35 votes
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What is the idea behind Green's function? What does it do?

Green's function, also called a response function, is a device that would allow you to deal with linear boundary value problems (in the literature there are also Green's functions for the initial ...
  • 13.6k
23 votes
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Fundamental solution for Helmholtz equation in higher dimensions

We first assume that the point source is located at $\vec r'=0$, $\vec r'\in\mathbb{R}^n$. Spherical symmetry implies that the Green (or "Green's") Function, $G(\vec r|\vec r'=0)$, for the Helmholtz ...
  • 168k
7 votes

Examples of Greens functions for Laplace's equation with Neumann boundary conditions.

I'm afraid that what I'm about to say is not quite you what to hear: your Neumann PDE does not have a solution for arbitrary choices of $g$. To see why, let's compute the integral of $g$ over the ...
  • 25.4k
6 votes
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A question on Green's functions & integral operators

The keyword here is linearity. A physicist would say that you are employing the superposition principle. You are decomposing your source term $f$ into a superposition of localized impulses: $$ f(x)=\...
6 votes
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Green’s Function for the Heat Equation

Consider the Cauchy problem for the heat equation: \begin{align*} \left\{ \begin{array}{r l} \frac{\partial u}{\partial t} - \Delta u = 0 & \text{in} \, \, \mathbb{R}^{d} \times (0,\infty) \\ u(x,...
  • 3,159
5 votes
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Green's function Laplace equation through Fourier space

$$\mathscr{F}\{\nabla^2 G\}(\vec k)=\iiint_{\mathbb{R}^3}\nabla^2G(\vec r)e^{i\vec k\cdot\vec r}\,dx\,dy\,dz=-k^2\mathscr{F}\{G(\vec r)\}=-1$$ Hence, $$\mathscr{F}\{G\}(\vec k)=\frac{1}{ k^2}$$ ...
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5 votes
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Initial Value Problem of a Differential Equation Involving a Step Function

This isn't really an answer to your question, but the inverse transform of $$Y(s) = \frac{1 - e^{-s}}{s(s+a)}$$ isn't too bad. We see $$\frac{1}{s(s+a)} = \frac{1}a \left(\frac 1 s - \frac 1 {s+a} \...
  • 14.8k
5 votes

Green's Function for 2D Poisson Equation

It suffices to show \begin{align} \int_{\mathbb{R}^2} G(\textbf{r}, \textbf{r}') \nabla^2f(\textbf{r}')\ d^2\textbf{r}' = f(\textbf{r}). \end{align} Observe we have \begin{align} \int \log|\textbf{r}...
  • 24.8k
5 votes
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Constructing a green function

You can solve the equation $y''+4y=f$ subject to $y(0)=y(\pi/4)=0$ using variation of parameters: $$ y = a(x)\sin(2x)+b(x)\sin(2x-\pi/2). $$ The first solution $\sin(2x)$ of $y''+4y=0$ ...
4 votes

Equivalent IVPs for the Wave Equation (moving the delta function) Kevorkian.

A proof from the book (Erdős used this to mean "an elegant -- even divine -- proof", here I prosaically mean "Elements of Green's functions and propagation - potentials, diffusion, and waves" (OUP, ...
  • 3,007
4 votes
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Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$

Note: I am finally correcting my mistake pointed out by Mark. For $G(x,y) = \sum_{n=1}^\infty \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y) $, since (here's where my mistake was: I had cos-cos ...
4 votes
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Green's Function for Laplacian on $S^1 \times S^2$

Here's a way to get a representation as series. It works in a more general situation. Let $M$ and $N$ be smooth closed Riemannian manifolds. Denote by $\{\varphi_i\}_{i=1}^\infty$, $\{\psi_j\}_{i=j}^\...
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4 votes

Green's function for Laplacian in the plane. Why does the constant matter?

You need distribution theory to really know what's going on. In the sense of distribution theory, the equation $\Delta G(x) = \delta(x)$ means that whenever $\phi$ is a smooth function with compact ...
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4 votes
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Green's function for Laplacian in the plane. Why does the constant matter?

We can redefine the problem to circumvent use of the Dirac Delta. So, rather than describing the problem by the expression $\nabla^2 G(\vec \rho|\vec \rho')=\delta(\vec \rho-\vec \rho')$ we describe ...
  • 168k
4 votes
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How to prove Green's function of $\partial_x^2 + \partial_y^2$ is $\frac{1}{2\pi}\log(\sqrt{x^2+y^2})$?

