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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Issues with Convergence of Finite Difference Method on Uniform Grid

I've written some code that carries out the central finite difference equation to solve this poisson system: $\Delta u(x,y) = f, u(x,y) = g$ in domain $[0,1]^2$. where $u(x,y) = \cos(4𝜋𝑥) \cos(4𝜋𝑦)...
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Lax-Milgran theorem with the Poisson equation

Let $ \Omega $ be a bounded domain in $\mathbb R^3$ with smooth boundary. Consider the Poisson equation $$ -\Delta u=f $$ where $ f\in C_0^{\infty}(\Omega) $ and $f$ is null outside $\Omega$. I'm not ...
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Maximum principle for a strong solution to non-homogenous Laplace equation

I am searching for a reference for this apparently well known fact (the part below Theorem 1.1 in the picture i.e. the equation $(6)$): This screenshot is from https://math.aalto.fi/~astalak2/files/...
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reference for poisson equation on unbounded domain

Let $\Omega\subset\mathbb{R}^3$ be some unbounded exterior domain, with $C^{\infty}$ smooth boundary. Consider the Poisson equation with Neumann boundary condition \begin{align} &\Delta u = f &...
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Construct a solution of Laplace/Poisson problem with a non constant gradient jump

consider the square $[-1,1]^2$ and a ball of radius $R$ entered at the origin $B_R(0)$. The function $u(x,y)=- \frac{\ln(\max(r^2,R^2))}{2}$ solves the Laplace problem $-\Delta u=0$, and the jump of ...
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Analytical solution of 2D Poisson's equation with Dirichlet boundray condition

I am trying to find the analytical solution to the following 2D Poisson's equation whose domain is a unit circle: \begin{align} -\Delta u = 1&, x \in \Omega \\ u = 0&, x \in \partial \Omega \...
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Numerical Solution of nonlinear P-B Equation in unbounded domain for determining the EDL potential distributions around a spherical particle

For my project I am studying a paper, namely "Perturbation solutions for the nonlinear Poisson–Boltzmann equation with a higher order-accuracy Debye–Huckel approximation" by Cunlu Zhao, ...
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Eigenmodes of 2D Poisson matirix with Neumann boundary conditions

I am trying to find analytic expressions for the eigenvectors (and eigenvalues) of the 2D discrete Poisson matrix, in the case of zero Neumann boundary conditions. [ pic 1 ] In my case, I'm using a ...
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Fractional Sobolev regularity of the solution to the fractional Dirichlet problem on a ball

While reading some notes due to Markus Faustmann, I came across the following exercise regarding the fractional Dirichlet problem on the unit ball. \begin{equation*} \left\{ \begin{aligned} (-\Delta)^\...
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$\nabla^2 f =\delta$ 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 ...
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Holder regularity for solutions to the Poisson equation

I am dealing with a nonlinear PDE of the form $$-\Delta u=f(u),\quad in\,\,\,\mathbb{R}^n$$ (where $f(u)$ is a nonliear function) I would like to ask you which regularity results do exist in the case ...
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Derivation of adjoint poisson equation with convective boundary condition

I am trying to derive the adjoint Poisson equation for the following problem to find the sensitivity of an objective function with respect to a decision variable, but I get stuck in the middle of the ...
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Eigenvalues for discretization matrix in Poisson equation with finite difference

I am trying to find the eigenvalues for the discretization matrix in the Poisson equation using the Chebyshev polynomials, i.e. $$ -u''(x) = f(x), x \in [0,1],\;\; u(0)=u(1)=0 $$ Discretize the ...
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Poisson's equation - error estimate

Background info for excercise Previously we have been asked to show that $||{u'-u_N'}||\leq \inf_{v \in V_N} ||{u'-v'}||$ and $|{u-u_N}|^2\leq \min \{x,1-x\}(\inf_{v \in V_N} (||{u'-v'}||)^2.$ We are ...
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Finite elements method: why test fuction vanishes on boundaries

I am trying to understand why (and exactly when) the test functions must vanish at the boundaries when Dirichles conditions are applied to a PDE. The context is the learning of the finite element ...
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What does the wavenumber refer to when solving the Poisson equation with fourier transforms?

I am working on my senior project in college on $n$-body gravitational simulations solving the Poisson equation using Fast Fourier Transforms. After realizing that I did not understand the math well ...
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Inequality between $\|u\|_{H^3}$ and $\|\Delta u\|_{H^1}$

Consider the Poisson equation $$-\Delta u =f$$ I have read a paper stating that $$ \|u\|_{H^3} \leq C \|f\|_{H^1} $$ for some constant $C$. Here $u$ and $f$ are given sufficient smoothness so that the ...
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Solving Laplace's Equation on Complex Geometries Using Conformal Maps

I am interested in solving Laplace's equation $$\Delta u = 0.$$ with Dirichlet boundary conditions but on complex geometries. To my understanding, the standard approach is to (if possible) map your ...
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Generalized mean value property for the Poisson equation

It is well known that solutions to the Laplace equation in a region $\Omega\subseteq\mathbb{R}^n$, $\nabla^2u=0$ satisfy the mean value property, namely for all $x\in\Omega$, and for all $r>0$, $$ ...
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inverse laplace operator bounded

Given $\Omega \subset \mathbb R^d $ open and bounded, we define the operator $T:L^2(\Omega) \to H_0^1 (\Omega)$ as $T(f)=u$ where $u \in H_0^1(\Omega)$ is the unique weak solution to the Poisson ...
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How can I derive the equation of this paper(Markov Chains, Poisson Equation)?

