Greens function Definition and 16 Threads

So I've just started learning about Greens functions and I think there is some confusion. We start with the Stokes equations in Cartesian coords for a point force. $$-\nabla \textbf{P} + \nu \nabla^2 \textbf{u} + \textbf{F}\delta(\textbf{x})=0$$ $$\nabla \cdot \textbf{u}=0$$ We can apply the...

I am currently trying to compute the Green's function matrix of an infinite lattice with a periodicity in 1 dimension in the tight binding model. I have matrix ##V## that describes the hopping of electrons within each unit cell, and a matrix ##W## that describes the hopping between unit cells...

This is Jackson's 3rd edition 1.12 problem. So, for both ## \phi ## and ## \phi' ##, I started from Green's second identity: ## \int_V ( \phi \nabla^2 \phi' - \phi' \nabla^2 \phi )dV = \int_S ( \phi \frac {\partial \phi'} {\partial n} - \phi' \frac {\partial \phi} {\partial n} ) dS ## And...

Hello (I'm reposting this from stack exchange, and thought this site may be more appropriate, so if you see it that's why), I'm working through this paper, and have encountered "a little algebra shows that...", yet I'm not familiar enough with the topic at hand to figure this out. Here is the...

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...

Homework Statement I was a bit confused when reading my notes.After some derivation it states that G_h = e^(ikr)G_p,where G_h = Helmoltz Green's function G_p = Poisson Green's function = - 1 /(4πr) By definition (D^2)(G_p) = δ^3 (r) (D^2 = Laplacian) Please see the attached (D^2)(G_p) = 0...

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\...

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 $ G(\vec{r_1},\vec{r_2})=...

Not following this example (PDE for Greens function) in my book: Book states: $ \left( {\nabla}^{2} +{k}^{2}\right)G(r, r_2)=-\delta(r-r_2) = -\int \frac{e^{ip.(r-r_2)}}{\left(2\pi\right)^3} {d}^{3}p$ I recognised this as the Hemlmholtz eqtn, but cannot find where the 3rd term comes from? It...

Homework Statement A potential ##\phi(\rho, \phi ,z)## satisfies ##\nabla^2 \phi=0## in the volume ##V={z\geqslant a}## with boundary condition ##\partial \phi / \partial n =F_{s}(\rho, \phi)## on the surface ##S={z=0}##. a) write the Neumann Green's function ##G_N (x,x')## within V in...

I can't think of a situation where we can utilize greens function without the presence of a point charge. let's consider the following equation: \Phi=\frac{1}{4\pi \epsilon} \int dv \rho(x')G_{N} (x,x')+ \frac{1}{4\pi} \int da F_{s}(\rho , \phi) G_{N} + <\phi>_S Here we see that a volume...

In my book the path integral representation of the green's function is given as that on the attached picture. But how do you go from the usual trace formula for the Green's function 2.6 to this equation?

In this note (http://sgovindarajan.wdfiles.com/local--files/serc2009/greenfunction.pdf) the Klein-Gordon retarded green function is derived on the form $$G_{ret}(x − x′) = \theta(t − t') \int \frac{d^3 \vec k}{(2\pi)^3 \omega_k} \sin \omega_k (t − t′) e^{i \vec{k}\cdot (\vec x - \vec x')}$$...

Hey all, some weeks ago in a tutorial our TA solved Poissons equation with Greens functions..would be very short, but he also derived the Greens function using a Fourier transform. Two points I really don't get and he could also not explain it. Maybe you can help me? There might be even a short...

So I just recently learned about how to use Greens Function to solve a differential equation. The formula was derived and it said the main goal was to find an integral representation to the solution. It seems to me, however that Greens Function is nothing more than variation of parameters with...

Homework Statement use the greens function G(x,z) to solve inhomogeneous problem: (1-x2 ) y'' - x y' + y = f(x) y(0) = y(1) = 0 Homework Equations the answer is: G(x,z)= -x for x<z and -z(1-x2 ) 1/2 (1-z2 ) 1/2 The Attempt at a Solution the general solution to the...