Green's function Definition and 185 Threads

Boundary Value Problem + Green's Function Consider the BVP y''+4y=e^x y(0)=0 y'(1)=0 Find the Green's function for this problem. I am completely lost can someone help me out?

I have a serious blind-spot with mathematics (but I keep trying) Can someone help me with this. I have a relation A = \mu_{0}/4\pi\int J/r \ dVol Which (apparently!) can be written \nabla^{2} A = - \mu_{0} J I know that \nabla^{2} A = 1/r \ \delta^{2} ( r A ) / \delta r^{2} which is...

Consider the BVP y''+4y=f(x) (0\leqx\leq1) y(0)=0 y'(1)=0 Find the Green's function (two-sided) for this problem. Working: So firstly, I let y(x)=Asin2x+Bcos2x Then using the boundary conditions, Asin(2.0)+Bcos(2.0)=0 => B=0 y'(x)=2Acos(2x)-2Asin(2x) y'(0)=2A=0...

Hello! Something about N-point Green's function in QFT really troubles me... In the path-integral formalism,why will we introduce the N-point Green's function? I mean is it enough because we have calculated the 2-point green's function. And in the canonical formalism, it seems we can finish...

Is it possible to define unambiguously retarded and advanced Green's function in spacetime without timelike Killing vector. Most often e.g. retarded Green function G_R(t,\vec{x},t',\vec{x}') is defined to be 0 unless t'<t but maybe one can express this condition using only casual structure...

Is it true that there always exists Green's function for Dirichlet boundary problem. I mean a function G(r,r') which fullfils the following conditions: div (\epsilon grad G(r,r')) =- \delta(r,r') inside volume V and G(r,r') is 0 on boundary of V. If V is whole space there exists obvious...

The differential equation is as follows: [d/dx^2 + k^2 - tau * dirac_delta(x-x') ] * G(x,x') = dirac_delta(x-x') where tau is a complex valued scattering strength, and assuming scattering waves at infinity. The problem asks to derive the solution to this equation. I've looked over...

Homework Statement If a hollow spherical shell of radius a is held at potential \Phi(a, \theta ', \phi '), then the potential at an arbitrary point is given by, \Phi(r, \theta, \phi)=\frac{1}{4 \pi} \oint \Phi(a, \theta ', \phi ') \frac{\delta G(r, r')}{\delta n '}dS' where G(r...

Is there a physical unit related to the Green's function of the wave equation? In particular, let \nabla^2 P -\frac{1}{c^2}\frac{\partial^2 P}{\partial t^2} = f(t) where P is pressure in Pa. Since the Green's function solves the PDE when f(t) is the delta function, the Green's function G...

Say we have a 3D function, p(x,y,z) and we define it in terms of another function f(x,y,z) via, \nabla ^2 p = f. I know that if we are working in R^3 space (with no boundaries) we can say that, p= \frac{-1}{4\pi}\iiint \limits_R \frac{f(x',y',z')}{\sqrt{(x-x')^2 +(y-y')^2+(z-z')^2}} dx'...

Matrix "Green's function" Hi. If you have a differential equation \mathcal L y=f where \mathcal L is some linear differential operator, then you can find a particular solution using the Green's function technique. It is then said that the Green's function is kind of the inverse to \mathcal...

Hi, I have been trying to find the (causal) Green's function of \frac{\partial^2 \phi}{\partial t^2} + \frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + m^2 \phi = 0. What would be a good way to approach this? I have initial values for t=0, so I use...

Hello Forum, given a input=delta located at time t=0, the system will respond generating a function h(t). If the delta is instead located at t=t0 (delayed by tau), the system will respond with a function g(t)=h(t-tau), just a shifted version of the response for the delta a t=0... If...

Here are some pages of Arfken's “Mathematical Methods for Physicists ” I don't How to work out the Green's function! Can anyone explain (9.174)and(9.175) for me ? I'm hoping for your help, Thank you !

Use Green's Functions to solve: \frac{d^{2}y}{dx^{2}} + y = cosec x Subject to the boundary conditions: y\left(0\right) = y\left(\frac{\pi}{2}\right) = 0 Attempt: \frac{d^{2}G\left(x,z\right)}{dx^{2}} + G\left(x,z\right) = \delta\left(x-z\right) For x\neq z the RHS is zero...

By taking the Fourier transform of the fundamental Helmholtz equation (\nabla^2+k^2)G(\vec{x})=-\delta(\vec{x}), one finds that G(\vec{x})=\frac{e^{ikr}}{r} and \tilde{G}(\vec{\xi})=\frac{1}{k^2-\xi^2}. However, I can't figure out how to directly confirm that this Fourier...

This comes from Jackson's Classical Electrodynamics 3rd edition, page 613. He finds the Green's function for the covariant form of the wave equation as: D(z) = -1/(2\pi)^{4}\int d^{4}k\: \frac{e^{-ik\cdot z}}{k\cdot k} Where z = x - x' the 4 vector difference, k\cdot z = k_0z_0 -...

