Green's function Definition and 213 Threads

In mathematics, a Green's function is the impulse response of an inhomogeneous linear differential operator defined on a domain with specified initial conditions or boundary conditions.
This means that if L is the linear differential operator, then

the Green's function G is the solution of the equation LG = δ, where δ is Dirac's delta function;
the solution of the initial-value problem Ly = f is the convolution (G * f ), where G is the Green's function.Through the superposition principle, 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.
Green's functions are named after the British mathematician George Green, who first developed the concept in the 1820s. In the modern study of linear partial differential equations, Green's functions are studied largely from the point of view of fundamental solutions instead.
Under 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 functions, even those that do not fit the mathematical definition. In quantum field theory, Green's functions take the roles of propagators.

View More On Wikipedia.org
Recent contents Top users
  1. W

    Green's Function: Solving a Differential Equation with a Green's Function

    Homework Statement I need to solve the following D.E for ##\phi(x,t)## $$[\frac{\partial}{\partial t} - D \frac{\partial ^2}{\partial x^2}]\phi (x,t) = f(x,t)$$ with the help of the following DE with a Green's function $$[\frac{\partial}{\partial t} - D \frac{\partial ^2}{\partial x^2}]G...
  2. W

    Green's functions: Logic behind this step

    Homework Statement Hi all, I came across these steps in my notes, relating to a step whereby, $$\hat{G} (k, t - t') = \int_{-\infty}^{\infty} e^{-ik(x - x')}G(x-x' , t-t')dx$$ and performing the following operation on ##\hat{G}## gives the following expression, $$[\frac{\partial}{\partial t}...
  3. amjad-sh

    Green's function in electrostatics

    Sorry it may seem that my question is a homework question but it is not since I have the solution of the problem. It is about obtaining Green function and using it to calculate the potential in space, provided the boundary conditions are satisfied. the questions are like below (It is a problem...
  4. DrClaude

    Green's function for the Helmholtz equation

    Homework Statement Show that $$ G(x,x') = \left\{ \begin{array}{ll} \frac{1}{2ik} e^{i k (x-x')} & x > x' \\ \frac{1}{2ik} e^{-i k (x-x')} & x < x' \end{array} \right. $$ is a Green's function for the 1D Helmholtz equation, i.e., $$ \left( \frac{\partial^2}{\partial x^2} + k^2 \right) G(x,x') =...
  5. G

    General solution for the heat equation of a 1-D circle

    Homework Statement Modify the initial conditions (for the diffusion equation of a circle) to have the initial conditions ## g(\theta)= \sum_{n=-\infty}^{\infty}d_{n}e^{2\pi in\theta} ## Using the method of Green's functions, and ## S(\theta,t)= \frac{1}{\sqrt{4\pi...
  6. A

    Classical Please recommend two textbookss about the Poisson equation and Green's function

    Please recommend two textbooks about Poisson equation, Green's function and Green's theorem for a theoretical physics student. One is easy to read so that I can have an overall understanding of the topics, another is mathematically rigorous and has a deep and modern exploration of these topics...
  7. redtree

    I Green's function and the evolution operator

    The Green's function is defined as follows, where ##\hat{L}_{\textbf{r}}## is a differential operator: \begin{equation} \begin{split} \hat{L}_{\textbf{r}} \hat{G}(\textbf{r},\textbf{r}_0)&=\delta(\textbf{r}-\textbf{r}_0) \end{split} \end{equation}However, I have seen the following...
  8. BiGyElLoWhAt

    Solving for the Green's Function and Using It to Solve an Differential Equation

    Homework Statement Find the green's function for y'' +4y' +3y = 0 with y(0)=y'(0)=0 and use it to solve y'' +4y' +3' =e^-2x Homework Equations ##y = \int_a^b G*f(z)dz## The Attempt at a Solution ##\lambda^2 + 4\lambda + 3 = 0 \to \lambda = -1,-3## ##G(x,z) = \left\{ \begin{array}{ll} Ae^{-x}...
  9. BiGyElLoWhAt

    How Do You Solve a Differential Equation Using Green's Function?

