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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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I'm looking for resources that involve concretely taking Lebesgue integral of functions (non-axiomatic and computation focused)

I want to practice finding the Lebesgue integrals of certain functions. My source of inspiration is integrating Dirac delta functions and anything relating to differential equations like Green's ...
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Method of images for a sphere

I know that for a circle, the process for modifying Green's function to account for the boundary condition $u(a,\theta) = 0$ is to (from geometric arguments) create an image at the point $\textbf{x}_0*...
Researcher R's user avatar
1 vote
2 answers
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Given Green's function, can I find the corresponding operator?

Green's function is the solution to the equation $L G(x;x') = \delta(x-x')$, where $L$ is a linear differential operator. Usually, we want to find the Green's function of a given $L$. Instead, if we ...
Sean's user avatar
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Question about Evans’ derivation of a Green's function

At page 34 of "Partial Differential Equations" by Evans, in order to define the Green function for the set $U$, the author defines a family of functions as the solutions of the boundary ...
Lorenzo Vanni's user avatar
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The Helmholtz equation for the spherical harmonics with delta functions

In three dimensions, the Green’s function for the Helmholtz equation with a radiating point source $$ (\nabla^{2}+k_{0}^{2})g(\textbf{r},\textbf{r}')=\delta(\textbf{r}-\textbf{r}') $$ is $$ g(\textbf{...
Chris's user avatar
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Integral of Poisson Kernel

This doubt comes from Dupaigne's book named stable solutions of elliptic partial differential equations. The Poisson Kernel is \begin{equation} P(x,y)=\frac{\partial G(x,y)}{\partial n_{y}}=\frac{1-|x|...
Richard's user avatar
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1 answer
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Green function of a differential equation

Let $$ Lu(x)=a(x)\frac{d^2u(x)}{dx^2}+b(x)\frac{du(x)}{dx}+c(x)u(x), $$ We define its Green function $G_0(x,y)$ by $$ LG_0(x,y)=\delta_x(y) $$ in the sense of distribution. It's esay to get this Green ...
Rayyyyy's user avatar
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How to get the sign right for branch-cut contour integration of the standard free-field propagator

(Apologies for any awkwardness. This is my very first post.) This is a question about how to get the sign right for the classic integral dealt with here Keyhole Contour with Square Root Branch Cut on ...
Alred's user avatar
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1 answer
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Where does this expression for Green's function come from?

In the context of $2$nd order ODEs, I found in some solution sheet that they computed the Green's function using the following expression $$G(x;x')=\dfrac{y_2(x)y_1(x')-y_1(x)y_2(x')}{\overline{\...
Conreu's user avatar
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Deriving the Green's function for the D'Alembert operator from the Helmholtz equation with treatment of $\frac{\partial_t}{c}$ as a scalar

The 3-dimensional Green's function for the Helmholtz operator $$(\Delta_x + \omega^2)G(x,x') = \delta(x-x')$$ is given by $$G(x,x') = G(x-x') = -\frac{e^{\pm i \omega \|x-x'\| }}{4\pi \|x-x'\|}.$$ In ...
theta_phi's user avatar
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What is the procedure for obtaining Green's function in a spherical (or non-Cartesian) coordinate scheme?

With the assistance of those on this board and Haberman's Intro to PDEs, I've managed to use Green's Function to solve 2D PDEs in Cartesian coordinates, now I need to move to a transformed coordinate ...
Researcher R's user avatar
2 votes
0 answers
45 views

Neumann Greens function for the exterior of a ball.