I can't figure out how to use the Green's function approach rigorously, i.e., taking into account the fact that the Dirac Delta function is not a function on the reals. Suppose we have Laplace's Equation: \nabla^2 \phi(\vec{x})=f(\vec{x}) The solution, for "well-behaved" f(\vec{x}) is...

Homework Statement I'm asked to calculate Green's function's real and imaginary parts. The expression for the given Green's function is: g00=[1-(1-4t2(z-E0)-2)1/2]/2t2(z-E0)-1 (1) Where, z is the complex variable: z= E+iO+ (2) Homework Equations...

Hey Guys; I'm solving PDE's with the use of Green's function where all the boundary conditions are homogeneous. However, how do you solve ones in which we have non-homogeneous b.c's. In case it helps, the particular PDE I'm looking at is: y'' = -x^2 y(0) + y'(0) = 4, y'(1)= 2...

Homework Statement Consider a potential problem in the half space z>=0 with Dirichlet boundary conditions on the plane z=0. If the potential on the plane z=0 is specified to be V inside a circle of radius a centered at the origin, and Phi=0 outside that circle, show that along the axis of...

Hello! I have problem with my homework, but what I'm going to ask you is not homework problem so I hope it is OK I'm writing it here :) I need to find Green's function for differential operator L=a\frac{d^2}{dx^2}+b\frac{d}{dx}+c i.e. find solution for differential equation equation...

i find that most books on green's function are burdened with too much formalism i am now reading the book by Rammer, which deals with non-equilibrium physics. The formalism is so lengthy and so confusing. You have to strive hard to remenber the various green's functions, and only to find...

Griffiths develops an intelgral equation for Scrödinger equation in his QM book. As doing so, he requires Green's function for Helmholtz equation (k^2 + \nabla^2) G( \mathbf r) = \delta^3(\mathbf r) A rigourious series of steps, including Fourier transforms and residue integrals follow...

How does the method of images work? I can see why it works (by going back to the form of the Green function and differentiating) but don't see how useful it is to solve problems. At the moment I am basically memorising the different images for each boundary condition which I am sure is not the...

Homework Statement Consider \nabla ^2 u = Q\left( {x,y,z} \right) in the half space region z > 0 where u(x,y,o) = 0. The relevant Green's function is G(x,y,z|x',y',z'). Find the solution to the PDE in terms of G. If Q\left( {x,y,z} \right) = x^2 e^{ - z} \delta \left( {x - 2}...

Okey Dokey, so I'm bored and decided to play around with some math. I've got a problem that I can't figure out now; I have the green's function for the laplacian G(\vec{x}, \vec{x'}) = - \frac{1}{4\pi |\vec{x} - \vec{x'}|} There are no boundary conditions. Is there any lazy way to figure out...

Homework Statement The homogeneous Helmholtz equation \bigtriangledown^2\psi+\lambda^2\psi=0 has eigenvalues \lambda^2_i and eigenfunctions \psi_i. Show that the corresponding Green's function that satisfies \bigtriangledown^2 G(\vec{r}_1, \vec{r}_2)+\lambda^2 G(\vec{r}_1...

URGENT - Green's Function Solution to Poisson/Helmholtz equations hey, i have an exam pretty soon and couldn't find any answers/hints on how to do this: 1.How do you express the solution f(x') of the Helmholtz equation in terms of the green function g(x,x') in integral form, with dirichlet...

help about the Nonequilibrium Green's Function in H. Haug and A.-P. Jauho Book Quantum kinetics in transport and Optics of Semiconductors Eq.(4.31) C^r(t,t')=A^<(t,t')B^r(t,t')+A^r(t,t')B^<(t,t')+A^r(t,t')B^r(t,t') I can not derive this equation, my result has a extra term \theta(t-t') i.e...

Question a) Find two linearly independent solutions of t^2x''+tx' - x = 0 b) Calculate Green's Function for the equation t^2x''+tx' - x = 0, and use it to find a particular solution to the following inhomogeneous differential equation. t^2x''+tx'-x = t^4 c) Explain why the global...

anybody can recommend a good introducotry book on "advanced and retarded Green's function" and its application to QM, particularly transport problems. Thanks. :smile:

Hi all...need a little help with this one... I need to find the Green's function for the half space Neumann problem in the domain z>0. i.e. Laplacian u=f in D, du/dn=h on the boundary of D. Any ideas?

I'm wondering if the general method I'm using for getting greens function solutions is wrong, because it's not giving me the right answer. Here's what I do. Starting with a differential equation: a(x) \frac{d^2 y(x)}{dx^2} + b(x) \frac{dy(x)}{dx} +c(x) y(x) = d(x) the green's...

Hi! I encountered the problem that I need to decompose the Green function into a set of eigenfunction. Particularly, I have the free space Green function G(\vec r; \vec r') = \frac {e^{i k | \vec r - \vec r'|} } {4 \pi | \vec r - \vec r'|} and I need to express it into series of...