    Homework Statement Find the green's function for y'' +2y' +2y = 0 with boundary conditions y(0)=y'(0)=0 and use it to solve y'' + 2y' +2y = e^(-2x) Homework Equations ##y = \int_a^b G(x,z)f(z)dz## The Attempt at a Solution I'm going to rush through the first bit. If you need a specific step...
  10. BiGyElLoWhAt

    I A question about boundary conditions in Green's functions

    I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function. I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...
  11. BiGyElLoWhAt

    I A somewhat conceptual question about Green's functions

    I just did a problem for a final that required us to use a green's function to solve a diff eq. y'' +y/4 = sin(2x) I went through and solved it and got a really nasty looking thing, but I checked it in wolfram and it works out. Now, my question is this: After I got the solution from my greens...
  12. F

    A Physical interpretation of correlator

    Consider the 2-point correlator of a real scalar field ##\hat{\phi}(t,\mathbf{x})##, $$\langle\hat{\phi}(t,\mathbf{x})\hat{\phi}(t,\mathbf{y})\rangle$$ How does one interpret this quantity physically? Is it quantifying the probability amplitude for a particle to be created at space-time point...
  13. M

    A Green's function at boundaries

    The derivative of the Green's function is: i \dfrac{dG_{A,B}(t)}{dt} =\delta(t) \left< {[A,B]}\right>+G_{[A,H],B}(t) the Fourier transform is: \omega G_{A,B}(t)=\left< {[A,B]}\right>+G_{[A,H],B}(\omega) but this would require that the Green's function is 0 for t->inf. Why is that the case...
  14. AwesomeTrains

    Green's function differential equation

    Hello I'm doing some problems in QM scattering regarding the Green's function. Homework Statement Determine the differential equation of G(\vec{r},\vec{r}',\omega) Homework Equations I've been given the Fourier transform for the case where the Hamiltonian is time independent...
  15. Summer95

    I Confused about Finding the Green Function

    Suppose we have a differential equation with initial conditions ##y_{0}=y^{\prime}_{0}=0## and we need to solve it using a Green Function. Then we set up our differential equation with the right side "forcing function" as ##\delta(t^{\prime}-t)## (or with ##t^{\prime}## and ##t## switched I'm a...
  16. P

    Application of boundary conditions in determining the Green's function

    Homework Statement Find the Green's function $G(t,\tau)$ that satisfies $$\frac{\text{d}^2G(t,\tau)}{\text{d}t^2}+\alpha\frac{\text{d}G(t,\tau)}{\text{d}t}=\delta(t-\tau)$$ under the boundary conditions $$G(0,\tau)=0~~~\text{ and }~~~\frac{\text{d}G(t,\tau)}{\text{d}t}=0\big|_{t=0}$$ Then...
  17. Y

    Potential and charge on a plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  18. Y

    Potential and total charge on plane

    Homework Statement An infinite plane in z=0 is held in potential 0, except a square sheet -2a<x,y<2a which is held in potential V. Above it in z=d there is a grounded plane. Find: a) the potential in 0<z<d? b) the total induced charge on the z=0 plane. Homework Equations Green's function for a...
  19. A

    Understanding the method of Green's function

    I'm trying to understand the derivation for methods of Greens functions for PDEs but I can't get my head around some parts. I'm starting to feel comfortable with the method itself but I want to understand why it works. The thing I have problem with is quite crucial and it is the following: I...
  20. askhetan

    Understanding operators for Green's function derivation

    Dear All, I am trying to understand what operators actually mean when deriving the definition of green's function. Is this integral representation of an operator in the ##x-basis## correct ? ## D = <x|\int dx|D|x>## I am asking this because the identity operator for non-denumerable or...
  21. Einj

    Is There a Rigorous Method to Regularize Green's Functions in Coordinate Space?

    Hello everyone, I would like to know if there is a known, rigorous way to regularize a Green's function in coordinate space. In particular, it is known that the Green's function for a circle of radius R and source located at \vec x_0 is given by: $$ G(\vec x,\vec...
  22. evinda

    MHB Constructing the Green's Function for Non-trivial Solutions

    Hello! (Wave) Can the Green's function be contructed in the case when the homogeneous problem has non-trivial solutions? Justify your answer. Try to construct the Green's function for the following problem:$$y''+y= \cos x , y(0)=y(\pi)=0$$ The corresponding homogeneous problem has solutions...
  23. Einj

    Green's function in elliptic box

    Hello everyone! Does anyone know if there is a know expression for the Green's function for Poisson's equation that vanishes on an ellipse in 2 dimensions? I'm essentially looking for a solution to: $$ \nabla^2G(\vec x-\vec x_0)=\delta^2(\vec x-\vec x_0) $$ in 2 dimensions where $$G(\vec x-\vec...
  24. Andreol263

    Green's Function for Spherical Problem(Jackson)

    Hello guys, here's my question is how the book managed to solve this boundary value problem?? can anyone explain it to me in detail? thanks in advance.
  25. RUber

