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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11 views

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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41 views

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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18 views

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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2answers
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How to make sense of the Green's function of the 4D wave equation?

In the paper "Wakes and waves in N dimensions" by Harry Soodak and Martin S.Tiersten, equation $(36)$ gives the Green's function for the 4D wave equation in the following form: $$G_4(r,t)=\...
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9 views

Why is this constraint justified when constructing a Green's function

Let's consider a linear operator $L$ such that $$Ly = f$$ subject to some homogeneous boundary conditions on the interval $[a,b]$. The homogeneous solution is $$y_h(x) = c_1y_1(x) + c_2y_2(x)$$ for ...
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17 views

How to find Green's function for 2-point BVP where none of the complementary functions satisfy a BC

I have an inhomogeneous differential equation: $$\left(\frac{d^2}{dr^2}+\frac{2}{r}\frac{d}{dr}+ \lambda\right)\rho(r) = f(r),$$ where $0 \leq R \leq b$, $f(r=0,R)=0$. BC: $\rho(r=0,R)=0$. The two ...
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28 views

How to find the right combination of general solutions to fulfill nonhomogeneous boundary conditions when solving heat equation in spherical polars

This question below is a deeper dive to the problem described here, but it should stand alone on its own as well. I am trying to solve $$\dot{u} = \alpha \,\Delta u,$$ where$$u=u(r,t) \text{ and } R ...
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29 views

Fourier Transform Involving Logarithm in Denominator

In a theoretical physics paper, I came across the following Green's function $$ G(\vec{r}, t)=\int \frac{d^4 p}{(2 \pi)^4} e^{i (p_0 t - \mathbf{p} \cdot \vec{r})} \frac{1}{p^2 \text{ln}(\frac{p}{k})},...
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36 views

Heat Equation with Initial Data in $L^1.$

In Friedman's book PDE’s of Parabolic Type, the Green function $G$ for the heat equation is said to satisfy \begin{equation*} G_t - \Delta G = 0 \end{equation*} in $\Omega\times(0,T)$ and $G=0$ on $\...
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114 views

Solve the wave equation $ (\partial^2_t- \nabla^2) u(t,\mathbf{x}) = f(t,\mathbf{x})$ for a source that is always there

Consider the inhomogeneous wave-equation (with units $c=1$ and in 3 spatial dimensions) $$ \frac{\partial^2 u}{\partial t^2} - \nabla^2 u(t,\mathbf{x}) = f(t,\mathbf{x}) \tag{1} $$ where $f(t,\mathbf{...
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1answer
73 views

Applying Green's function for one dimensional wave equation

The Green's function of the one dimensional wave equation $$ (\partial_t^2-\partial_z^2)\phi=0 $$ fulfills $$ (\partial_t^2-\partial_z^2)G(z,t)=\delta(z)\delta(t) $$ I calculated that its retarded ...
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30 views

Inverse Fast Fourier Transform in Matlab

I have a function to invert $\hat{f}(\omega)=-\dfrac{i}{\sqrt{2\pi}\,\omega}$, but I am not sure why this is not coded correctly. So the inverse is supposed to be $f(x)=-\text{Sign}(x)/\sqrt{2}$ but ...
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1answer
36 views

Green function and operator Sturm-Liouville

Consider $$\mathcal{L}=\sigma_y - c\mathbb{I}$$ Where $\sigma_y$ is the Pauli matrix, and $\mathbb{I} =\left[\begin{array}{ccc} 1 & 0 \\ 0 & 1 \\ \end{array}\right]$ is the identity matrix. ...
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1answer
18 views

ODE made of Green's functions

I need to solve the following equation $$\dot G + G =\delta (t)$$ $$G(-\infty)=0$$ What could it physically mean that $G(-\infty)=0$ in the context of wave propagation?
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22 views

Can we assure $\frac{d^2}{dx^2} G(\textbf{r},\textbf{r}') = \frac{1}{3} \delta (\textbf{r}-\textbf{r}')$? [duplicate]