The Dirac delta is a distribution characterized by the property that $\newcommand{\rx}{x}$ $$ \int f(\rx) \delta(\rx) \, \mathrm{d}\rx = f(0) $$ holds for all nice test functions $f$. (Or more ...
4 votes
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What does "translation invariant" mean for linear differential functionals?

First question - what is the exact meaning of "translation invariant" with regard to functionals? Many, many people write "invariant" when they mean "equivariant," and ...
4 votes
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Green's function for Laplace equation in the exterior of disks

There is a Moebius transformation $\psi(z)$ of the Riemann sphere that maps the unit disk $D_1(0)$ to itself, and maps $D_1(a)$ to the complement of $D_R(0)$ for some $R>1$. To find $\psi$, first ...
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3 votes
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Green's function For Helmholtz Equation in 1 Dimension

Taking the Fourier transform of both sides, $$ (k^2-(2\pi \omega)^2) \tilde{g}(\omega) = 1, $$ so $$ \tilde{g}(\omega) = \frac{1}{k^2-(2\pi \omega)^2} = \frac{1}{2k}\left(\frac{1}{2\pi \omega-k}-\frac{...
  • 65.5k
3 votes
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Estimate the Green function for the Laplace equation in 2D

First let's find out where the logarithm changes sign. We can rewrite the logarithm to make it $$ -\frac{1}{4\pi} \log{\left( \frac{\lvert x \rvert^2 \big\lvert y-x/\lvert x \rvert^2 \big\rvert^2 }{\...
  • 65.5k
3 votes
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Green's function using method of images

Finally figured out how to apply the method of images for this problem, it's similar as described previously in the same book (see page 83). Consider the domain $D = \{(x_1,x_2): x_1 \in (-\infty,\...
  • 1,442
3 votes

PDE: Fokker-Planck equation with time-dependent boundary conditions

You can reduce this problem to a more familiar one with the following trick. Consider the problem $$ q_t(x,t) = -au_x(x,t) + \frac{D}{2}q_{xx}(x,t) \quad (0 < t, 0 < x < 2L)\\ q(x,0) = 0, \...
  • 14.4k
3 votes
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Green's function for Helmholtz equation for the plane with a hole

The Green's Function $G(\vec \rho,\vec \rho')$ that satisfies the $2$-D Helmholtz equation $$\nabla^2G(\vec \rho,\vec \rho')+k^2G(\vec \rho,\vec \rho')=\frac{\delta(\rho-\rho')\delta(\phi-\phi')}{\...
  • 168k
3 votes
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Fundamental solution to the Poisson equation by Fourier transform

The following argument works for d>3. From Fourier transform of $1/|x|^{\alpha}$. we know that if $f(x) = 1/|x|^{d-2}$ then $$\widehat f(x) = \frac{\pi^{(d-2)/2}}{\pi \Gamma((d-2)/2)}\frac{1}{x^2}.$$ ...
3 votes
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Uniqueness of the Greens function

The Wronskian $$W=u_1u_2'-u_1'u_2$$ is calculated with respect to the same solutions $u_1,u_2$ that are used to define the Green function. This corresponds to a specific value of the constant $C$ and ...
3 votes
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Sign of Laplacian Green's function in 3D

If you apply maximum principle on $\Omega \setminus B_{\epsilon}(\mathbf{x}_0)$, there are two boundaries. As you have said Green function takes zero on $\partial \Omega$. For another boundary, note ...
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3 votes

Computing an infinite trigonometric sum $\sum \frac{2}{n^2 \pi^2} \cos(n \pi x)\cos(n \pi y)$

Marty Cohen is correct in his approach to this $G(x,y)$ but there is the sign error in the trig expressions. Defined on the interval $0 < t < P$, it is fairly easy to demonstrate that the ...
3 votes
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Greens function for 2d laplace equation with neumann boundary conditions

We present an outline only for obtaining the solution to the posed problem for $u$ and provide an outline of a way for verifying the solution. We proceed by using Green's Identity to write $$\...
  • 168k
3 votes

What is the idea behind Green's function? What does it do?

I don't have enough reputation to comment, but in response to OP's request for an ODE example, check out this recent Mathematica blog post which uses Green's function to solve a RLC circuit problem ...
3 votes

Is it possible to have a $f(\vec{r})$ satisfy this relation?

Fourier transforming your equation, you can check that the function $f({\bf r})$ is given by $$f({\bf r}) = \int \frac{d^3 q}{(2\pi)^3} \frac{e^{i {\bf q} \cdot {\bf r}}}{(q_3+\alpha)^2 - q_1^2 -q_2^...
  • 22.6k

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