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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How do I prove $\operatorname{div} (f \nabla g) = f\Delta g + \nabla f \cdot\nabla g$?

Let $f\in C^1$ and $g\in C^2$ be scalar functions. How do I prove the identity $$\operatorname{div} (f \nabla g) = f\Delta g + \nabla f \cdot\nabla g$$ ?
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Conjugate of the discrete Laplacian Green's function on a square lattice

I have an engineering background and I am faced with the following problem. Green's function for the discrete Laplacian on a square lattice is well known and I think it is a discrete harmonic function....
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What's going on with the two-dimensional Helmholtz equation?

I've come to realize that its somehow harder to find results for this equation than for the three-dimensional one. For example the wikipedia article on Green's functions has a list of green functions ...
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solve this fourth order differential equation using green's function

I need to solve this equation using the green's function EIw''''-pw''=F(Dirac delta symbol)(x-L/2) EI,P and F are constant. the 'Dirac delta' is for the Dirac delta symbol to show impulse. F* the ...
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Fourth-order PDE of beam with complex boundary conditions.

I am trying to solve the following differential problem: \begin{equation} \frac{\partial^4 u(x,t)}{\partial x^4}+a\cdot u(x,t) = b\cdot g(x,t)\\ u(0,t) = u'(0,t)=0\\ u''(l,t)=\alpha_1\frac{\...
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Choose Green's function of Possion equation on half-space

Background: Consider the Dirichlet problem (A) defined in $\mathbb{R}^3$ $$-\Delta u=0~~~~\text{in $\mathbb{R}_+^3$}=\{x=(x_1,x_2,x_3):x_3>0\},$$ $$u=g~~~~\text{in $\partial \mathbb{R}_+^3$ }.$$ We ...
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Why do Green's Functions satisfy homogeneous boundary condition?

Most the textbooks introduce Green's function by considering the following boundary value problem: $$ \frac{d}{dx}\left[p(x)\frac{dy(x)}{dx}\right]+q(x)y(x)=-f(x), $$ $$ \alpha_1y(a)+\alpha_2y'(a)=0,\...
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Give an example of a Green’s function for a specific domain

Definition: The Green’s function G(x) for the operator $\Delta$ and the domain $D$ at the point $x_0 ∈ D$ is a function defined for $x ∈D$ such that: (i) $G(x)$ possesses continuous second ...
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Question about solving for Green's Function

I arm reading Agarwal and O'Regan's Ordinary and Partial Differential Equations, Lecture 16. In it, they derive the following whilst constructing Green's Function, where $$l_1(y)=a_0y(\alpha)+a_1y'(\...
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How the green function for the relativistic heat equation converges to the green function of the heat equation?

The relativistic heat equation or telegraphers equation is: $$ (\alpha\partial_t^2 + \beta\partial_t - \omega\,\nabla^2_{\text{3D}})G_R = \delta $$ if $\alpha \rightarrow 0$ the solution must ...
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Green's Function for 1st order ODE

I'm trying to verify another result from Coddington-Levinson's ODE: Given $$Lx=ix'=lx, \qquad x(0)-x(1)=0$$ We seek to find Green's function. The author gives it as: $$G(t,s,l)=\begin{cases} \frac{ie^{...
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Finding Green function and solution of Dirichlet problem

For unit open ball $D=\{{x}\in\mathbb{R}:\lvert{x}\rvert<1\}$, I need to find Green function $G$ with Dirichlet Problem $$ \Delta u =0\,\, \text{ in }D,\qquad u=g\,\, \text{ on }\partial D $$ Let $\...
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Integrate $\frac{ e^{i k | \mathbf{r} - \mathbf{r'} |} }{|\mathbf{r} - \mathbf{r'} |}$ in a spherical shell

How can we compute the following triple integral (electromagnetic diffusion in a sphericall shell)? $ E(\mathbf{r}) = \int_0^{2 \pi} d\phi' \int_0^{\pi} \sin \theta' d\theta' \int_R^{R+h} d r' r'^2 \...
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Solving a multivariate recurrence relation (or its dual PDE)

I have a two-variable recurrence relation of the form, \begin{align} -&[(N+1)n+N(n+1)+(M+1)m+M(m+1)]p(n,m)\\ -&\epsilon[(n+1)m+(m+1)n)]p(n,m)\\ +&(N+1)(n+1)p(n+1,m)+(M+1)(m+1)p(n,m+1)\\ +&...
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Discretization and numerical evaluation of Green's function in scattering problems

I'm trying to discretize and evaluate the following integral involving a Green's function in free space: $ p(\mathbf{r}) = \int_V G(\mathbf{r}-\mathbf{r}') o(\mathbf{r}') d\mathbf{r}'$ for $G(\mathbf{...
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Green function of $\left(\nabla^2+k^2\right)\!\psi=-4\pi\,\delta\!\left(\vec{r}\right)$?

