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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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Error estimates in the $L^2$ norm of conforming FEM about Poisson’s equation with mixed boundary conditions

Consider Poisson’s equation $$- \Delta u = f{\rm\qquad{ in }}\;\Omega $$ with following mixed boundary cconditions $$u = g{\rm\qquad{ on }}\;\Gamma \subset \partial \Omega $$ $$\frac{{\...
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Prove an inequality in Poisson equation

Let $u$ be a $C^2$ solution for $-\Delta u=f$ in a bounded set $\Omega$. Show that $$sup_{\bar \Omega} \vert u\vert\le sup_{\bar \Omega}\,f+sup_{\partial\Omega} \vert u\vert$$ If the maximum of $u$ ...
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How to compute the actual current of Poisson equation

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
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Bounds for Poisson solution with Dirichlet B.C using Maximum Principle

Let $U$ be a bounded, open set in $\mathbb{R}^n$. For any $u \in \mathcal{C}^2(\overline{U})$ show that $0\leq u \leq 1$ if it satisfies the Poisson problem with Dirichlet B.C. as \begin{equation} \...
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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 ...
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Solving Poisson's equation for the current streamlines [closed]

Consider Poisson equation $\nabla \cdot (\sigma(x)\nabla u)=0$ in a domain $D$, where $\sigma(x)$ is the spatially dependent conductivity. On the boundary we have $n$ electrodes (Dirichlet BC $u=\text{...
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Unclear inequality of L2 norms (Poisson equation for modeling flow)

I encountered a problem working through a paper about modeling flow with the use of the Poisson equation (source given below). There appears an inequality of L2 norms I don't understand so far. Your ...
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Existence of closed solution of a certain type of Poisson equations

I want to check whether my numerical FEM solver is correct or not. So I seek a solution of \begin{equation} -\Delta u(x,y)=f(x,y) \text{ on } [-1,1]\times[-1,1]\\ u(-1,y)=1\\ u(x,y)=0\text{ on other ...
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Maximum Principle For Gradient Of Poisson Equation

Ran across this problem for a poisson equation. Looks like some sort ofaximum principle for the gradient. Can't quite wrap my head around it. Let $\Omega$ be an open subset of $\mathbb{R}^n$. Suppose ...
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Give a bound of $\|(u_{xx}+u_{yy})\|_p$ by $\|\nabla(\nabla u)\|_p$

Consider twice differentiable $u$ defined on unit disk, that is constantly zero on the boundary. Then, is there fixed $a>0$ s.t. $a\|\nabla(\nabla u)\|_p\le \|u_{xx}+u_{yy}\|_p$? Here, you may ...
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Obtaining simultaneous equations from a dicretized Poisson's equation.

I'm trying to simulate fluids using Moving particle semi-implicit method(MPS). Basically, for incompressible fluids, at every time step, a pressure term is calculated for each fluid particle to ...
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Green's function of 2D Poisson equation in the lower half-plane with Neumann boundary

The solution of the 2D Poisson equation: $$ \frac{\partial^2 V}{\partial x^2}+\frac{\partial^2 V}{\partial y^2}=\delta(x-x_s)\delta(y-y_s) \tag{*} $$ subjected to the boundary condition $V(\infty)=0$ ...
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Source term of Poisson's equation in finite-difference modeling

The Poisson's equation: $$ \nabla^2\phi=I\delta(\vec{r}-\vec{r_s}) $$ with the boundary condition $\phi(\infty)=0$, where $I$ is a constant and $\vec{r_s}$ is the source location, has the solution: $$ ...
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What does “smooth” mean? (Numerical Analysis)

I know a notion of smoothness for functions, say, in $\mathbb{R}^n$, which simply means of class $C^\infty$. But in studying Numerical Analysis I sometimes read the term smooth for discrete functions, ...
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Variant of Poisson equation (biharmonic equation)

Consider the equation $\Delta^2 f=0$ for $f \in C^4(\mathbb R^n)$ $f$ is supposed to be rotationally invariant $f(x)=g(\vert x \vert)$ Prove that all rotation-invariant solutions are of the form $...
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Solve Green function of an annulus to calculate the shape of a clamped elastic sheet

I am trying to solve the shape of an elastic sheet clamped at $r=1$ and $r=b<1$. $$\left\{ \begin{array}{c l} \Delta u = \rho(r,\phi) \quad (a<r<1)\\ u(a)=0\\ u(1)=1 \end{...
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Weak form of Poisson-equation in 3D-cylindrical space (heat conduction)

I'm currently trying to code a FEM-program (finite element method) to solve a heat conduction problem including a few tricky things like phase change and a chemical reaction. In a FEM-book [REDDY - ...
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Solution to Poisson's equation.

