Questions tagged [poissons-equation]
In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. (Def: https://en.wikipedia.org/wiki/Poisson%27s_equation)
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convergence of Poisson equations
Let $E$ be a locally compact separable metric space and $B(E)$ be a Banach space of bounded measurable functions on $E$. For each $n \in \mathbb{N} $, $L^n$ and $L$ are linear operators on $B(E)$ ...
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Where can i find a reference with explanation of each term about Poisson equation in anisotropic media?
I studied about Poisson equation on anisotropic media with equation as such
$$
k_{11}\dfrac{\partial ^2 u}{\partial x^2} + (k_{12} + k_{21})\dfrac{\partial ^2 u}{\partial x \partial y} + k_{22}\dfrac{\...
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Gradient estimate for second-order uniformly elliptic equation in divergence form
Motivation for the following question: I am currently reading this paper by Bernard and Rivière. In Lemma VI.1 in equation (VI.10), they seem to apply a certain estimate for the gradient of an ...
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How to correctly calculate Poisson's equation for electric potential using FFT with zero-padding?
I'm working on a program that simulates the electrostatic field in 3D using FFT to solve Poisson's equations based on the following formulas:
$$
\phi_{(k)} = \frac{\rho_{(k)}}{\epsilon_0 \times K^2}
$$...
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What is wrong with using the $H^1_0$ inner product here?
This question is a problem I am toying with.
Consider the Poisson equation,
$$-\Delta u =f \ \text{in} \ \Omega \times (0,\infty).$$
$$u=0 \ \text{on} \ \partial \Omega \times (0,\infty)$$
Suppose the ...
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Solving the Radially Symmetric Poisson Equation with Exponential Source Term
I want to solve the poisson equation
$$
-\Delta u(\mathbf{x})=\rho(\mathbf{x})=\frac{e^{-|\mathbf{x}|^2}}{|\mathbf{x}|^2-1}
$$
The problem want me to use the fundamental solution of laplace operator, ...
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Singularity in Poisson's Equation
Consider an instance of Poisson's equation in spherical coordinates for the radial dimension:
$$
\nabla \cdot \nabla \phi(r) = \frac{1}{r^2} \frac{d}{dr}\left(r^2 \frac{d\phi}{dr} \right) = -\sin(r).
$...
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How to use Fourier method to solve the Poisson equation $-\Delta v=x y$?
Suppose that $B_{1}$ is a ball centered at $0$ with radius $1$, consider the equation $-\Delta v=x y$ and $v=0$ on boundary, I want to use Fourier method to solve it, but the Fourier transform is ...
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Proving a bound on $|\nabla w(0)|$ for a solution to $\Delta w = f(w)$
Let $f \in C_c^{\infty}(\mathbb R)$ with $0 \leq f \leq 1$ on $\mathbb R$. I am trying to prove the following:
Suppose $w \in C^\infty(B_3(0))$, $w \geq 0$, and solves $\Delta w = f(w)$ on $B_3(0)$. ...
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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 ...
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Help me intuit Equal Probabilities in Poisson Distribution for $k = λ$ and $k = λ-1$
I was trying to understand Poisson distribution and I'm confused as to why the likelihood of $k=λ-1$ is equal to $k=λ$. Here is my understanding and where my confusion is:
For a given time interval, ...
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Solving the Poisson equation $-\Delta u=f$ on a domain $G$
Let $G$ be a domain and $f\in C_c^1(\mathbb{R}^n)$. We want to solve
$$-\Delta u =f \text{ in } G$$
without any boundary condition. Can we just take $\Delta^{-1}$ the inverse Laplacian on $\mathbb{R}^...
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Solution of $-\Delta u=f$
Fact 1. The function $1/|\xi|^{s}$ locally integrable (in the unit ball) if and only if $s<n$.
Fact 2. If $f\in L^1(\mathbb{R}^n)$, then $\widehat{f}\in\mathcal{C}^{\infty}(\mathbb{R}^n)$ (bounded)....
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Kronecker product and finite difference discretization for poisson equation
In this notebook from MIT's Intro to Linear PDEs course, it is unclear to me why the Kronecker product is used to formulate the coefficients matrix $A$ for solving the linear system of equations $A u =...
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Maximum principle to get $C^0$ estimates in terms of $L^1$ norms
I'm reading this paper https://arxiv.org/abs/1401.7366 and trying to prove Corollary 4.6.
The result essentially says the following. Let $B_r \subseteq \mathbb R^4$ be the ball of radius $r$ with the ...
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