I would like some clarification on the Neumann Green's function for the following Poisson problem: $\nabla^2\phi(x) = f(x) \;\;\;\; x\in R^2/ B(0,1)$ $\hat{n} \cdot\nabla\phi = 0 \;\;\;\; x\in \...
antoniosgeme's user avatar
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Undergraduate references on Sturm-Liouville Theory and Green's Functions

Does anybody know of any clear treatments of Sturm-Liouville theory and Greens functions suitable to accompany undergraduate courses on the subject material? Thanks.
Noah 's user avatar
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Interpreting Green's function in Evans' Partial Differential Equations

Pictures below is from Evans' Partial Differential Equations. $U\subset \mathbb R^n$ is open, bounded. And $\nu$ is the normal vector of $\partial U$. I want to get the last red box from the first ...
Enhao Lan's user avatar
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Needing reference on fundamental set of solution for differential equation on vector fields

Let $L$ be a differential operator on functions $f:\mathbb{R}^N\mapsto\mathbb{R}$. Under some assumptions, we can find the solutions of $Lf = g$ where $f$ and $g$ are both functions from $\mathbb{R}^N$...
Oersted's user avatar
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Trouble on transforming a PDE into an ODE and solving it

I have encountered an issue in a PDE (A Green's function actually). I am solving it in (d+1)-dimensions and I use Poincare coordinates, meaning I have a dimension "z" and I also have d-...
Βασίλης Γερμανίδης's user avatar
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1 answer
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Contour integral of complex generalized function

I'm reading a derivation of the Green function for the hyperbolic PDE and got to the point where I need to evaluate the integral: $$\int_{-\infty}^{+\infty}e^{i\omega t}(\frac{1}{\omega-kc-i0}-\frac{1}...
Krum Kutsarov's user avatar
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1 answer
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Is this about notation of Heaviside function or is my simplification in-correct?

I am reading some introductory material on Green's function from the book "Applications of Green's functions in science and engineering",Greenberg. I have a question related to a boundary ...
ishan_ae's user avatar
1 vote
1 answer
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Calculate the integral with the Gauss Green formula

Calculate the following double integral (directly and with the Gauss Green formula) $$\iint_D xy\,\mathrm{d}x\,\mathrm{dy}$$$$D={(x,y)|y\le x \le \sqrt{y+2}, 0\le y \le 2}$$ directly: $$\int^2_0\int^\...
Pizza's user avatar
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1 answer
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Solution of the parabolic PDE using Green's function

Green's function for the parabolic PDE is defined as: $$\Delta G(\vec{x},t,\vec{\xi},\theta)=\delta(\vec{x}-\vec{\xi},t-\theta)$$ Where $G$ satifies the homogeneous initial and boundary conditions. ...
Krum Kutsarov's user avatar
1 vote
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Numerical integration with modified Bessel function of second kind

I am working with the so-called screened Poisson PDE, whose solutions in two-dimensions are given in terms of the modified Bessel function of the second kind, $K_0$, for Dirichlet boundary conditions ...
Woe's user avatar
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1 answer
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What is monopole, mathematically?

$\newcommand{\realset}{\mathbb{R}}$ $\newcommand{\lapop}[1]{\nabla^{2} #1}$ $\newcommand{\norm}[1]{\left\lVert #1 \right\rVert}$ $\newcommand{\shift}[1]{\tau_{#1}}$ Recently, I wrote down the ...
Ziqi Fan's user avatar
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1 answer
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What justifies the $\epsilon \rightarrow 0$ limit in the domain of this integral?

I am following these notes on Green's function for Poisson's equation, which are based on Evan's PDE book. Let $\Omega \subset \mathbb{R}^n$ be open and bounded. Let $u \in C^2(\overline{\Omega})$ be ...
CBBAM's user avatar
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5 answers
185 views

How to get/determine the Green's Function?

I want to determine the Green's function of $$t y^{\prime \prime}(t)+y^{\prime}(t)=t, \quad y(1)=y(e)=0.$$ I have even solved the ODE with the initial conditions, but I do not know how to determine ...
Barbara María's user avatar
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37 views

About kernel, Green functions.

Consider the heat equation \begin{align} \partial_t u(t,x)&=\Delta u(t,x),\quad t>0,\, x\in\mathbb{R}^n\\ u(0,x)&=u_0(x) \end{align} The Green function (or kernel) is $G_t(x)=\int_{\mathbb{...
eraldcoil's user avatar
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Are the Particular Integral and Complementary Function orthogonal?

