# 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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### evaluate integral with the kernel of Green function (fundamental solution of Poisson equation) of infinite domain using conformal map

I must find the following definite integral $$\phi(x,y) = \int_{0}^{\alpha} f(\theta)\frac{-1}{2\pi} \ln \sqrt{(x-\rho\cos\theta)^2+(y-\rho\sin\theta)^2} \rho d\theta$$ for a given $f(\theta)$ which ...
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### Find the Green's function of the differential operator for periodic boundaries.

On the space of functions $f(x)$ where $x \in (-\pi,\pi]$ satisfying $\int^{\pi}_{-\pi} f(x)dx = 0$ and periodic boundary conditions, find the Green's function for the operator $-\frac{d^2}{dx^2}$. I ...
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### Fundamental Solution of a heat equation with a drift term

Let consider the following equation on $[0,T]\times \mathbb{R}$: $$\partial_t u + \beta \partial_x u = \alpha \partial_{xx} u.$$ Question: Is it possible to construct a fundamental solution for ...
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### On the solution of the heat equation using distribution theory

The Green's function $$\tag{1} \displaystyle K(t,x,y)={\frac {1}{(4\pi t)^{d/2}}}e^{-\|x-y\|^{2}/4t}$$ solves the heat equation $${\frac {\partial K}{\partial t}}(t,x,y)=\Delta _{x}K(t,x,y)\$$ for ...
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### Use Green's Function to solve ODE

I have the following ODE to solve via Green's function: $$\frac{d^2y}{dx^2} - y = F(x)$$ for $0<x<1$ with boundary conditions $y(0)=y(1)=0$. So far I have done the following: I try a Green's ...
1 vote
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### integral over Green's function of a $d$-Ball

I'm studying for an exam on PDE's and I'm stuck trying to solve the following problem. This is in context of the following boundary value problem: \begin{align} \begin{cases} -\Delta u \equiv 0,&\...
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### A priori estimate of solution of Poisson equation

I'm studying for an exam on PDE's and came across the below problem. I found an approach for solving it, but am missing a last argument. Can you help me conclude my proof? This is related to this ...
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### Green function of the 2-dimensional "curl" operator

I want to find the solution to the equation: $$x \partial_y f - y \partial_x f = \delta^{(2)}(x,y) \,,$$ where $f : \mathbb{R}^2 \to \mathbb{R}$ is a scalar function, and $\delta^{(2)}(x,y)$ is the ...
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Consider the Poincare disk $\mathbb{H}^2$, that is the unit disk endowed with the following metric $$g_{ij}=\frac{2}{1-|x|^2}\delta_{ij}.$$ I believe there exists a global Green's function on \$\...