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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PDE Poisson-Dirichlet problem on quarter/octave of a 3D space

Given the following Poisson PDE equation $$ \Delta u = f(x) \quad\ \text{for}\ x \in V $$ $$ u = g(x) \quad\ \text{for}\ x \in \partial V $$ where the domain $V$ is equals to the following cases: $$ (...
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21 views

Constants in the fundamental solution of Laplace's equation

I'm taking a course of PDE. I have a problem with the choice of the constants in the fundamental solution of the Laplace equation. In particular, observing that the equation is rotation invariant, we ...
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15 views

Solution of the poisson equation on a rectangle

I have the solution $u(x,y) = \sin(k\pi x) \sinh(k\pi (1-y))$ of the poisson problem $ \nabla^2u(x,y) = 0 \\ u(x, 1) = u(1, y) = u(0, y) = 0 \\ u(x, 0) = \sin(k\pi x) $ Now I thought when my domain is ...
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Is the solution of Poisson equation analytic?

I have two questions about the analyticity of solutions of Poisson equation: Let $f$ be a real-valued smooth function on $\Omega \subset \mathbb{R}^2$, as we know that the Poisson equation $$\Delta u =...
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31 views

Inequality of Poisson boundary value problem

I want to show that the solution $u\in \mathcal{C}^2(\Omega)\cap\mathcal{C}^0(\bar{\Omega})$ to the following Poisson boundary value problem $$ \begin{cases} r=-\Delta u & \textrm{in } \Omega\\ ...
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How can I apply 4th order Runge-Kutta to a Laplacian equation in spherical coordinates?

Can someone please help me solve the Poisson-Boltzmann equation in spherical coordinates over the domain $r\in(r_0, \infty)$ with Runge-Kutta: $$ \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{...
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Solving 1D Poisson equation using finite element method and understanding the Galerkin orthogonality

Let's consider the following test problem $$ u'' = 12x^2 - 36x + 18 \qquad u(0) = u(3) = 0 $$ Analytical solution is $$ u(x) = (x-3)^2 x^2 $$ I'm solving this using the finite element method, ...
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If $\nabla\cdot v=0$ and $w=\nabla^\perp\cdot v$, then $v=\nabla^\perp g\ast w$, where $g$ is the fundamental solution of the Poission equation

Let $\Omega\subseteq\mathbb R^2$ be open, $v:\Omega\to\mathbb R^2$ with $\nabla\cdot v=0$ (in a sense to be specified later), $$\nabla^\perp:=\left(-\frac\partial{\partial x_2},\frac\partial{\partial ...
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How to offset a known solution to the poisson equation?

Given the (2D) poisson equation $\nabla^2u=x+C$ and given $x$ and a known solution $u$ and constant $C$, what is the simplest function $v=f(u)$, such that $\nabla^2v=x$ ? In my concrete case ...
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Numerical grid generation for non-rectangular grids for solving the Poisson equation using finite difference method.

I am trying to solve the Poisson equation using a central difference scheme on a non-rectangular domain. $$ \Delta u = f$$ I came across an online article which has outlined the following method for ...
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Solvability of Poisson equation with Cauchy boundary condition

I am interested in any explanation/comment/reference you could provide me regarding the solvability of the Poisson problem with Cauchy boundary data $$ \begin{cases} -\Delta u = f \ &\textrm{in}\ ...
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The laplacian of the solution of FastPoisson is off by the mean of the RHS

I'm trying to implement a FastPoisson solver to solve the (two-dimensional) equation $\nabla^2u=x$ with Neumann boundary conditions. The algorithm goes like this ($N_x$ and $N_y$ are the width and ...
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Planet Zog: Gravity of a Sphere with Spherical Deletions

Years ago I was given this problem to do. I couldn't manage it at the time but was given the broad strokes of the solution. I came across it again recently and decided to have a go. Could someone ...
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FFT solve Poisson, why amplitude not match

I'm testing the FFT method, use a Poission equation $\Delta u = -16\pi\sin(4\pi x)$ The analytical solution should be $u = \sin(4\pi x)$ My result is more or less similarly, however the amplitude ...
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31 views

Why is $a$ not coercive when it is defined on $H^1(\Omega)$?

Given the Poisson's equation with homogeneous Neumann boundary conditions and the associated bilinear form $$a(u,v) = \int_{\Omega}\nabla u \cdot \nabla v \, dx$$ on $H^1(\Omega)$, why is $a$ not ...
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PDE-Poisson Equation $u_{xx}+u_{yy}=-1$?? [closed]

I want to solve the partial differential equation $$u_{xx}+u_{yy}=-1$$ in the region $0<x<1, y>0$ subject to the boundary conditions $u(0,y)=0, u(1,y)=1$ and $u(x,0)=0$.
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Analytic and harmonic functions in the unit disc (Question 4.1.1 of “Complex Polynomials” by Sheil-Small)