I am a physics student so please forgive my lack of rigour. Suppose I have a Linear, Inhomogenous O.D.E given by Ly(x) = f(x). I may construct a solution from the Greens function, G(x,z), by ...
Oliver Gregory's user avatar
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Solve a trick differential partial equation

So basically I have came across a partial equation in the $AdS_{d+1}$ spacetime, $$z_0^{d+1} (z_0^{-d+1} \phi_{,0})_{,0} + z_0^2 \phi_{,jj} = \delta^{d+1}(x-v)$$ Where $,k$ means derivative wrt $k$. ...
Lac's user avatar
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Solution to nonhomogeneous heat equation via Green function

Consider the heat equation $$ u_t = \epsilon u_{xx} \quad x\in (0,1)$$ where $\epsilon >0$ is constant. Suppose that this equation is subject to the boundary conditions $$ u(0,t) = \alpha (t), \...
Galois's user avatar
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4 votes
4 answers
88 views

Transition probability density function for a non-trivial diffusion process.

Let $\mu$ and $\sigma > 0$ and $\beta_1 \ge 0 $ and $\beta_2 \ge 0$ be real numbers. Consider a stochastic process $X_t$ that satisfies the following stochastic differential equation: \begin{...
Przemo's user avatar
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1 vote
1 answer
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jump condition of a Green's function

Find the Green's function for the BVP $$y''-\frac1xy'=0 \ \ ; \ \ y(0)=y(1)=0$$ Clearly the operator is not self-adjoint, so the equivalent self-adjoint equation is $$\left(\frac{y'}{x}\right)'=0$$ ...
am_11235...'s user avatar
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0 answers
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Regularisation for a differential equation

When stuying a 1D differential equation, I tried to solve the problem by finding the Green's function and then solving with the inverse Fourier transform. To make the problem well-behaved, I used a ...
AxelT's user avatar
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1 vote
2 answers
97 views

Potential density of killed brownian by local time

Suppose $B_t$ is a standard Brownian motion on $\mathbb{R}$ and let $L_t$ be its local time at zero. Let $p_t(x,y)$ be the transition density of $B_t$, i.e. $p_t(x,y) = \frac{1}{\sqrt{2\pi}}\exp\left(-...
JY0's user avatar
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0 answers
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Conormal Derivative and Green Function in integral

I have the following question that starts from this expression: $$ u(x)=\int_{\partial B}\left(-\frac{\partial u}{\partial n}(y) \Gamma(y-x)+u(y) \frac{\partial \Gamma}{\partial n}(y-x)\right) d \...
Richard's user avatar
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1 answer
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The Green function of Schrodinger operator is positive for non-negative q(x)

Consider $\left(E_0\right): L y:=-y^{\prime \prime}+q(x) y=0$ with non-negative function $q \in C(\mathbb{R})$. Let $y_1$ be the solution of $\left(E_0\right)$ satisfying $y_1(0)=0, y_1^{\prime}(0)=1$;...
YuerCauchy's user avatar
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0 answers
17 views

Is there an analytical solution for the three-dimensional time-dependent parabolic equation (using Green's functions)?

problem: $\frac{\partial w}{\partial t}=a_1(x,t) \frac{\partial^2 w}{\partial x^2}+\Phi(x, y, z, t)$ with $w=f(x, y, z) \quad$ at $\quad t=0$.(initial condition) $\frac{\partial w}{\partial x}+k(\...
Yilin Cheng's user avatar
2 votes
2 answers
203 views

Symmetry of Green's Functions

In the book Introduction to Partial Differential Equations by Folland, he states the claim Let $ \Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $ S$. The Green's function $G$ for $...
newbie's user avatar
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2 votes
1 answer
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Integral of the fundamental solution of the laplace equation over a circle

I asked a similar but different question earlier here (this is not a duplicate) I'm interested in solving this problem in closed form, if such a solution exists, in $2d$. $$\int_{\Gamma_R} G\; d\...
Cedric Martens's user avatar
2 votes
0 answers
51 views

Integral of the normal derivative of the fundamental solution of the laplace equation over a circle

I'm interested in solving this problem in closed form, if such a solution exists, in $2d$. $$\int_{\Gamma_R} \frac{\partial G}{\partial \vec{n}} d\Gamma_R$$ Where $\Gamma_R$ is a circle of radius $R$ ...
Cedric Martens's user avatar
0 votes
1 answer
36 views