I was revisiting an old paper by Leslie L Foldy where he states that the solution to $$\left(\nabla^2+k^2\right)\!\psi\!\left(\vec{r}\right)=-4\pi\,\delta\!\left(\vec{r}\right)$$ is $$\psi\!\left(\vec{...
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Obtaining the Green's function for a 2D Poisson equation ( in polar coordinates)

I am trying to solve the following BVP within an annular region of radii $r_1$, and $r_2$ : $$ \begin{cases} \nabla^2u=f\\ u(r_1) = p\\ u(r_2) = q \end{cases} $$ If we define an auxiliary problem in ...
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Green function Stokes equation

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, where $\textbf{P}=\textbf{P}(x,y,z)$,...
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Green Function for Different a Differential Operator

Good afternoon everyone. I am solving a hard PDE and, in doing so, I ended up needing to solve $${\bf r}\cdot\nabla\psi = f({\bf r})$$ where $\nabla$ and ${\bf r}$ are in 2D and the source term $f$ ...
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2 votes
1 answer
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Solving Simple PDE by Green's Function, Very Confused By Some Mistake

Suppose I want to solve $u_{xy} = xy$ via Green's Function. This will correspond to the associated PDE $G_{xy} = \delta(x - x_G,\ y - y_G)$, and I want my boundary conditions for this Green problem ...
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Boundedness of solutions to an ODE and their derivatives

The Scenario: Consider the ODE given by $$ \left\{ \begin{alignat*}{99} &-g'' + g = F(x,f(x)) \qquad &&x \in [0,L] \\ &g(0) = g(L) = 0 \end{alignat*} \right. \tag{1} $$ where: $F$ is ...
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Green's Function Computation (Fourier Transform Edition)

I want to calculate Green's Function to solve $\triangle u = f(x,\ y)$, using Fourier Transforms. Because the Laplacian is self-adjoint, my associated Green's Function equation can immediately be ...
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1 answer
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Resolvent of Dirichlet Laplacian via Fourier Transform

For $-\Delta$ considered as a self-adjoint operator on $L^2(\mathbb{R}^d)$, one may write it's resolvent as the Fourier multiplier $g(\xi)=(|\xi|^2-z)^{-1}$. Now let $\Omega\subset \mathbb{R}^d$ be a ...
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1 vote
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Green's function for $Au=((1-x^2)u')'$ [duplicate]

I am trying to find the Green's function for the operator $Au=((1-x^2)u')'$ with boundary conditions $|u(\pm 1)|<\infty$. The general solution of $Au=0$ gives $u=c_1+c_2\log{\frac{1-x}{1+x}}$. To ...
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Green's Function Computation

I want to calculate Green's Function to solve $\triangle u = f(x,\ y)$, using Laplace Transforms. My plan was to tailor boundary conditions to the problem which simplify the computation. Because the ...
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Is $0$ the null space of the integral operator with kernel $G(r,r') = \frac{\exp(-ik|r-r'|)} {|r-r'|}$?

Let $G(r,r') = \frac{\exp(-ik|r-r'|)} {|r-r'|} $, where $r$ and $r'$ are position vectors in a domain $D$ of $\mathbb R^3$ and $k$ is a positive real constant. Suppose that $h$ is a continuous real ...
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2 answers
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How to solve homogeneous differential equation with initial value conditions using Green's function?

Solve the differential equation $$xy'' + y' = 0$$ using the Green’s function satisfying the initial condition $y(1) = y'(1)$. Generally, Green's functions are used to solve nonhomogeneous differential ...
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1 answer
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Green's function simplification

I'm reading the chapter on Green's functions in Strauss' PDE book. He gives the Green's function for the sphere as: $$ G(\textbf{x}, \textbf{x}_o) = - \frac{1}{4 \pi |\textbf{x}-\textbf{x}_o|}+ \frac{...
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Construct the Green's function for the following boundary value problem and use it to find the solution $y''-y'=t^2$

Construct the Green's function for the following boundary value problem and use it to find the solution $$y''-y'=t^2,\quad y(0)=0,\: y(1)=0$$ I know that, $$ \begin{aligned} &\langle L u, G\...
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What's a "Green's second theorem" that is applied to Laplace's equation?

What's a "Green's second theorem" that is applied to Laplace's equation? To get a formula of the kind: $$\int_S \frac{\partial G}{\partial n_q} (p,q) \phi(q)dS_q+\frac{1}{2} \phi(q)=\int_S G(...
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Given a Green’s function how can I find an operator that would be solved by it?

I have a Greens function G, I’m looking for an operator such that OoG(x,x’)=δ(x-x’). Not for one specific function but rather a general one. How might I do this? Is there any formula or technic?
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1 vote
1 answer
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How to interpret the definition of a Green's function?

