# Questions tagged [greens-function]

This tag is for questions about a Green's function which is the impulse response of an inhomogeneous differential equation defined on a domain, with specified initial conditions or boundary conditions.

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### Green's Function: Notation in Evans versus Others

Suppose we want to find Green's function in one of the $2$-dimensional quadrants, say the first one $D = \{(x,y) \in \mathbb{R}^n : x > 0, y > 0\}$. Let $x = (x_1, x_2)$ and $y = (y_1, y_2)$. ...
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### Green's function for a second order ode

I want to find the Green's function for $$\frac{1}{1+x^2}y''-\frac{2x}{(1+x^2)^2}y'=f(x).$$ I have found that the solution to the homogeneous case is $A(x+x^3/3)+C$ but I'm unsure of how to turn ...
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### Determining the Green's function and solution for $f''(x)=-g(x)$ with boundary conditions $f(0)=f(1)=0$.

I am trying to solve the Poisson's equation in one dimension using Green's function: $$f''(x)=-g(x)$$ With the boundary conditions $f(0)=f(1)=0$. I know that the Green's function is going to satisfy ...
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### Green's functions for noncompact finite-volume quotient

I am very unfamiliar with the theory of differential equations, so apologies if this question is very standard or too vague; I'd be happy just for a reference that is supposed to explain the theory. ...
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### Is there a Green function for the p-Laplacian?

The Green's function is defined for a linear differential operator $L$ as the solution of the equation $LG = \delta$, where $\delta$ is Dirac's delta function. A direct consequence of the definition ...
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### Representation formula using Green's function (Evans)

From the Green's function definition, it seems we only require $G(x,y)\in C(\bar{U}) \cap C^2(U)$. In the theorem 12, we have a term $\frac{\partial G}{\partial v}(x,y)$. Since it is a directional ...
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### Checking uniqueness of solution to a Laplace equation. Related to minimal surface modelling

hope you all are doing well. I am working on minimal surfaces (Chemical engineering background), and I am stuck at a particular problem. I need to solve Laplace equation with the following boundary ...
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### What is the correct choice of the contour in the case of undamped forced harmonic oscillator?

I am interested in finding the Green's function (GF) for the undamped forced harmonic oscillator equation: $$\Big(\frac{d^2}{dx^2}+\omega_0^2\Big)x(t)=f(t).$$ In order to find the GF, start by define ...
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### Green's functions with cylindrical boundary conditions that cover the entire interior of the cylinder

I am trying to compute the Green's function $\mathrm{G}\left(x,x'\right)$, $\ \nabla_{x}^{2}\mathrm{G}\left(x,x'\right) = 4\pi\,\delta\left(x - x'\right)$ and $\mathrm{G}\left(x,x'\right) = 0$ for $x$ ...
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### How to check if a differential operator is translation invariant practically?

Recently I was calculating some stuff in curved (ADs to be exact) spaces, when the following question came to my mind, Suppose you have in general a differential operator $\hat{D}$ acting on the ...
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### Inhomogeneous wave equation with periodic BC

I'm looking for the solution of the inhomogeneous 3D wave equation $\frac{\partial^2\rho}{\partial{t}^2} - c^2\nabla^2\rho = S(x,y,z,t)$ with periodic boundary condition in all three directions in a ...
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### Fundamental solution with initial conditions

For a linear partial differential equation $\mathcal{L}u=f$ on an open set of $\mathbb{R}^{n}$, the fundamental solution is defined as the distribution $F$ such that $\mathcal{L}F=\delta(x)$. The ...
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