I am currently reading the paper titled Actor-Critic Algorithms. I can not understand part of paper saying 'We are interested in minimizing $\lambda(\theta)$ over all $ \theta $. For each $\theta \in \...
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Existence of Laplacian solution inside a high-dimensional hypercube

Consider this "Neumann problem" with Laplace's Equation: Find a solution to $\nabla^2\phi=0$ inside an N-dimensional hypercube which has a given (on the $2N$ faces of the hypercube) normal ...
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Behavior of u if initial data are compactly supported

Recall the Kirchhoff's $$ u(x, t)=\mathrel{\int\!\!\!\!\!\!-}_{\partial B(x, t)} t h(y)+D g(y) \cdot(y-x)+g(y) \mathrm{d} \sigma_{y}, \quad x \in \mathbb{R}^{3}, t>0 $$ and Poisson's fomulas $$ u(x,...
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Stationnay profile for diffusion-consumption equation in 2 dimensions with a point source

I'm looking for the stationnary solution of the following equation in 2 dimensions : $$\nabla^2 f(r)=f(r) $$ $$f(0)=f_c$$ $$f(+\infty)=0 $$ in radial symmetry. $\nabla^2$ corresponds to the Laplacian, ...
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Solve Laplacian equation with Green's function in 1D

I have a 1D Laplacian equation $\nabla^2 u(x) = 0$ with some boundary condition $u|_{\partial \Omega} = g$, where the domain $\Omega = [0,L]$ and the boundary condition is Dirichlet, that for some $x$,...
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Range $u_{\mathrm{max}}-u_{\mathrm{min}}$ of the solution to graph Poisson equation $Lu=b$

I found out an interesting phenomenon when trying to solve the linear equation $Lu=b$, and I don't know how to interpret such phenomenon. Goal: Fit $u(x, y)=\cos(x)$ for $x,y \in [0,\pi]$. We draw ...
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Solution of Poisson equation for $f(x,y)=1$

Consider Poisson equation $$ \Delta u=f $$ To be more specific, $$ \frac{\partial^2u}{\partial x^2}+\frac{\partial^2u}{\partial y^2}=f(x,y) $$ What would be the analytical(Exact) solution of the above ...
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Understating the proof of the existence theorem of Poisson integral from Rudin

I have some problem with the understanding of the following theorem from rudin: Theorem $11.7$: If $f \in C(T)$ and if $Hf$ is defined on the closed unit disc $\bar U$ by $$(Hf)(re^{i\theta})=\begin{...
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Neumann Boundary Condition at the corner points of a domain

Let's say there's a Poisson equation to solve. The domain is a unit square $\Omega=(0,1)\times(0,1)$. It is subject to Neumann boundary condition $\partial_nu=g(x,y)$, for $(x,y)\in\partial\Omega$, ...
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Neumann Boundary Condition around a square

Let's say there's a Poisson equation to solve. The domain is a unit square $\Omega=(0,1)\times(0,1)$. It is subject to Neumann boundary condition $\partial_nu=g(x,y)$, for $(x,y)\in\partial\Omega$, ...
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Bounding the supremum norm by the $L^2$-norm of the Laplacian

For a smooth, bounded domain $\mathcal{O} \subset \mathbb{R}^n$, consider $u$ the solution to $$ \begin{cases} \Delta u &= f \qquad \text{ on } \mathcal{O}\\ u &= 0 \qquad \text{ on } \partial ...
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How to solve given Poisson's equation to get shown answer?

I tried for a week but couldn't get the answer as shown. Can someone clarify ? Poisson equation and it's domain shown in the fig. Answer is shown in the last. Click here to see image.
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Poisson's equation in polar coordinates: non-homogeneous PDE with homogeneous Dirichlet BC

In polar coordinates, I am interested in the particular solution of $\Delta u=f$ on a disk of radius $a$ with boundary conditions $u(a,\theta)=0$. There are well-known solutions using Green's function ...
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How do Green's functions satisfy boundary conditions?

