) = \int_V \dfrac{\rho(\mathbf{x}')}{4 \pi \varepsilon \left|\mathbf{x} - \mathbf{x}'\right|} \, d^3\mathbf{x}' \, .</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

{{further|Poisson's equation}}

==Example==
Find the Green function for the following problem, whose [[Green's function number]] is X11:
<math display="block">\begin{align}
 Lu & = u'' + k^2 u = f(x) \\
 u(0)& = 0, \quad u{\left(\tfrac{\pi}{2k}\right)} = 0.
\end{align}</math>

'''First step:''' The Green's function for the linear operator at hand is defined as the solution to
{{NumBlk||<math display="block">G''(x,s) + k^2 G(x,s) = \delta(x-s). </math>|Eq. {{EquationRef|<nowiki>*</nowiki>}}}}

If <math>x\ne s</math>, then the delta function gives zero, and the general solution is
<math display="block">G(x,s)=c_1 \cos kx+c_2 \sin kx.</math>

For <math>x < s</math>, the boundary condition at <math>x=0</math> implies
<math display="block">G(0,s)=c_1 \cdot 1+c_2 \cdot 0=0, \quad c_1 = 0</math>
if <math>x < s</math> and <math>s \ne \tfrac{\pi}{2k}</math>.

For <math>x>s</math>, the boundary condition at <math>x = \tfrac{\pi}{2k}</math> implies
<math display="block">G{\left(\tfrac{\pi}{2k},s\right)} = c_3 \cdot 0+c_4 \cdot 1=0, \quad c_4 = 0 </math>

The equation of <math>G(0,s) = 0</math> is skipped for similar reasons.

To summarize the results thus far:
<math display="block">G(x,s) = \begin{cases}
 c_2 \sin kx, & \text{for } x < s, \\[0.4ex]
 c_3 \cos kx, & \text{for } s < x.
\end{cases}</math>

'''Second step:''' The next task is to determine <math>c_2</math> and {{Nowrap|<math>c_3</math>.}}

Ensuring continuity in the Green's function at <math>x = s</math> implies
<math display="block">c_2 \sin ks=c_3 \cos ks</math>

One can ensure proper discontinuity in the first derivative by integrating the defining differential equation (i.e., {{EquationNote|*|Eq. *}}) from <math>x=s-\varepsilon</math> to <math>x=s+\varepsilon</math> and taking the limit as <math>\varepsilon</math> goes to zero. Note that we only integrate the second derivative as the remaining term will be continuous by construction.
<math display="block">c_3 \cdot (-k \sin ks)-c_2 \cdot (k \cos ks)=1</math>

The two (dis)continuity equations can be solved for <math>c_2</math> and <math>c_3</math> to obtain
<math display="block">c_2 = -\frac{\cos ks}{k} \quad;\quad c_3 = -\frac{\sin ks}{k}</math>

So Green's function for this problem is:
<math display="block">G(x,s) = \begin{cases}
 -\frac{\cos ks}{k} \sin kx, & x<s, \\
 -\frac{\sin ks}{k} \cos kx, & s<x.
\end{cases}</math>

==Further examples==
* Let {{math|1=''n'' = 1}} and let the subset be all of {{math|'''R'''}}. Let {{mvar|L}} be <math display="inline">\frac{d}{dx}</math>. Then, the [[Heaviside step function]] {{math|Θ(''x'' − ''x''<sub>0</sub>)}} is a Green's function of {{math|''L''}} at {{math|''x''<sub>0</sub>}}.
* Let {{math|1=''n'' = 2}} and let the subset be the quarter-plane {{math|1={(''x'', ''y'') : ''x'', ''y'' ≥ 0}<nowiki/>}} and {{mvar|L}} be the [[Laplacian]]. Also, assume a [[Dirichlet boundary condition]] is imposed at {{math|1=''x'' = 0}} and a [[Neumann boundary condition]] is imposed at {{math|1=''y'' = 0}}. Then the X10Y20 Green's function is <math display="block"> \begin{align}
G(x, y, x_0, y_0) =
\dfrac{1}{2\pi}
&\left[\ln\sqrt{\left(x-x_0\right)^2+\left(y-y_0\right)^2} - \ln\sqrt{\left(x+x_0\right)^2 + \left(y-y_0\right)^2} \right. \\[5pt]
&\left. {} + \ln\sqrt{\left(x-x_0\right)^2 + \left(y+y_0\right)^2}- \ln\sqrt{\left(x+x_0\right)^2 + \left(y+y_0\right)^2} \, \right].
\end{align}</math>
* Let <math> a < x < b </math>, and all three are elements of the real numbers. Then, for any function <math>f:\mathbb{R}\to\mathbb{R}</math> with an <math>n</math>-th derivative that is integrable over the interval <math>[a, b]</math>: <math display="block">
f(x) = \sum_{m=0}^{n-1} \frac{(x - a)^m}{m!} \left[ \frac{d^m f}{d x^m} \right]_{x=a} + \int_a^b \left[\frac{(x - s)^{n-1}}{(n-1)!} \Theta(x - s)\right] \left[ \frac{d^n f}{dx^n} \right]_{x=s} ds \,.</math> The Green's function in the above equation, <math>G(x,s) = \frac{(x - s)^{n-1}}{(n-1)!} \Theta(x - s)</math>, is not unique. How is the equation modified if <math>g(x-s)</math> is added to <math>G(x,s)</math>, where <math>g(x)</math> satisfies <math display="inline">\frac{d^n g}{d x^n} = 0</math> for all <math>x \in [a, b]</math> (for example, <math>g(x) = -x/2</math> with {{nowrap|<math>n=2</math>)?}} Also, compare the above equation to the form of a [[Taylor series]] centered at <math>x = a</math>.