    Differentiating Integral with Green's function

    Hello, I am having trouble finding the proper justification for being able to pass the derivative through the integral in the following: ## u(x,y) = \frac{\partial}{\partial y} \int_0^\infty\int_{-\infty}^\infty f(x') K_0( \sqrt{ (x - x')^2 + (y-y')^2 } \, dx' dy' ## ##K_0## is the Modified...
  26. ShayanJ

    Path integral propagator as Schr's eq Green's function

    Its usually said that the propagator ## K(\mathbf x'',t;\mathbf x',t_0) ## that appears as an integral kernel in integrals in the path integral formulation of QM, is actually the Green's function for the Schrodinger equation and satisfies the equation below: ## \left[ -\frac{\hbar^2}{2m}...
  27. D

    Green's functions for translationally invariant systems

    As I understand it a Green's function ##G(x,y)## for a translationally invariant differential equation satisfies $$G(x+a,y+a)=G(x,y)\qquad\Rightarrow\qquad G(x,y)=G(x-y)$$ (where ##a## is an arbitrary constant shift.) My question is, given such a translationally invariant system, how does one...
  28. ognik

    Fourier Transforms, Green's function, Helmholtz

    Homework Statement I've gotten myself mixed up here , appreciate some insights ... Using Fourier Transforms, shows that Greens function satisfying the nonhomogeneous Helmholtz eqtn $$ \left(\nabla ^2 +k_0^2 \right) G(\vec{r_1},\vec{r_2})= -\delta (\vec{r_1} -\vec{r_2}) \:is\...
  29. T

    Green's Function with repulsive force

    Homework Statement Consider an object subject to a linear repulsive force, ##F = kx##. Show that the Green’s function for this object is given by: ##G(t-t^\prime)=\frac{1}{m\gamma}sinh(\gamma(t-t^\prime))## where ##\gamma=\sqrt{\frac{k}{m}}## Homework Equations ##sinhx=\frac{e^x+e^{-x}}{2}##...
  30. Pierre13

    Poisson's equation with Green's function for point charge

    Hello! I'm having a problem with the Green's function solution of the simplest case of Poisson's equation, namely a single test charge ##q## located at ##\boldsymbol r = \boldsymbol r'##. I've read the related posts on Poisson's equation via Green's function formalism, but they do not answer my...
  31. K

    Green's function and density of states

    Dear all, In his book chapter " Green’s Function Methods for Phonon Transport Through Nano-Contacts", Mingo arrives at the Green's function for the end atom of a one dimensional lattice chain (each atom modeled as a mass connected to neighbouring atoms through springs). He gives the green...
  32. L

    Retarded Green's Function for D'Alembertian

    Hey All, I recently posted this in another area but was suggested to put it here instead. Here is my original post:
  33. T

    Green's function with same time and spatial arguments

    Concerning green's function with the same time and spatial argument(i.e. ##G_0(x,t;x,t)##, mostly in QFT), I have the following question 1. Is green's function well defined at this point? 2. if green's function is well defined at this point, is it continuous here? 3. In quantum many body...
  34. A

    Green's function for resonant level

    I would really like some help for exercise 2 in the attached pdf. I know it's a lot asking you to read through all the pages but maybe you can skim them and catch the main points leading to exercise 2. What I don't understand is pretty basic. What is meant by the Green's function g(l,ikn)? In...
  35. T

    Relation between adiabatic approximation and imaginary time

    Regarding interacting green's function, I found two different description: 1. usually in QFT: <\Omega|T\{ABC\}|\Omega>=\lim\limits_{T \to \infty(1-i\epsilon)}\frac{<0|T\{A_IB_I U(-T,T)\}|0>}{<0|T\{U(-T,T)\}|0>} 2. usually in quantum many body systems...
  36. L

    Retarded Green's Function for D'Alembertian

    Homework Statement Hi all, I'm currently reviewing for a final and would like some help understanding a certain part of this particular problem: Determine the retarded Green's Function for the D'Alembertian operator ##D = \partial_s^2 - \Delta##, where ##\Delta \equiv \nabla \cdot \nabla## ...
  37. M

    Finding the potential by Green's function

    Homework Statement An infinite plane at z=0 is divided into two: the right half of it (y>0) is held at potential zero and the left half of it is held at potential \phi_0 . Over this surface lies a point charge q at (0,y_0,z_0) . Use Green's function to calculate the potential at z>0. The...
  38. Coffee_

    Is the Solution sin(t)H(t) for SHO a Particular Solution?

