Talk:Green's function

message: The use of the adjoint operator in the first equation of the article is wrong. It should be the operator itself, not the adjoint. Anyone care to defend the current version? Holmansf (talk) 21:04, 5 February 2014 (UTC)
thread_title:  Adjoint 

message: "we leave it to the reader to fill in the in-between steps" Is something contained in bad textbooks and bad lectures. Can anyone fill in the blanks here?
That sentence has been replaced by in-between steps --Wolfch 04:53, 5 September 2007 (UTC)
thread_title:  bad form 

message: shouldn't there be a minus sign in the Laplacian Green's function 1/|x-x'| (and in the expressionwith the charge density \rho)? maybe I'm confused at the moment, but I'm pretty sure there should be... If no one answers I'll change it. Dan Gluck 16:35, 10 October 2006 (UTC)
replies: text: If you are going to refer to the solution to electrostatic problems them you will need to insert the proper constants to make the units correct.  The sign depends on the choice of the Poisson problem that you are trying to solve.  In the case of electrostatics in the CGS system you have \nabla^2\Phi=-4\pi\rho where \Phi is the electrostatic potential electrostatic units (esu) per centimeter and \rho is the charge density in esu per centimeter.  However, in the MKS unit system the Poisson equation for electrostatics is given by \nabla^2\Phi=-\frac{\rho}{\epsilon_0} where \Phi is given in terms of Coulombs per meter, \rho is given in terms of Coulombs per cubic meter, and \epsilon_0 is the permittivity of free space given in terms of Farads per meter.  In both cases there is a negative sign in the Poisson equation and hence the solutions to these Poisson equations will be \Phi(x)=\int\frac{\rho(x^\prime)}{|x-x^\prime|}d^3x^\prime and \Phi(x)=\frac{1}{4\pi\epsilon_0}\int\frac{\rho(x^\prime)}{|x-x^\prime|}d^3x^\prime. So the (positive) sign is correct in both unit systems and the units are correct if working in the cgs unit system. HowiAuckland 00:20, 11 September 2007 (UTC)
thread_title: Minus sign missing

message: I think we all know the best way to learn stuff is Examples, Like actual problems. I think it would be great if we could get many examples from some text books or somewhere, to give readers (and hopless students) an idea of how to use the this function. this is a more General statement to the applications to Quantum Mechanics Section.24.25.211.241 04:48, 24 October 2006 (UTC)
thread_title: More Examples

message: I think there's a little confusion with regards to this section. Until someone disagrees (Andywall?), I'd like to replace it with a section explaining the connections and differences between this kind of Green('s) function — the inverse of a linear operator — and what's meant by the term in quantum/statistical field theory, where it essentially just means correlation function. Stevvers 03:15, 5 November 2006 (UTC)
replies: text: I have added a sentence in the introduction to mention this distinction and will add something to Correlation function (quantum field theory).Stevvers 01:16, 10 November 2006 (UTC)
thread_title:  Green's functions in condensed matter physics 

message: After looking at my math text, I think I just "got" this topic, at least for the case of -u=f. I would add my understanding to the page, but am not sure it is a complete understanding. Is this right?:
: The Green function, \mathcal{G}(x,y), can be thought of as the response of the solution variable, u, to a delta function, \delta(x-y). Because the system is linear, we can apply superposition: we can represent our given f as the sum of delta functions and expect our solution to be a sum of the responses to each of those delta functions. 
: For example, suppose we have a horizontal string with distributed and point weights on it and we want to know its deflection along its length. Assume small deflection (so that the problem is linear). We could hang a unit mass at every point, y, along the string and record the deflection as a function of x. This would be our \mathcal{G}(x,y). (Obviously we would expect a V shape centered on the point). The the solution will be
:: u(x) = \int_a^b \mathcal{G}(x,y) f(y)\,dy.
Is that about right? —Ben FrantzDale 00:16, 13 November 2006 (UTC)
replies: text:  Yes, that sounds exactly right. If you put this into the page, make sure you explicitly write the differential equation that the string satisfies. Stevvers 21:27, 15 November 2006 (UTC)
thread_title:  Intuition 