The question (not homework) is Let $f\in\mathcal{H}$ and suppose that $f(0) = 0$ and $\left|f(z)\right|\leq 1$ for $z\in\mathbb{U}$. Show that $$\left|f(z)\right|\leq \frac{2}{\pi} arg\left(\frac{1+...
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Exercise on Poisson processes

I've been trying to solve an exercise related to Poisson. This is the exercise: This is what i did (i am not sure about my answers from points 4 to 10) and honestly i do not understand point 1,2 ...
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Proof of uniqueness of solution of the Poisson's equation for given boundary conditions

I would like to show that the Poisson's equation, i.e., $\nabla^2 \Phi = \rho$, has a unique solution for given boundary conditions, namely, Dirichlet and Neumann boundary conditions. To this end, ...
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Prove that Poisson kernel integrates to $1$: $\frac{1}{2 \pi}\int_{-\pi}^{\pi} P(r,\theta - \Phi) d \Phi=1$

I've been reading a lot about Poisson Kernel and there's always a property that I understand but can't proove. First, $$P(r,\theta)=\frac{1-r^2}{1-2r\cos\theta+r^2}$$ and $$U(r,\theta)=\frac{1}{...
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Existence and uniqueness of solenoidal decomposition $f=f^{\text{s}}+\nabla\phi$ for a vector field $f\colon\mathbb{R}^n\rightarrow\mathbb{R}^n$?

I am interested in the existence of a global solution to the Poisson equation $$ \Delta\phi=g \quad \text{in} \ \mathbb{R}^n $$ such that $\phi\rightarrow 0$ when $|x|\rightarrow\infty$. My motivation ...
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Regularity of the one dimensional Poisson equation

Let $-\infty < a < b < \infty$ and set $U = (a,b)$. A weak solution of the Poisson's equation $\Delta u = f$ subject to $u = 0$ on the boundary with $f \in L^2(U)$, is a function $u \in H_0^1(...
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55 views

Green's Function for Dirichlet problems

I have been studying Green's functions for Laplace/Poisson's equation and have been having some trouble on a few things. In Strauss's book he claims the solution to the Dirichlet problem is: $$u(\bf ...
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Multipole expansion of solution to the Poisson equation

In electrodynamics I have seen the following: Let $\phi$ be a solution to the Poisson equation $-\Delta \phi= \rho$, and assume that $\rho$ is compactly supported. Then we can expand $\phi$ as the ...
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A continuous function for which Poisson's equation has no C^2 solutions

I am trying to solve exercise 4.9 of Gilbarg and Trudinger, and in particular need to show that for the function $f(x)=\sum_{k=0}^{\infty}\frac{1}{k}\Delta(\eta{P})(2^kx)$ the problem $\Delta{u}=f$ ...
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Analytical Solution to Poisson's equation with sinusoidal source

In a domain $\Omega=[-1,1]\times[-1,1]$, consider the Poisson equation \begin{align} -\Delta u=\sin(\pi x)\sin(\pi y) \end{align} inside $\Omega$, and $u(x,y)=0$ for $(x,y)$ in the boundary of $\...
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A saddle point problem for Poisson's equation

I want to prove the well-posedness of the following saddle point problem: Given $f\in L^{2}(\Omega)$ , $\alpha \in \mathbb{R}$. Consider the problem of finding $(u,\lambda)\in H_{0}^{1}\times \mathbb{...
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44 views

Solution to a Poisson equation

I was trying to find the solution $u(x,y)$ to the following Dirichlet problem for the Poisson's equation \begin{cases} -\dfrac{\partial^2 u}{\partial x^2 } - \dfrac{\partial^2 u}{\partial y^2 }= \sin(...
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Solving the Poisson Equation with Mixed Boundary Conditions

I am trying to solve the 2D Poisson equation $$ \triangle u(x, y) = -1$$ with mixed boundary conditions on the edges of a rectangle $[0, a]\times [0, b] $: $$u(x, b) = 0 \qquad u_y(x, 0) = 0$$ $$u_x(...
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31 views

Poisson's Equation on an Ellipse with Constant Source

Suppose I have the problem: $$\nabla^2\phi=1$$ but at the ellipse $\bigg(\dfrac{x}{a}\bigg)^2+\bigg(\dfrac{y}{b}\bigg)^2=1$, $\nabla^2\phi=0$. I have an inclination that the solution may be concentric ...
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19 views

The series expression of Green's function.

I'm new to partial differential equation and I have a question about the Dirichlet Poisson problem $$\left\{\begin{array}{ll} \nabla^{2} u=h \quad \text { in } D=\{0<x<1,0<y<\pi\} \\ u(x,0)...
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37 views

Why is the $1/r^2$ factor retained in Laplace's equation in spherical coordinates?