Recently, I've made a question regarding the proof of $\nabla ^2 G(\textbf{r},\textbf{r}') = \delta (\textbf{r}-\textbf{r}')$ for $G(\textbf{r},\textbf{r}')=\frac{1}{4\pi|\textbf{r}-\textbf{r}'|}$. ...
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23 views

Regularity for a Nonlinear Heat Equation

Let $\Omega \subset \mathbb{R}^n$ be a bounded open set with smooth boundary and let $G$ denote the Green function of the heat equation with Dirichlet boundary condition, i.e., $G_t(x,y,t)-\Delta G(x,...
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2answers
84 views

Confusion regarding the solution to $\nabla ^2 \phi(\textbf{r}) = \rho(\textbf{r})$ using Green's function [duplicate]

We know we can solve one of the Maxwell's equation using Green's function. More specifically, we can solve $$\nabla ^2 \phi(\textbf{r}) = \rho(\textbf{r})$$ using $$\phi(\textbf{r}) = \int d\textbf{r}'...
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1answer
33 views

On Green functions for recurrent processes

Green function $G(x,y)$ is a function satisfying $\Delta G(x,y)=\delta (x-y)$ where $\delta$ is Dirac's delta. Brownian motion on $\mathbb{R}^2$ is recurrent, so we have $G(x,y)=\int _0^\infty p(t,x,y)...
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1answer
48 views

Wave equation and Green's function

For a simple linear inhomogeneous ODE, it's easy to derive that the Green's function should satisfies $$L_xG(x) = \delta(x-x')\tag{1}$$ where $L_x$ is the differential operator. However, for the ...
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60 views

Green's function for random walks on a network

My question comes from section 9.4. of the book "Markov Chains and Mixing Times (2nd edition)" written by David A.Levin and Yuval Peres. Specifically, in page 119, the author defines the $\...
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15 views

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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1answer
67 views

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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57 views

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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31 views

Green's functions (Multivariable Calculus)

This is an edited excerpt from a textbook that I'm using. I had a question, which I've emphasized in boldface. Anyway, consider the equation (for $a \leq x \leq b$ ) $$ a_{n}(x) \frac{d^{n} y}{d x^{n}}...
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1answer
57 views

Solve the ODE $(x-1)y'' - xy' + y = 1$ subject to boundary conditions $y(0)=0,y(1)=2$ using Greens function

I'm looking to solve the BVP $$(x-1)\frac{d^2y}{dx^2} - x\frac{dy}{dx} + y = 1$$ subject to the conditions $y(0)=0,y(1)=2$, firstly by getting the problem into self adjoint form and then by finding ...
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23 views

About Green's function

Let $I,R: \Omega\times\mathbb{R}\to \mathbb{R}$ ($\Omega$ is a bounded domain of $\mathbb{R}^{d}$) functions such that $$\dfrac{\partial I}{\partial t} - \triangle I=\alpha I - \gamma\dfrac{I(I+R)}{N},...
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28 views

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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1answer
32 views

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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47 views

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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25 views

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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1answer
110 views

Find Greens function associated with quadratic form

Let $D$ be the open disc in $\mathbb{R}^2$ of radius $R$ (for some $R>0$). Let $\theta:D\to[0,\pi]$ be given (constraints on it are postponed for now). Consider the quadratic form $$ Q^\theta(\...
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16 views

Is this equality about integral on dyadic product true?

Is this equality in the attached picture true? The curl is is done over $r$ not $r_0$, unlike the integral. The symbol $G$ with two pars, is a dyadic.
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Green function for the 1d transport equation

This is very basic, but I need some re-assurance, and I couldn't find this despite googling for a while. Given the differential operator $$ \mathcal L = \frac{d}{dt}- v \frac{d}{dx}, $$ I know that ...
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85 views

Green Function on $S^2$

I am trying to prove the Green's representation formula on $S^2$. Let $x \in S^2$ be fixed, consider \begin{equation*} G(x,y) = \frac{1}{8\pi}\log[4\sin^4(\frac{r}{2})] \end{equation*} where $r = ...
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1answer
93 views

Helmholtz-Dirichlet Green's Function for unit ball - Explicit formula?

