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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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Relation between Warping Function and Solution of Poisson's Equation

In the solution of torsion problem for non-circular cross-sections, the warping function is defined. Without going into the details, the torsion constant is defined as follows: $$ J = \int_A \left[ \...
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Integral of $fg$ over area $A$ vanishes for every $g \Rightarrow f = 0$

I am asked to produce a rigorous proof of the following: Let $A$ be some area in $\mathbb{R}^2$ whose boundary $\partial A$ is smooth, let $g$ be a $C^2$ function on $\bar A$ (the closure of $A$) ...
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Solving a variant of the Poisson Boltzmann equation

The following is an equation I've derived in my personal research: $$ \frac{d^2V}{dx^2}=e^{\alpha x} \sinh(V) $$ I'm looking to solve it explicitly for V(x). It's a variant of the Poisson-...
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$-\Delta u=f$ in $L^1$ but $u_{x_ix_j}$ not in $L^1$ ($i\neq j$)

I want to show the following function is a counterexample for Poisson equation with $L^1$ RHS but its solution is not in $W^{2,1}$: Let $n\geq3$ and $x\in\mathbb{R}^n$. Let $f(x)=|x|^{-n}(\log|x|)^{-...
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Is the solution to the Poisson equation an analytic function in general?

I am wondering this because I just came from my PDE's 2 class and we talked about the regularity of the Laplace equation and elliptic functions DE's. My professor the stated the following which we ...
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Laplace equaion with integral source terms

I have the following coupled PDEs: \begin{eqnarray} \frac{\partial \theta_h}{\partial x} + \beta_h (\theta_h - \theta_w) &=& 0,\\ \frac{\partial \theta_c}{\partial y} + \beta_c (\theta_c - ...
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Poisson parameter 𝜇 and expected occurrence of an event

In a CSV file there is data that shows only months and number of traffic accidents in each month. In an assignment about Poisson distribution and traffic accidents, they ask: What is the expected ...
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Axial Green’s function

Which is the solution of the Poisson equation in axial symmetric coordinates (r,phi,z)? I would like to know the potential generated in the point (r_f,phi_f,z_f) by a circular loop (the source) ...
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What are the differences between Heat equations and Poisson Equations?

Am fairly new into heat equations and wanted to have some clarifications. What are the distinguishing features between the heat equation and the Poisson equation?
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Understanding a proof exercise Evans PDE mean value formula

I am trying to understand a solution to exercise 2.5 (3) in Evans' PDE pag 85-86, which tates the following: The part in the Oval is the one I have trouble understanding. Thanks in advance for any ...
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Poisson problem, minimun value of subharmonic function in the interior.

Consider the problem: $$ -\Delta u = - f(x_1,x_2), \text{in } \Omega$$ $$ u = f(x_1,x_2), \text{in } \partial\Omega$$ Where $\Omega=B(0;1) \subset \mathbb{R}^2$ and $f(x_1,x_2)=1-x_1$. Is it ...
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Using MATLAB to solve Poisson matrix equation

I'm not sure if this is the correct forum to post my question. I'm working on a Poisson-based maths assignment and am stuck as regards finding the solution to the Poisson matrix equation. The matrix ...
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Multigrid with different stencil

I have a code for MultiGrid in 1D. The code solves the Poisson equation with the central difference approximation. $uL-2u+uR=f$. I would like to generalize this approach and use it for the $uL-a u+...
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PDE with Robin Boundary Condition at alpha Solving with Poisson's Equation

Using a code like this, I am having a hard time applying a Robin Boundary condition for a instead of a dirichlet for the following problem: IMG OF QUESTION FOR POISSON ROBIN BC Nx = ...
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Theoretical Solution for this Poisson Equation Problem

$\frac{\partial^2 \phi}{\partial x^2} + \frac{\partial^2 \phi}{\partial y^2} + 1 = 0$ with the following boundary conditions: $\phi(\pm 1,y)=0 \ and \ \phi(x,\pm 1) = 0$ I was able to solve this ...
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Ahlfors page 171 Poisson Integral

Tl;dr : compute the last integral with $z$ fixed. If $C_1$ and $C_2$ are complementary arcs on the unit circlw, set $U = 1$ on $C_1$ and $U=0$ on $C_2$. Let $0 \leq \theta_0 < \theta_1 \leq 2 \...
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Poisson equation on semi-infinite strip

Function $u(x,y)$ satisfy the equation: $$\Delta u = e^{-2y}\sin x$$ in the semi-infinite strip: $$0<x<\pi, y>0$$ and the boundary condition: $$u(0,y) = u(\pi,y),\text{ }u(x,0)=\sin(3x),\...
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Pulling the Laplacian into an Integral (Poisson's equation)

So I'm reading a book about PDE and I have following question: For Poisson's equation in $\mathbb{R}^2: -\bigtriangleup u=f$ The solution is $u(x)=-\frac{1}{2\pi}\int_{\mathbb{R}^2} log(|x-y|)f(y)dy$ ...
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Poisson Kernel $L^p$-functions

Suppose $f \in L^p(\partial \mathbb{D})$ where $\mathbb{D}$ is the unit disk in $\mathbb{R}^2$. Let $\Pi(x, y)$ be the Poisson kernel (of the unit disk). Is it possible to bound the $L^p(\partial\...
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Poisson Equation with Boundary Value Problem

I am trying to learn about solving boundary value problems, but I stuck when I came to finding BVP of $1$D Poisson equation on $[0,1]$: $$ \begin{cases} \dfrac{\mathrm{d}^2}{\mathrm{d}x^2}u(x)=-g(x) \\...
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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, \\ \...