Consider Laplace's equation in spherical coordinates $$\nabla^2 f = \frac{1}{r^2}\frac{\partial }{\partial r}\left(r^2 \frac{\partial f}{\partial r}\right)+\frac{1}{r^2\sin\theta}\frac{\partial}{\...
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PMF of an awkward combination of Poisson processes

I have two independent Poisson processes A and B, and have a model I wish to fit to the combination A+2B using a maximum likelihood method. I understand that this combination is no longer a Poisson ...
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24 views

Poisson equation on $M\times \mathbb{R}$ with exponential asymptotic condition

Let $M$ be a closed Riemannian manifold, $g$ be a smooth function with compact support on $M\times\mathbb{R}$. Equip $M\times \mathbb{R}$ with the standard product metric. My problem : Determine ...
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Integral formula for Dirichlet pb for Poisson equation in the upper half plane

In order to solve the Dirichlet problem for the Poisson equation $\Delta u(x_1,x_2)=f(x_1,x_2)$ in the upper half plane for a function $$ f(x_1,x_2)=\sin(x_1)\exp(2x_2)$$ with boundary condition $$g(...
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Elliptic regularity Neumann problem in non-Lipschitz domain

Let $\Omega$ be a bounded open connected domain of $\mathbb R^3$ with polyhedral, but not Lipschitz in general, boundary. Let $f\in L^2(\Omega)/ \mathbb R$. We consider the following. Find $\varphi:\...
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Solutions of p-Laplace equation

I found that for the following problem \begin{cases} -\Delta_p u = 1,&x\in B_1(0)\\ u = 0,\quad &x\in\partial B_1(0) \end{cases} where $B_1(0)$ is the unitary ball of $\mathbb{R}^N$ and $\...
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Evans PDE, Problem 5, Chapter 2

I'm taking my first theoretical PDE course in a year and am bashing my head against a rock with this problem. Prove that there exists a constant $C$, depending only on $n$, such that $\max_{B(0,1)}|...
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Finding solutions to $ \int_{\mathbb{R}^d} \Psi(x) \Delta f(x) dx = f(0). $

Define $\Psi(x) = v(\lvert x \rvert)$. Find all the functions $v:(0, \infty) \to \mathbb{R}$ such that for $x \neq 0$, $\Delta \Psi(x) = 0$. Determine all those solutions, if they exist, satisfying ...
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estimation on the Poisson equation

Let $\Omega \subset \mathbb{R}^3$ be a bounded region with the smooth boundary. Consider the Poisson equation with the Dirichlet boundary condition: $$\begin{align*} - \nabla^2 \phi &= f \\ \...
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Poisson PDE solution regularity

I want to know the required conditions for the solution of the Poisson equation to exist in $H^2(\Omega)$. I am considering two cases. Let $\Omega\subset[-1,1]^d$ be a bounded set with 1) $\partial\...
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Holder semi norm estimate for a solution of Poisson's equation in the half space

Let $u\in C^{2,\alpha}_0(\overline{\mathbb{R}^n_+})$, $0<\alpha<1$, such that $$-\Delta u=f \quad\text{in }\quad \overline{\mathbb{R}^n_+}$$ $$ u=0 \quad\text{in }\quad \partial\overline{\...
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37 views

Bound for the derivative of a solution of Poisson's equation

Let $u \in C^\infty(\mathbb{R}^n) $, such that $$-\Delta u=f$$ For a smooth $f$. Show that $\forall r>0$ and $B=B(x,r)$ one has: $$|\frac{\partial u}{\partial x_i}(x)|\leq \frac{N}{r} \text{osc}...
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214 views

Estimate solution of Poisson equation on unit ball

Consider the following boundary value problem where $U=\{x \in \mathbb{R}^3 \mid |x|<1\}$ and $g$ is some nice bounded function, $$\Delta u = g ~~~ \text{on}~U\\ u=0 ~~~\text{on} ~\partial U.$$ ...
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65 views

Probability of a Poisson variable being greater than k others

If one has multiple Poisson random variables (with potentially different lambda parameter), how can one theoretically find the probability that one of those variables is greater than the others? That ...
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26 views

Determining green's function inside a circle for Poisson/Laplacean PDE

In the 2D circle of radius $A$, how does one determine the Green's function for the 2D Poisson Equation? I recognize that you have to use a reflection technique so that the function is 0 on the ...
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42 views

Poisson Equation Solution in Python

I want to solve the Poisson Equation for potential, $V(r)$ at $\delta$. But I am not sure how to do it. I have values of charge density, $\rho(r)$ at discrete values of $r$ stored in an array. Values ...
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77 views

Variational formulation Poisson equation (1d FEM)

I have the same question of this one $\bullet $ In the answer I've seen that a "Lifting" function is used, is it to have a formulation where the test functions are in the same space of the solution? ...
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49 views

Boundary derivative for Poisson equation in the square

Considering the Poisson Equation $-u_{xx} - u_{yy}=1$ in the unit square $\Omega = [0,1] \times [0,1]$, with homogeneous Dirichlet boundary conditions, I am required to compute the second derivative ...
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22 views

Extending harmonic function from half-space to whole space

$u$ is harmonic in $\mathbb{R}^n_+$ and $u=0$ on the boundary. I wish to extend $u$ to a harmonic function on $\mathbb{R}^n$. Suppose I defined $u(x_1,...,x_{n-1},x_n)=u(x_1,...,x_{n-1},-x_n)$ for $...

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