message: Currently "kernel" links to the "kernel (algebra)" article i.e. the article on the kernel of a homomorphism.  I don't see the connection between the two uses of the word "kernel," and I'm wondering if the link should be removed.
Atmd 01:33, 29 January 2007 (UTC)
replies: text: I am confused by you confusion :) So let me try to clear this up. A green function is associated to a linear operator L. Linear operators are group morphism (among other things) and thus we can define a kernel for them which is exactly what is done at Kernel_%28algebra%29#Linear_operators. Perhaps you could describe more clearly what exactly in the text confuses you so we can try to clear things up ? MathMartin 15:04, 21 March 2007 (UTC)

text: u(x) = \int G(x,y) f(y)dy
It is good for us physics folks to be made aware of the deeper connections between the tools we use.
Nickvence (talk) 19:33, 28 May 2008 (UTC)
thread_title:  kernel 

message: This very specific, (previously) uncategorized article doesn't stand on its own.  It needs to move to this article (or to Laplace's equation or Poisson's equation).  Lunch 03:50, 8 August 2007 (UTC)
thread_title: Merge from [[Green's function for the three variable Laplace equation]]

message: Is it possible to give a definition without distributions? This should be helpful for the general reader, I think. Ulner 15:19, 8 November 2007 (UTC)
thread_title:  Change of introduction? 

message: The 'example' section on this page refers to "demand-2","demand-3",and "demand-4" without any actual reference to what these entities are. Have I missed something here? Is it referring to the constraints? In any case I think it should be made more clear. Gazzo 144.32.126.14 (talk) 17:43, 3 June 2008 (UTC)
replies: text: I think the equation numbering got messed up. Does someone know how to fix this? mike40033 (talk) 05:12, 20 June 2008 (UTC)
thread_title:  example 

message: Throughout the article x and s are used as points on a manifold, so x-s, which is also used, makes no sense. Jjauregui (talk) 15:42, 16 September 2008 (UTC)
replies: text: Your absolutely right, for the moment I am changing the article remove this discrepancy. Thenub314 (talk) 18:52, 17 September 2008 (UTC)
thread_title:  Manifolds vs. R3 

message: The paragraph mentioning convolution is not quite right either. Presently it reads
"Convolving with a Green's function gives solutions to inhomogeneous differential-integral equations, most commonly a Sturm-Liouville problem. If G is the Green's function of an operator L, then the solution for u of the equation Lu = f is given by
:    u(x) = \int{ f(s) G(x,s) \, ds} .
This can be thought of as an expansion of f according to a Dirac delta function basis (projecting f over δ(x − s)) and a superposition of the solution on each projection. Such an integral is known as a Fredholm integral equation, the study of which constitutes Fredholm theory."
First of all, 'convolution' should be referenced. Second, it is appropriate here only if the operator L(x) is invariant under displacement - i.e. does not include an explicit reference to x. If so then
:    L(x) = L(x-s) => L(x-s)G(x,s) = \delta(x-s) => G(x,s) -> G(x-s)  \, .
and now u is a convolution of G with f:
:    u(x) = \int G(x-s)f(s) ds 
I'll change it to this on 3rd December if no one objects.
--Mike (talk) 22:04, 30 October 2008 (UTC)
thread_title:  Convolution 

message: There are a couple of somewhat confusing things in this article.  
First, why does the definition begin with "In addition..." In addition to what?
Second, why do we move from G(x,s) to G(x,x') ?  The notation G(x,s) is much less confusing considering that prime is overloaded and typically means differentiation.
Third, I know it's common to say that distributions are not properly functions, but are they not linear functionals over a vector subspace of infinitely differentiable functions? So technically, they are functions, just not functions from \mathbb{R}^m to \mathbb{R}^n.  I think this perhaps could be clarified since technically it's wrong.
Fourth, what is with the hats on the A and \sigma in the section on Green's Theorem for the Laplacian?  Is this standard in physics?  It looks a bit odd and cluttered, as if A was defined somewhere else and \hat {A} is some modification of A.  —Preceding unsigned comment added by 129.2.175.79 (talk) 17:09, 16 August 2009 (UTC)
thread_title: Notation And Clarity