The formula for the Green's function for Laplace's equation on the unit ball with Dirichlet boundary conditions is well-known: $$ G(x,y) = \frac{1}{4\pi}\left(\frac{1}{|x-y|} - \frac{1}{\Big|x|y|-\...
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35 views

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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29 views

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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If $\nabla\cdot v=0$ and $w=\nabla^\perp\cdot v$, then $v=\nabla^\perp g\ast w$, where $g$ is the fundamental solution of the Poission equation

Let $\Omega\subseteq\mathbb R^2$ be open, $v:\Omega\to\mathbb R^2$ with $\nabla\cdot v=0$ (in a sense to be specified later), $$\nabla^\perp:=\left(-\frac\partial{\partial x_2},\frac\partial{\partial ...
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1answer
30 views

relations between Bessel functions J0 and K0 (to find the Green's function for the 2D screened Poisson equation)

I'm trying to find the Green's function for the screened Poisson equation in two dimensions, i.e. the $G(\mathbf{r})$ that solves $$(\nabla^2-1/\rho^2) G(\mathbf{r}) = \delta(\mathbf{r}), \qquad \...
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16 views

Fourier Transform for a variable dependent on another

I am dealing with finding the green function for a PDE where by change of variables I could make it linear. At first, I had a function of $f(x,y,t)$ which I change it to $f(x,z,t)$ considering $z=x^2+...
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33 views

Sink term in mass transfer

Assuming I have a model with initial concentration of $c_0(x)$, I would like to model a point sink for this case (considering the velocity is neglected :D) $$\frac{\partial c}{\partial t}=D\frac{\...
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11 views

parabolic complete manifold

A complete Riemannian manifold is parabolic if it does not admit a positive Green's function, which is equivalent to the fact that there is no positive non-constant superharmonic function whose ...
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19 views

Green's function of the Dirichlet problem for the Laplace equation

I have to find the Green function of the Dirichlet problem for the Laplace equation $$\Delta u = 0$$ in $\mathcal{B}:=\{x\in\mathbb{R}^3: |x|<R\}$ How can i attack this problem?
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20 views

Green function on finite-dimensional spaces

I know that Green functions are usually used when we're dealing with distributions and differential equations but how can we define it in terms of operators defined on finite-dimensional spaces? To be ...
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18 views

How to determine the Green's function for wave equation

$$\begin{align} u_{tt}&=c^2u_{xx}+ Q(x,t), \quad x>0 \\ \\ u(x,0)&=f(x)\\ u_t(x,0)&=g(x)\\ u(0,t) &= h(t) \end{align}$$ Question: How should you determine the Green's function?
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34 views

Laplace transform of an unusual convolution

I'm trying to solve a (linear, homogeneous) ODE (and find the function f(t)) which includes the convolution \begin{equation} \begin{aligned} (G*f)(t) := \int_{-\infty}^tG(t - u)f(u)du + \int_{t}^\...
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12 views

Conditions for realness of Green function for $SL_2(\mathbb{R})$

Assume I consider the Green function attached to the resulvant of the differential operator $\Delta + \lambda I$. What conditions on $\lambda$ imply that $G_{\lambda}(g,h)$ is a real valued function? ...
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26 views

Computing the Green function

Consider the topological space $SL_2(\mathbb{R})$, and the linear differential operator given by $L = \Delta + \lambda\cdot I + \dfrac{\partial^2}{\partial\theta^2}$. I would like to compute its ...
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37 views

Periodic Green's function for Laplacian in $\mathbb{R}^3$

The periodic Green's function $G^p$ for Laplacian $\Delta$ in $\mathbb{R}^3$ is defined by lattice sum: $$G^p(x)=-\sum_{n\in \mathbb{Z}^3-\{0\}}\frac{e^{2\pi i n\cdot x}}{4\pi^2|n|^2}$$ and it is well ...
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28 views

Numerical integration of singularity in green's function

Let's say for arguments sake, or as a way to understand how integrals of singularities work, we want to numerically verify that \begin{equation} \int_{V}{[\nabla^{2}G(r,r')A(r')]}dV' = A(r) \end{...

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