Mathematics Stack Exchange
Mathematics Stack Exchange

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)

In mathematics, Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. It is given by $\nabla^2\varphi=f$ where $\varphi,f$ are real- or complex-valued functions on a manifold. Reference: Wikipedia.

It is used, for instance, to describe the potential energy field caused by a given charge or mass density distribution.

483 questions
3
votes
0 answers

Existence of classical solution for Poisson's equation

I read a bit through uniqueness of the solution to Poisson's equation with boundary data and also existence/uniqueness of the variational formulation. But under what conditions can one guarantee the existence of a classical solution for Poissons…
Tesla
  • 1,380
3
votes
1 answer

Conditional Probability in Poisson- How To Calculate The Probability That A Soccer Team Wins From Behind

I have been cracking my head trying to get a solution to this problem for awhile, but I can't seem to get it: "In a soccer match, given team A and team B are both expected to score 1.3 goals each, and that goals follow a poisson distribution, what…
clattenburg cake
  • 211
2
votes
0 answers

Variance of two poisson processes

I have two Poisson processes with the same rate $\lambda_1$. If I choose to model them using single superposed process, it will have rate $\lambda_2 = 2 \lambda_1$. From here it is obvious that the combined distribution will have $2\!\times$ higher…
Morty
  • 21
1
vote
0 answers

$\nabla^2 f =\delta$ with two boundaries

I was wondering what are the techniques ideas to solve the following problem, consider Poisson equation for a delta source $$\nabla^2 f(x,y,z)=\delta(x-x_0)\delta(y)\delta(z)$$ where $x,y,z$ are Cartesian coordinates, $0
Mauricio
  • 385
1
vote
1 answer

Solution of Poisson's equation in 3D, where $f(\vec{x})=\chi_{B(0,1)}(\vec{x})$

Find a solution $u$ of Poisson's equation $-\Delta u=f$ in 3D that corresponds to $f(\vec{x})=\chi_{B(0,1)}(\vec{x})=\begin{cases}f=1 & \vec{x}\in B(0,1)\\f=0&\hbox{otherwise}\end{cases}$. The general/explicit solution for Poisson's equation is…
Gregoire Rocheteau
  • 1,019
1
vote
0 answers

Poisson's equation / Helmholtz-Hodge decomposition on a sphere

Given a vector field $X^a$ on a sphere, I want to decompose it into a surface divergence-free component and a surface curl-free component (similar to the Helmholtz decomposition, but on the 2-dimensional sphere $S^2$ instead of $\mathbb{R}^3$). The…
Anastasia
  • 21
1
vote
0 answers

How to express a Poisson regression equation as a quasi-Poisson

I was reading this paper: http://www.researchgate.net/publication/13878515_A_simple_non-linear_model_in_incidence_prediction and was wondering whether the equations 1 and 3 presented in the paper to represent modeling of cancer incidence using a…
Kaleb
  • 105
1
vote
1 answer

Poisson equation with a source term that is a full divergence

Can the Poisson equation be solved exactly when we know that the source term is a divergence? $\nabla^2 \psi = \nabla \cdot \mathbf{F}$
user250673
  • 11
1
vote
1 answer

2D Poisson equation with Dirichlet and Neumann boundary conditions

The problem is given as follows \begin{align} -\Delta u &= f, \text{in} \: \Omega \\ u &= 0, \text{on} \: \delta \Omega_D \\ H(u) &= 0, \text{on} \: \delta \Omega_N, \end{align} where $\Omega$ is the domain to be considered and the Dirichlet and…
harisf
  • 329
0
votes
0 answers

Solution of $-\Delta u=f$

Fact 1. The function $1/|\xi|^{s}$ locally integrable (in the unit ball) if and only if $s
eraldcoil
  • 3,508
0
votes
0 answers

Poisson's equation and Fourier transform.

Is it possible to solve the Poisson's equation with the Fourier transform? Let \begin{align} -\partial_x^2 u(x)=f(x),\quad x\in\mathbb{R} \end{align} then $u(x)=\mathcal{F}^{-1}\left( \frac{1}{\xi^2}\widehat{f}(\xi)\right)$?
eraldcoil
  • 3,508
0
votes
0 answers

How to approach a Poisson's Equation of a square domain

As someone with severe Asperger's I can't understand programming and after 100s of hours trying to learn it just doesn't click. I am looking for any suggestions on material to learn to solve an equation such as Poisson equation on a square domain of…
David Scidmore
  • 345
0
votes
1 answer

Discretization of 1D Poisson Equation

Consider the one-dimensional Poisson’s equation $$−u''(x) + u(x) = f(x), \hspace{5mm} x \in (a, b),$$ with $u(a) = g_{1}$, $u'(b) = g_2$. Discretize the equation using the finite element method with piecewise linear basis functions. I am not sure…
taupi
  • 377
0
votes
1 answer

Solving Poisson equation on square column of infinite length

My professor personally asked me to solve the following problem : Let $\Omega=(0,1)\times(0,1)\times\mathbb{R}$. \begin{cases} -\triangle u=f & \text{in }\Omega,\\ u=0 & \text{on }\partial\Omega, \end{cases} which is a Poisson equation on square…
kayak
  • 1,031
0
votes
0 answers

Show the solution to Poisson's equation in $\mathbb{R}^2$ is well-defined

In Mathematical Theory of Incompressible Nonviscous Fluids by Marchioro and Pulvirenti, the authors define a function $$ \Phi(x) = -\frac{1}{2\pi}\int_{\mathbb{R}^2} \log|x-y|\omega(y)\ dy$$ (the solution to Poisson's equation $-\Delta \Phi =…
Relativity
  • 11
1
2