message: The article starts well givng general statements, but in the Framework subsection, it restricts to second order diferential equations, is there any reason for that? It exists Green's function for higher order?
Paranoidhuman (talk) 00:11, 13 January 2011 (UTC)
thread_title: Second Order

message: This page has some major errors.  The definition of the Green's function given is only for self-adjoint operators (fortunately all of the examples have self-adjoint operators), and will not work for other operators.  It should be the adjoint of the operator that appears in the definition of the Green's function.  Also taking the operator outside the integral is not ok with the assumptions given.  This should be done by repeated "integration by parts" (or application of the divergence theorem) from which the definition of the adjoint operator arises.
I also can't see why there should be any linear operator without a Green's function (it might not be found analytically, but it will still exist and could be found numerically).
Source: Applied Partial Differential Equations: S. Howison, A. Lacey, A. Movchan, J. Ockendon.
I will try and make changes when I have timeAklamas (talk) 16:25, 21 January 2011 (UTC)
thread_title: Only for self-adjoint

message: The introduction ends with: "Green's functions are studied largely from the point of view of fundamental solutions instead." Instead of what? I find this unclear.  — Preceding unsigned comment added by 171.64.163.233 (talk) 02:41, 24 January 2012 (UTC)
thread_title:  Instead of what? 

message: I would propose adding the following to the end of the "Definitions and Uses" part:
--
i.e., Green's functions give the wavefield response in one place (x) as a result of an impulse-like source in another place (s).
— Preceding unsigned comment added by Frankdj100 (talk • contribs) 10:47, 26 March 2012 (UTC)
thread_title:  Add use in wave physics 

message: The article's structure seems messy to me. Could someone (native speaker) try to restructure the whole article? (And help me, where my table of Green's functions fits the best. I added it because I suppose that's the kind of information many people are looking for.) Thanks, --PassPort (talk) 00:29, 2 April 2013 (UTC)
replies: text: Oh thanks, I didn't notice that the references tag was missing since there was already a references section. Anyone comments on the article's structure? --PassPort (talk) 21:42, 2 April 2013 (UTC)
thread_title:  Messy structure 

message: I think that there is an error in the "Table of Green's Functions" section, unless I'm reading something wrong.  The claimed solution for the 2D Laplacian (\Delta _{2D} = \partial_x^2+\partial_y^2) appears to be wrong, according to this source, doing out myself, and basic logic e.g. (for physicists) it makes it look like an election and a proton will attract in 3D but repel in 2D.
Unless I hear someone with an argument not to, I'm going to change it to \frac{1}{2\pi} \ln \rho in a few days. TChapProctor (talk) 16:04, 15 July 2013 (UTC)
thread_title:  Another missing minus sign (or just an error) 

message: The Table of Green's Functions uses θ(t) in several places, but as far as I can see, this notation is not defined on this page. 172.56.9.253 (talk) 16:07, 24 October 2013 (UTC)
replies: text: Just above the table of Green's Functions it is defined as 'θ(t) is the Heaviside step function' Johnkeevil (talk) 17:09, 24 October 2022 (UTC)
thread_title:  What is θ(t)? 