Suppose we are interested in solving Poisson's equation. The approach behind solving this problem is to use Green's functions, and so far this is an informal summary of what I have understood of the ...
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How to show using the chain rule that $u_{x x}+u_{y y}=u_{r r}+\frac{1}{r} u_{r}+\frac{1}{r^{2}} u_{\theta \theta}$? [closed]

In solving a potential problem by the separation of variables method in a circula it is necessary to express the problem in polar coordinates. By setting $$ x=r \cos \theta, \quad y=r \sin \theta $$ ...
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Looking for book on using 2D Fast Fourier Transforms

I’m looking for a book on how to use 2D Fourier Transforms. One with examples in code or involving finite data is preferred, but I’ll take one that doesn’t so long as its a good book that uses 2D ...
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Solving Poisson's equation $∇^2w+1=0$

I'm trying to solve Poisson's equation $∇^2w+1=0$, where $w=0$ on a 2 by 1 rectangle boundary. I just learned Poisson's equation so I'm really not familiar how to do that (I wish I could show what I'...
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$L^p$ estimate for Poisson's equation, Gilbarg's 1998 ed

I am reading this classic to learn elliptic equations. Corollary 9.10 on pp. 235 states for $u\in W_0^{2,p}(\Omega)$, $1<p<\infty$: First, $$ (9.33) \quad \|D^2 u\|_p\le C(n,p) \|\Delta u\|_p $$ ...
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Why the inlet and outlet fluxes are different when I solve the Poisson equation by FEM over a trapezoidal domain?

I solved the Poission equation which is given by \begin{equation} \Delta h = 0, \end{equation} where $h$, in my case, is the hydraulic pressure. Because I want to solve steady-state flow, the $\nabla ...
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Does this function live in $H^{\frac{1}{2}}(\partial \Omega)$

In the Poisson equation $$-\Delta u=f$$ $$u=g \in \partial \Omega$$ we usually see in textbooks the requirement $g \in H^{\frac{1}{2}}(\partial \Omega)$ and $u \in H^1(\Omega)$. In my case, $\Omega = [...
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How to solve a pair of coupled Poisson equations with inhomogeneous boundary conditions?

I am trying to make some code that will solve the following 2D Poisson equations: $$\left(\frac{\partial^2}{\partial x^2}+\frac{\partial}{\partial y^2}\right)P(x,y) = f(x,y),$$ $$\left(\frac{\partial^...
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Why do we write $H_0^1(\Omega) \cap H^2$ instead of only $H^2_0(\Omega)$?

I've seen that when we deal with Poisson equation with homogeneous boundary conditions, let's say in 2D with a convex domain $\Omega$, we write that the regularity of $u$ is $H^2 \cap H_0^1(\Omega)$. ...
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Green’s function of poisson’s equation in general dimensions

Green’s function of the Helmholtz equation, in the general dimension D, can be calculated as follows: \begin{align} G(\mathbf{r})&=\int \frac{d^D{k}}{(2\pi)^D} \frac{e^{-i \mathbf{k}\mathbf{r}}}{\...
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non-homogeneous laplace equation with mixed boundary condition

Consider this problem $$ \begin{cases} -\Delta u=10 \hspace{6mm} \mbox{in} \hspace{6mm} \Omega \\ u=0 \hspace{6mm}\mbox{on}\hspace{6mm}\Gamma_d \\ \frac{\partial u}{\partial n}=-\sqrt{4x^2+64y^2} \...
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Non-linear, non-constant coefficient, Laplace equation

I have the original Poisson/Laplace equation: $ \nabla^2V = 0$, and I want to break that down (in the context of electrostatics in Physics), using $V(x,y) = I(x,y)\rho(x,y)$, where $x$, and $y$, are ...
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References for Poisson's equation with datum in weighted $L^2$ space

I'm trying to figure out some basic properties of the following problem. Fix the ball $B_1 \subset \mathbb{R}^2$, and let us look at \begin{cases} \hfill -\Delta u = g\,, \qquad &x\in B_1\,,\\ \...
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3 votes
2 answers
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Laplace equation for circular symmetry: given a potential $\phi (z,0)$, how to find $\phi (0,r)$?

The potential function $\phi(z,r)$ has circular symmetry with respect to $r$. It also satisfies the Laplace equation $\partial \phi^2/\partial z^2 + \partial \phi^2/\partial r^2 + (1/r)\partial \phi/\...
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2 votes
1 answer
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Poisson's Discrete Equation for 2 dimensions with Interfaces

I'm trying to solve the Poisson's Discrete Equation when there are interfaces. I'm solving it for the electromagnetic potential for a given grid. When developing my equations, I've found the following ...
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Fourier series as a solution of Poisson's equation

on a tutorial my class solved the Poisson's equation on a square as on the . My question is: if we have a Poisson's equation as $$ u_{xx} + u_{yy} = f(x,y) $$ on a square $ D = [a,b] \times[c,d]$ then ...
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Poisson equation with radial function

while reading the book, I ran into a problem with the partial differential equation. The problem is following: Find positive radially symmetric function $w$ such that $-\Delta w = \Phi (r) \ (r=|x|)$ ...
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