==See also==
{{div col}}
* [[Bessel potential]]
* [[Discrete Laplace operator#Discrete Green's function|Discrete Green's functions]] – defined on graphs and grids
* [[Impulse response]] – the analog of a Green's function in signal processing
* [[Transfer function]]
* [[Fundamental solution]]
* [[Green's function (many-body theory)|Green's function in many-body theory]]
* [[Correlation function (quantum field theory)|Correlation function]]
* [[Propagator]]
* [[Green's identities]]
* [[Parametrix]]
* [[Volterra integral equation]]
* [[Resolvent formalism]]
* [[Keldysh formalism]]
* [[Spectral theory]]
* [[Multiscale Green's function]]
{{div col end}}

==Footnotes==
{{notelist|1}}

==References==
{{reflist}}
===Cited works===
{{refbegin}}
* {{cite book |first=S.S. |last=Bayin |year=2006 |title=Mathematical Methods in Science and Engineering |publisher=Wiley |at=Chapters 18 and 19}}
* {{cite book |last=Eyges |first=Leonard |title=The Classical Electromagnetic Field |publisher=Dover Publications |location=New York, NY |year=1972 |isbn=0-486-63947-9}}<br>''Chapter&nbsp;5 contains a very readable account of using Green's functions to solve boundary value problems in electrostatics.''
* {{cite book |first1=A.D. |last1=Polyanin |first2=V.F. |last2=Zaitsev |title=Handbook of Exact Solutions for Ordinary Differential Equations |edition=2nd |publisher=Chapman & Hall/CRC Press |location=Boca Raton, FL |year=2003 |isbn=1-58488-297-2}}
* {{Cite book |last=Barton |first=Gabriel |title=Elements of Green's functions and propagation: potentials, diffusion, and waves |date=1989 |publisher=Clarendon Press ; Oxford University Press |isbn=978-0-19-851988-1 |series=Oxford science publications |location=Oxford : New York}}<br>''Textbook on Green's function with worked-out steps.''
* {{cite book |first=A.D. |last=Polyanin |title=Handbook of Linear Partial Differential Equations for Engineers and Scientists |publisher=Chapman & Hall/CRC Press |location=Boca Raton, FL |year=2002 |isbn=1-58488-299-9}}
* {{cite book |last1=Mathews |first1=Jon |last2=Walker |first2=Robert L. |year=1970 |title=Mathematical methods of physics |edition=2nd |location=New York |publisher=W. A. Benjamin |isbn=0-8053-7002-1}}
* {{cite book |first=G.B. |last=Folland |author-link=Gerald Folland |title=Fourier Analysis and its Applications |publisher=Wadsworth and Brooks/Cole |series=Mathematics Series}}
* {{cite book |last=Green |first=G |title=An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism |location=Nottingham, England |publisher=T. Wheelhouse |year=1828 |at=[https://books.google.com/books?id=GwYXAAAAYAAJ&pg=PA10 pages 10-12]}}
* {{cite book |first1=M. |last1=Faryad and |author2-link=Akhlesh Lakhtakia |first2=A. |last2=Lakhtakia |title=Infinite-Space Dyadic Green Functions in Electromagnetism |location=London, UK / San Rafael, CA |publisher=IoP Science (UK) / Morgan and Claypool (US) |year=2018 |bibcode=2018idgf.book.....F |isbn=978-1-68174-557-2 |url=http://iopscience.iop.org/book/978-1-6817-4557-2}}
*{{Cite book |last=Şeremet |first=V. D. |title=Handbook of Green's functions and matrices |date=2003 |publisher=WIT Press |isbn=978-1-85312-933-9 |location=Southampton}}
{{refend}}

==External links==
* {{springer|title=Green function|id=p/g045090}}
* {{MathWorld | urlname=GreensFunction | title=Green's Function}}
* {{PlanetMath | urlname=GreensFunctionForDifferentialOperator | title=Green's function for differential operator}}
* {{PlanetMath | urlname=GreensFunction | title=Green's function}}
* {{PlanetMath | urlname=GreenFunctionsAndConformalMapping | title=Green functions and conformal mapping}}
* [http://nanohub.org/resources/1877 Introduction to the Keldysh Nonequilibrium Green Function Technique] by A. P. Jauho
* [https://www.engr.unl.edu/~glibrary/home/index.html Green's Function Library]
* [https://web.archive.org/web/20110905015156/http://www.boulder.nist.gov/div853/greenfn/tutorial.html Tutorial on Green's functions]
* [http://www.ntu.edu.sg/home/mwtang/bemsite.htm Boundary Element Method (for some idea on how Green's functions may be used with the boundary element method for solving potential problems numerically)] {{Webarchive|url=https://web.archive.org/web/20120207225549/http://www.ntu.edu.sg/home/mwtang/bemsite.htm |date=2012-02-07 }}
* [http://en.citizendium.org/wiki/Green%27s_function At Citizendium]
* [https://archive.today/20130101181958/http://academicearth.com/lectures/delta-function-and-greens-function MIT video lecture on Green's function]
* {{cite web|last=Bowley|first=Roger|title=George Green & Green's Functions|url=http://www.sixtysymbols.com/videos/georgegreen.htm|work=Sixty Symbols|publisher=[[Brady Haran]] for the [[University of Nottingham]]}}

{{Authority control}}

[[Category:Differential equations]]
[[Category:Generalized functions]]
[[Category:Equations of physics]]
[[Category:Mathematical physics]]
[[Category:Schwartz distributions]]