I am trying to figure out the solution for a Poisson's equation, which relates the scalar electric potential $\phi$ to the electric charge density $q$. I found the below equation and it's solution ...
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Equation $\nabla(\phi) = \nabla\times \bf{A}$ on a (2D) square.

In solving a PDE problem I managed to reduce it to following equation for the unknown functions $\phi(x,y)$ and ${\bf A}(x,y)$: $\nabla \phi(x,y) = \nabla \times {\bf A}(x,y)$, defined on a 2D ...
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Solving a Poisson Equation to Obtain the Flow Field Inside a Groove

I have a physical problem which involves a groove of width $w$ and height $h$ inscribed at the bottom of a flow channel. Inside the groove is fluid A with viscosity $\mu_0$, and outside the groove is ...
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Solution of Poisson's equation using separation of variables

I am attempting to solve the following question for practice: I know how to solve Laplace's equation using separation of variables. In this case, however, when I try a solution of the form $\Phi(r,\...
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Reference for poisson equation

I don't know if this is ok to ask in stack exchange but I've been looking for a reference book to solutions of partial differential equations for some time. Can anyone tell me their favorite? I am ...
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Evans Proof of Solution to Poisson's Equation

There already are many questions about the proof of Theorem 1 in chapter 2 of Evans' PDE on here. However, I don't think this specific question has been asked yet. On pages 23 and 24, Evans proves ...
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Let $X$ be a Poisson variable with… Obtain a closed-from expression for…

Let $X$ be a Poisson variable with $$P(X=n) = \frac{\lambda ^{n}}{n!}e^{-\lambda }, \lambda >0 , n = 0,1,2,...$$ Obtain a closed-form expression for $E[{e^{iuX}}]$ for $u$ real, and $i=\sqrt{-1}$. ...
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Method of Lagrange multipliers for constrained minimum of functional

I have this problem where I have to find the steepest descent direction of a functional. Basically, it comes down to solving a Poisson PDE subject to a certain constraint ($||v|| = 1$). The image ...
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Why Method of Images solves Laplace's Equation

We have been studying Laplace's and Poisson's equation in maths, and have derived fundamental Green's functions, eg. in 3D $$ G(\vec{r},\vec{r_0}) = \frac{-1}{4\pi |\vec{r} -\vec{r_0}|}$$ When we have ...
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Reference for Laplace and Poisson equation (PDE)

It would be helpful if someone could share some exercises, preferably with some solved examples, on solving Laplace and Poisson equation partial differential equations. I have searched various places ...
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Understanding the matrix equation used to solve 2-D Poisson Equation with non-uniform grid

I am working on a module in my class that is meant to teach us about Jacobi/Gauss-Seidel methods of solving matrix equations in conjunction with Poisson's equation, and am having trouble with setting ...
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Solving 2D Poisson Equation with one inhomogeneous Dirichlet Boundary Condition

I'm familiar how to go about solving problems with separation of variables, but I'm very confused on how to get started with a problem that has inhomogeneous boundary conditions. For example: Any ...
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Essential vs natural boundary conditions

This is from the DOLFIN tutorial (link): Mixed Poisson problem $$ \sigma - \nabla u = 0 , \text{ on } \Omega\\ \nabla \cdot \sigma = -f, \text{ on } \Omega $$ and $$ u=u_0 \text{ on } \Gamma_D, \\ \...
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Poisson equation - Markov chain path averages

I assume that $(X_k)_{k≥0}$ is a uniformly ergodic Markov chain on $X$ with transition density $q$ and stationary distribution $π$. Then for all $n ≥ 1$, let the $n$-step transition density $q^n$ be ...
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Difficulty in Solution of Poisson's equation using Fourier Transform

I wanted to find the potential on an infinite plane due to a point charge located at some point $(x_0,y_0,z_0)$. So I decided to solve the $2D$ Poisson's equation. $$\nabla^2 V(x,y)=-\frac{q}{\epsilon}...
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Solving Laplace Equation with two dielectrics in cylindrical coordinates

Suppose a concentric cylinder of height $c$ and radius $b$ with it's top cap held at $V=V_0$ and all other surfaces held at $V=0$. For calculating the potential inside the cylinder I use the Laplace ...
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Help with border conditions for 2D poisson 9-point finite difference