A Green's function for a linear differential operator $L$ on a domain $\Omega \subseteq \mathbb{R}^n$ is a function $G$ on $ \overline{\Omega} \times \overline{\Omega} $ that satisfies $$ L_xG(x, s) = ...
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Green's identity in multiply connected disk

Let $D_z, |z|<1$ be a disk with $M+1$ connectivity i.e. M smaller circular holes have been punched out of the disk. Let $G(z,a)$ be Green's function in this multiply connected domain $D_z$, i.e. $$ ...
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Four-dimensional Dirac delta from a Green's function

Consider four-dimensional space $\mathbb{R}^2 \times \mathbb{C}$ with coordinates $w=x,y,z,\bar{z}$. Define $D^i = (\partial_x,\partial_y,4\partial_z)$, and consider the following equation for a Green'...
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Green's generating function for random walk in $\mathbb{Z}^d$.

Let $(X_t)_{t\geq0}$ be a simple symmetric random walk in $\mathbb{Z}^d$, in continuous time (holding times are $Exp(1)$ random variables). Moreover, suppose $X_t$ is killed after walking for a time $...
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Ultra light renewal theorem for heavy tailed renewal processes

I am interested in some "ultra light renewal theorem" in the following sense: We look at the renewal process $$\mathcal R := \left\{ n: \sum_{i=1}^k R_i = n \mbox{ for some } k \right\}$$ ...
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How should I understand (8.4.6) from a book by G.Barton?

My question concerns trying to understand a paragraph of material from a book by Barton, Reference 1. This material is from Chapter 8 ‘The diffusion equation: I. Unbounded space’, the paragraph is ...
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Complex Conjugate of a Function for a given mapping

I have a complex function which actually is a Green Function of a Laplacian Operator in the Upper plane (y>0). I want to use a conformal mapping to map a circle into an upper half-plane (y>0) ...
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Asymptotics of Green's function for Laplace-Beltrami operator

In the following, let $M$ be a compact Riemannian manifold and $\Delta = \nabla \cdot\nabla $ a Laplace-Beltrami operator on $M$. Let $G_y(x)$ be Green's function for $\Delta$ (a.k.a fundamental ...
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Estimate for Green's function in $\Omega \subset \mathbb{R}^2$. Can we show that $|\nabla G(x,y)| \le \frac{C}{|x-y|} $?

Assume $\Omega$ is a smooth bouned domain of $\mathbb{R}^2$. Let $$G(x,y)=\frac{1}{2\pi} \log{\frac{1}{|x-y|}}+H(x,y)$$ be the standard Green's function on $\Omega$. I want to show that $$|\nabla H(x,...
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Solving the differential equation by constructing Greens function.

Given differential equation, $$\tag{I}y''= \sin(\pi x)$$ with boundary conditions \begin{align} \tag{1} y(0)+y(1)&=0\\ \tag{2}y'(0)+y'(1)&=0 \end{align} Solving the differential equation ...
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Trouble finding the Green function in a disk?

I have the following problem: Find the Green function to the diffusion equation in a disc: $ \phi_t=\pi^2\left(\frac{1}{r}(r\phi_r)_r + \frac{1}{r^2}\phi_{\theta \theta}\right) + r(1-r)\\ \phi (0,r,\...
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Cauchy function for difference equations

I am new on difference equations I am strugling in the definition of the Cauchy function and how to use it to find the Green's function for a difference equations. For example: for the deference ...
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Average behavior of Green's function near the boundary

Let $\Omega \subset \mathbb R^d$ be a Lipschitz domain. Let $g$ be the Green function of $\Omega$ for the operator $\operatorname{div}(A\nabla \cdot)$ ($A$ with $C^\infty$ coefficients or, for ...
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What is wrong with considering the $(n-2)th$ and the lower-order derivatives of Green’s function to be discontinuous?

Consider:$$\dfrac{d}{dx}\left( p\left( x\right) \dfrac{dG(x,t)}{dx}\right) +q\left( x\right) G(x,t)=\delta\left( t-x\right).$$ Integrating, we get $$ p\left( x\right) \dfrac{dG\left( x,t\right) }{dx}| ...
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Green's function for difference equation

Do we have the same Green's function the following difference equations: $$-\Delta^2u(t)=0\; t\in \{0,1,2,...,N+2\}$$ $$-\Delta^2u(t-1)=0\; t\in \{0,1,2,...,N+2\}$$ Where $$\Delta^2u(t)=u(t+2)+2u(t+1)+...
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Trying to Solve the Black Scholes PDE with the Green's Function

I have finished the transformation into the Heat Equation. And I am now at the point of establishing the initial conditions. The article I read said the $\max(S-K,0)$ is now the initial condition ...
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1 answer
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A distribution differential equation [closed]

Problem. Find a particular solution $v \in \mathcal{D}'(\mathbb{R})$ such that $$xv=\delta$$ knowing just the basic operations on distributions.
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On fundamental solutions

Definition. Let $L(D)$ be a differential operator with constant coefficients. We say that a distribution $E \in \mathcal{D}'(\mathbb{R}^n)$ is a fundamental solution of the differential operator $L(D)$...
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Dependence of Cauchy principal value in contour integration for Green's function

In evaluating the Green's function $G(x-x')$ (here $x$ and $x'$ are 4-vectors; $x-x'=(x^0-x'^0, \vec{r}-\vec{r}')$) of the wave operator $$\square=\Delta-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}$$ ...
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