    So we have derived that for the differential equation: ##x(t)''+x(t)=\delta(t)## The solution is given by ##x=sin(t)H(t)## where ##H## is the Heaviside function. To find this we assumed that the system was in rest before ##t=0## and that position and velocity are continious. QUESTION: I am...
  39. M

    Green's function + method of images

    Hello, I'm trying to understand the application of Green's function to find the potential better. I apologize in advance if I start mixing things up a little. From what I understood and seen, we use this method (Green and method of images) in known symmetries (cylindrical/spherical/planar) and...
  40. Exp HP

    Assuming separability when solving for a Green's Function

    Edit: I have substantially edited this post from its original form, as I realize that it might have fallen under the label of "textbook-style questions". Really, the heart of my issue here is that, anywhere I look, I can't seem to find a clear description anywhere of the limitations of the...
  41. Gyges

    Integral Equations - Green's Function

    Homework Statement [/B] I'm trying to show that, \phi(x')=b\frac{\sin kx'}{k}+a\cos kx'+\int_{0}^{x'}\frac{\sin k(x-x')}{k} f(x)dx is the solution of, \frac{d^{2}}{dx'^{2}}\phi(x')+k^{2}\phi(x')=f(x')dx where 0 \leq x'<\infty. 2. Homework Equations N/A The Attempt at a Solution [/B]...
  42. M

    Finding Green's Function for u''(x) + u(x) = f(x) with Boundary Conditions

    Homework Statement Find Green's function for ##u''(x) + u(x) = f(x)## subject to ##u(0) = A## and ##u(\pi) + u'(\pi) = B##. Homework Equations No set equation. The Attempt at a Solution I begin by recognizing that green's function ##G## satisfies ##G''(x) + G(x) = \delta(x - x_0)## subject to...
  43. R

    Green's Function - modified operator

    Hi, I'm stuck with a question from one of my examples sheets from uni. The question is as follows: If G(x,x') is a greens function for the linear operator L, then what is the corresponding greens function for the linear operator L'=f(x)L, where f(x) =/=0? So I've started by writing...
  44. K

    A question about Green's function

    There is a second order nonhomogeneous equation of motion with nonzero initial condition given at ##t=-\infty##: ##D^2 y(x)=f(x)## with ##y(-\infty)=e^{-i x}## where I have used the shorthand notation ##D^2## for the full differential operator. Also I have the two solutions ##y_1(x)## and...
  45. RUber

    2D Green's Function - Bessel function equivalence

    Homework Statement This is not a homework problem per se, but I have been working on it for a few days, and cannot make the logical connection, so here it is: -- The problem is to show that ##\frac{1}{4\pi} \int_{-\infty}^{\infty} \frac{ e^{-\sqrt{\xi ^2 + \alpha^2 } |y-y'| + i \xi (x-x')...
  46. J

    Calculation and application of dyadic Green's function

    Hello everyone: I'm confusing with the construction and application of dyadic green's function. If we are in the ideal resonant system where only certain resonant mode is supported in this space (such as cavity), the Green's function can be constructed by the mode expansion that is: Gij(r,r')...
  47. M

    Inverse Fourier Transform of ##1/k^2## in ##\mathbb{R}^N ##

    Homework Statement This comes up in the context of Poisson's equation Solve for ##\mathbf{x} \in \mathbb{R}^n ## $$ \nabla^2 G(\mathbf{x}) = \delta(\mathbf{x})$$ Homework Equations $$\int_0^\pi \sin\theta e^{ikr \cos\theta}\mathop{dk} = \int_{-1}^1 e^{ikr \cos\theta}\mathop{d\cos \theta }$$...
  48. Glomerular

    Finding Green's function for diffusion equation

    Homework Statement Find the solution for: ({\partial{}_t}^2 -D \Delta^2)G(\vec{r},t;\vec{r}_o,t_o)=\delta(\vec{r}-\vec{r}_o)\delta(t-t_o) In two dimensions. Homework EquationsThe Attempt at a Solution Am I supposed to use bessel eqs? I'm kind of stuck in starting the problem :L
  49. genxium

    How is (d^3)r in Green's Function equivalent to volume element?

    Homework Statement This is part of the online tutorial I'm reading: http://farside.ph.utexas.edu/teaching/em/lectures/node49.html I'm so confused about the notation of Dirac Delta. It's said that 3-dimensional delta function is denoted as \delta^3(x, y, z)=\delta(x)\delta(y)\delta(z) in...
  50. G

    Using the Green's function for Maxwell

    Homework Statement I need to calculate, in D=4, the time component of the vector potential, A_{0}, given the equation (below) for A_{\mu} with the Green's function, also given below. The answer is given to be A^{0}=+\frac{q}{4\pi}\frac{1}{\mid\overline{x}\mid} Homework Equations A_{\mu}=\int...
Back
Top