Solving homogeneous differential equation with boundary condition using Green functions

I have an homogeneous equation $Lf=0$ where $L$ is an operator and $f$ a function, with boundary conditions $f(\partial\Omega)=g(\partial\Omega)$ where $g$ is a known function. In the case when the ...
J.A's user avatar
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0 answers
20 views

Asymptotic behavior of a 2D integral: retarded/advanced green's function with saddle point dispersions

I want to analytically evaluate the 2D integral for some real $E< 1$ $I^\pm(E)=\lim_{\eta \rightarrow 0}\int_{-1}^{1} dx dy \frac{1}{E\pm i\eta -xy}$ In particular, I want to understand the ...
Yidan's user avatar
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1 answer
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I need to verify if I set up the following integral for $u(\textbf{x})= \int\int f(\textbf{x}_0)G(\textbf{x},\textbf{x}_0) dA_0$

I have a 2-D Poisson Equation with homogeneous boundary conditions, and I used an eigenfunction along x, and a Fourier sine series expands the solution to $u(x,y) = \sum_{n=1}^\infty a_n(y)\sin(\frac{...
Researcher R's user avatar
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0 answers
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How to solve out Green's function?

I am studying the Scattering theory by John R. Taylor but I met a mathematic problem in Section 11-h. $\phi_{lp}(r)$ is the solution of radial Schoedinger equation: $$ \Big[\frac{d^2}{dr^2}-\frac{l(l+...
Hsu Bill's user avatar
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1 vote
0 answers
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Reference request of the formula of Green function on compact manifold with no boundary

I just met a result in Lemma5.1 of ref1 which seems interesting and counterintuitive, it said that on 2-dimension torus $\Omega$, let $G$ be the Green's function to $-\Delta$ on $\Omega$, satisfying $\...
Elio Li's user avatar
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0 answers
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Question about the fomula of Green function of Laplacian on closed manifold.

I'm reading a paper which said that the Green function for $\left(-\Delta_g\right)^m$ on $2m-dimension$ closed manifold is of the form $$\tag{1} G_y(x)=\frac{2}{\Lambda_1} \log \frac{1}{d_g(x, y)}+\...
Elio Li's user avatar
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1 vote
0 answers
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How to set up integral for solution to PDE using Green's function?

I've found Green's function, the problem is I've never done the integration part to find the solution for a 2-D problem or higher before, so I'd like a little guidance for setting up the integral. The ...
Researcher R's user avatar
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0 answers
65 views

How to derive a key equation in Sturm-Liouville Theory

Recently bombed a quiz on Sturm-Liouville theory and orthogonal polynomials in my math methods for physics class, and I'm trying to go through the chapter on the theory and plug up the holes in the ...
oleosquarewave's user avatar
5 votes
0 answers
167 views

Basic properties of Green function and resolvent

I am trying to understand better the properties of the resolvent and Green function of a bounded self-adjoint operator $H$ at $z=E+i\eta$ when $\eta \to 0^+$. The resolvent is the operator $R(H;z)=(H-...
Keen-ameteur's user avatar
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Solving a 2-D BVP using Green's Function. Need guidance on obtaining $G(x,y,x',y')$

This is a generic example of a Poisson Equation with homogeneous BCs which I am reading the solution to in Haberman's Applied Partial Differential Equations with Fourier Series and Boundary Value ...
Researcher R's user avatar
1 vote
0 answers
93 views

Green's function for the screened Poisson equation in an n-dimensional ball

Consider the screened Poisson equation $\Delta u(r) - cu(r) = -g(r)$, in a ball (centered at the origin) of radius $R$, with boundary condition $u(R)=0$. In 3-dimensional space, the Green's function ...
ricm's user avatar
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0 answers
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Solving a boundary value problem (BVP) partial differential equation using Green's function

I have not seen a worked out solution to a 2-D BVP before, so I went over one of the exercises in Richard Haberman's Applied Partial Differential Eautions and picked a random example that I thought ...
Researcher R's user avatar

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