# Tag Info

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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
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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

### 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
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• 14.8k

### 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
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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 $y''+4y=0$ ...

### 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
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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 ...
• 103k
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• 168k
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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 $$\widehat f(x) = \frac{\pi^{(d-2)/2}}{\pi \Gamma((d-2)/2)}\frac{1}{x^2}.$$ ...
• 3,346
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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 ...
• 52.4k
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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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### 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 ...