message: The mistake is about halfway down the page, and this is it:
reduces to simply \phi(x) due to the defining property of the Dirac delta function and we have:
:\phi(x)=\int_V G(x,x') \rho(x')\ d^3x'+\int_S \left[\phi(x')\nabla' G(x,x')-G(x,x')\nabla'\phi(x')\right] \cdot d\hat\sigma'.
This must be incorrect, because here G(x,x')=1/( 4 \pi |x-x'| ) (or without the 4pi, it doesn't matter). But crucially, G must be positive here, because of the volume integral on the right-hand side of the equation. But, since G is positive, the surface integral term on the right hand side must instead be:
\int_S \left[ - \phi(x')\nabla' G(x,x')+G(x,x')\nabla'\phi(x')\right] \cdot d\hat\sigma'
(you can look on the wikipedia page for Helmholtz Decomposition for why it must be this way around). Anyway, this surface integral term needs a negative sign. I thought I would write this, in case anyone can think of an objection to changing it.— Preceding unsigned comment added by 151.228.183.81 (talk) 12:56, 16 March 2014 (UTC)
replies: text: I agree that the mistake has entered from a confusion about whether it's \nabla^2 \phi(x) = \rho(x) or \nabla^2 \phi(x) = - \rho(x). The article is also currently inconsistent about whether or not the Green's function should have a 1/(4 pi). I understand that in physics it can be normal to take the Green's function without the 4 pi, but then it's really the Green's function for 1/(4 pi) times the Laplacian. These choices of whether or not there is a 4 pi, and whether the Laplacian is positive or negative are not consistent through out the article ... .  Holmansf (talk) 10:38, 17 March 2014 (UTC)

text: Yes, I would endorse that change. Perhaps also a comment should be added explaining that there are differing conventions on whether or not Poissson's equation has a negative sign. Also, I think the Green's function later down in the same section should have the 4 pi.Holmansf (talk) 09:35, 19 March 2014 (UTC)
thread_title:  another missing minus sign 

message: In the subsection Further examples:
"Let L be d/dx"
Shouldn't that be d^2/dx^2 ?  — Preceding unsigned comment added by Ephecina (talk • contribs) 08:57, 4 July 2014 (UTC)
thread_title:  Laplace operator defined wrong? 

message: I feel that the Green's function for the 1 dimensional laplace operator should be mentioned. This is important for problems such finding the potential using Green's functions for an infinitely large capacitor.
I think it should be:
G = 1/2 |x| + Constant  — Preceding unsigned comment added by 84.148.212.147 (talk) 15:44, 25 September 2015 (UTC)
thread_title:  Table of Green's functions 

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thread_title:  External links modified 

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thread_title:  External links modified 

message: In other words, given a linear ordinary differential equation (ODE), L(solution) = source, one can first solve L(green) = δs, for each s, and realizing that, since the source is a sum of delta functions, the solution is a sum of Green's functions as well, by linearity of L.
Stripping away lower branches: 
One can solve [equation] and realizing that the solution is a sum.
I understand that jargon sometimes bends the rules, but this seems a little too bent, especially as we shouldn't be assuming that we're preaching to a choir of pre-existing jargon fu. — MaxEnt 16:27, 30 January 2019 (UTC)
replies: text: perhaps the fix looks like this: ... and in doing so, recognize that ... — MaxEnt 16:28, 30 January 2019 (UTC)
thread_title:  Strange sentence kernel 

message: What is a inhomogeneous linear differential operator? There may be non-linear ones, or there may be linear ones in an inhomogeneous linear differential equation...but the operator itself is not inhomogeneous. At least there is no mention of such thing in the Wiki article "Differential operator" and it is anything but intuitive what such an operator would look like. It would have to contain the inverse of the function I'm trying to find in a differential equation.
SomeGuy 14:33, 17 June 2020 (UTC)
thread_title: Inhomogeneous linear differential operator

message: The operator L occurs in this article in 4 different formatting styles:  L, , \operatorname{L}, and L. This makes things confusing as one might think they are not exactly the same. I might be missing simethingv, if there is a difference can someone clarify please? Tertionix (talk) 14:57, 15 July 2020 (UTC)
thread_title:  Usage of different formatting for L 

message: > Find the Green function for the following problem, whose Green's function number is X11:
What does X11 mean? there's no link or explanations 132.204.27.207 (talk) 16:10, 23 August 2023 (UTC)
thread_title:  What is X11 

message: The current formula in the table doesn't seem to make sense: the cross product of (x - x0) with itself will always yield the zero vector. 128.83.139.11 (talk) 18:03, 10 October 2023 (UTC)
thread_title:  Curl operator Green's function