I really need help with this. I am studying to become an engineer but my math skills are not very solid so please forgive me if I ask any stupid question. I have already programmed and researched for ...
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Cumulative function of a given function multiplied by a Poisson distribution

Can anyone help me solve the following issue? I have this expression: $$\tau_1 = \sum_{n=0}^\infty \bigg(\frac{e^{-\lambda}\lambda^n}{n!}\frac{C}{1+A(1-B)^n}\bigg) $$ where A ($A \approx 9$), B ($B &...
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Difference Quotients on Riemannian Manifolds

I want to define "the difference quotient of a differential form $ \omega $ in the direction of the vector field $ X $" on a Riemannian manifold. Let's call this object, if it can be defined, $ \...
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Poisson modification with regularity

Let $u:\mathbb{R}^n\rightarrow \mathbb{R}$ be a $C^\infty$ function. Let $r > 0$ be any radius and let $t < 0$ be some constant. Consider the Poisson modification $\tilde{u}$ of $u$ given by $$\...
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Multivariate Chain Rule & the Fundamental Theorem of Calculus

I am slowly reading through Evan's PDE and am having trouble with some multivariate integral manipulations. My background in multivariate calculus is undergraduate at best, you could call it my ...
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Prove that variable coefficient Poisson Equation has a unique solution for any choice of f.

I have the following problem that I have been thinking about for a week now. My thoughts so far. I think that I can show the 5 point discrete case has a unique solution in the following way. ...
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Exterior sphere condition for bounded domain

I read a theorem stating that if $\Omega$ is a bounded open set in $\mathbb R^n$ and $\partial \Omega$ is $C^2$ then $\Omega$ satisfies exterior sphere condition. So can anyone tell me some counter-...
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Is the vector field A equal to zero if following equations are satisfied?

$$\Omega : \nabla\cdot \textbf{A} = 0$$ $$\partial\Omega : \textbf{A} = 0$$ The vector field $\mathbf{A}$ with known divergence and curl is unique if either normal or tangential boundary condition ...
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Solution to Laplace's equation with inhomogeneous boundary values

Solve in cylindrical coordinates: PDE: $\Delta u(r,\theta)=0$ B.V: $u(R,\theta)=T_{0}\sin^{2}\theta$ , $u(2R,\theta)=T_{0}\cos^{2}\theta$ $R,T_{0}$ are constants. The problem describes a static ...
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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-...
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Variational Formulation - inhomogeneous Neumann boundary

I am not sure how to handle inhomogeneous Neumann boundary conditions in a week formulation of a pde. The problem is: Derive the variational formulation of $$ -u''=-e^x \; \;\; in \; \Omega \in (0,1) ...
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Estimate of integral of second order derivative

Let $\Omega \subseteq \mathbb{R}^n$ be a region, $u \in C^3(\Omega)$ satisfies \begin{cases} \Delta u=f,&\forall x\in \Omega \\ u|_{\partial\Omega}=0 \end{cases} Then \begin{equation} \sum_{i,j=1}^...
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Galerkin method + FEM - clarification for Poisson equation with mixed boundary conditions

I will be refering to this link, but I am interested in slightly easier equation: $$ -\Delta c = f, \quad (x, y) \in \Omega $$ with the following (mixed) boundary condition: $$ \begin{aligned} 1. &...
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Differentiation of the Newtonian potential of a function (Sandro Salsa's PDE book)

I am studying the PDE book Partial Differential Equations in Action, From Modelling to Theory, by Sandro Salsa. I would like to ask two questions about the differentiation of the Newtonian potential ...
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Solve $u_{xx} + u_{ yy} = 0\,\,$ in the disk $\{r < a\}\,\,$, with the boundary condition $u = 1 + 3 \sin θ$ on $r = a$.

Solve $u_{xx} + u_{ yy} = 0$ in the disk $\{r < a\}$ with the boundary condition $u = 1 + 3 \sin θ\,$ on $\,r = a.$ In this example, writer says that the full Fourier series for $h(\theta )=1+3\...
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Bounded solutions of Poisson's equation in $n = 2$

I want to prove is the following statement; Let $f \in C_c^2(\mathbb R^2)$ ($f$ is twice continuously differentiable function with compact support) then $u(x) = \int_{\mathbb R^2} \phi (x-y) f(y) ...
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Finding the General Solution to the Dirichlet Problem for the Poisson Equation

I am trying to show, with the following PDE: $$\begin{cases}\Delta u=f &\text{on}\>\>\Omega\\ u=g &\text{ on }\partial\Omega \end{cases}$$ that for all $x_0\in\Omega$, we have: $$u(x_0) =...