# 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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1 vote
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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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### 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 $\textbf{P}=\textbf{P}(x,y,z)$,...
1 vote
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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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### 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 ...
1 vote
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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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### 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 ...
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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### 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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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. $$... 2 votes 0 answers 40 views ### Four-dimensional Dirac delta from a Green's function Consider four-dimensional space \mathbb{R}^2 \times \mathbb{C} with coordinates w=x,y,z,\bar{z}. Define D^i = (\partial_x,\partial_y,4\partial_z), and consider the following equation for a Green'... 1 vote 0 answers 37 views ### Green's generating function for random walk in \mathbb{Z}^d. Let (X_t)_{t\geq0} be a simple symmetric random walk in \mathbb{Z}^d, in continuous time (holding times are Exp(1) random variables). Moreover, suppose X_t is killed after walking for a time ... 0 votes 0 answers 7 views ### Ultra light renewal theorem for heavy tailed renewal processes I am interested in some "ultra light renewal theorem" in the following sense: We look at the renewal process$$\mathcal R := \left\{ n: \sum_{i=1}^k R_i = n \mbox{ for some } k \right\}$$... 0 votes 0 answers 63 views ### How should I understand (8.4.6) from a book by G.Barton? My question concerns trying to understand a paragraph of material from a book by Barton, Reference 1. This material is from Chapter 8 ‘The diffusion equation: I. Unbounded space’, the paragraph is ... 0 votes 0 answers 17 views ### Complex Conjugate of a Function for a given mapping I have a complex function which actually is a Green Function of a Laplacian Operator in the Upper plane (y>0). I want to use a conformal mapping to map a circle into an upper half-plane (y>0) ... 1 vote 0 answers 61 views ### Asymptotics of Green's function for Laplace-Beltrami operator In the following, let M be a compact Riemannian manifold and \Delta = \nabla \cdot\nabla  a Laplace-Beltrami operator on M. Let G_y(x) be Green's function for \Delta (a.k.a fundamental ... 0 votes 0 answers 19 views ### Estimate for Green's function in \Omega \subset \mathbb{R}^2. Can we show that |\nabla G(x,y)| \le \frac{C}{|x-y|} ? Assume \Omega is a smooth bouned domain of \mathbb{R}^2. Let$$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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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 ...