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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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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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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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 gridWhen 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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Fourier transform on two variables

I have got a project to compute numericaly Poisson's and Laplace's equation with spectral method. I have implemented it and now I want to check it's error with analitical approach and here is a ...
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How does derivative transfer to test function?

I set out to find a fundamental solution $E$ for the Poisson equation, i.e. a distribution $\mathscr D'(\Bbb R^d)$ such that $\Delta E = \delta$; I'm almost done. The only thing I have left to do is ...
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Poisson's Equation with unusual conditions.

Solve the boundary value problem: $$Δu=f(r), 0<r<R, 0\leq θ\leq2 π,$$ $$u(R,θ)=g(θ), 0\leq θ\leq2 π.$$ Where $Δu$ denote $u_{rr}+\frac{1}{r}u_r+\frac{1}{r^2}u_{θθ}$ and $R$ is a fix value. I ...
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How does: $\sum^{\infty}_{i=0}\frac{i^2e^{-\lambda}\lambda^i}{i!}$ simplify to: $\lambda\sum^{\infty}_{i=1}\frac{ie^{-\lambda}\lambda^{i-1}}{(i-1)!}$

I understand that $\lambda$ is given as $np$ when $E[X]=np$. Though, I'm uncertain as to how the expression simplify's $i$ in the second expression for both the numerator and the denominator. I ...
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Normal derivative of the Green function on the boundary

In Lawrence C Evans' Partial Differential Equations, Chapter 2, he proves by construction that the representation formula for the solution to Poisson's equation with initial values: $$\begin{cases} -\...
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Upper bound of $\|u\|_{L^{\infty}}$ of solution to Poisson bvp

EDIT: I managed to solve it by defining $v(x):=\frac{C_1}{2}(x_1-\frac{1}{2})^2+C_2$ which is subharmonic and satisfies $v\geq u$, then generalizing $v$ by setting the constants to be the maxima of $f,...
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Poisson's equation estimate: $\Vert u\Vert_{L^{\infty}(\Omega)}\leq\frac{1}{8}A+B$

Define $\Omega=(0,1)^3$ and $A,B>0$. Let $u\in C^2(\Omega)\cap C^0(\bar{\Omega})$ be a solution of the equation \begin{equation} \begin{cases} -\Delta u=A&\text{ in }\Omega\\ \quad u=B& \...
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Calculate the Poisson bracket {A, H}, and check if there is a value of c for which A is a constant of motion?

Consider the Hamiltonian H given by$$H=(x,y,z,p_x,p_y,p_z)= \frac{p^2_x}{2m}+\frac{p^2_y}{2m}+\frac{p^2_z}{2m}-\frac{1}{\sqrt{x^2+y^2+z^2}}$$ where x(t), y(t) and z(t) give the location of a particle ...
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Is there any nonlinear Poisson equation with the form $-u''(x)=f(x, u(x))$?

everyone. Is there any nonlinear Poisson equation with the form $-u''(x)=f(x, u(x))$, where $f(x, u(x))$ is a nonlinear function. For example, the drift-diffusion model for simulating semiconductor ...
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Reference Request for Convergence Rate of the 1D Poisson Equation with the Finite Element Method

Consider the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [0,N]$. Im ...
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Evaluate integral $\int_{ |\vec{r}|\le R }\frac{y}{|\vec{r}-\vec{r}~'|}d^3V$

I'm working on an exercise in GR and it involves Poisson's equation with a spherical source which depends on the coordinate $y$: $$\nabla^2 h(\vec{r})=\begin{cases}16\pi \rho\Omega y, && |\vec{...
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Generalizations of Poisson's Equation

I'm currently reading a book in Multivariable Calculus, and there is a section on Applications of Calculus to Physics - Poisson's Equation. It states the following: We have the 3D version: Let $$u(x) =...
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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 ...
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Uniqueness of Poisson like equation with Neumann boundary conditions

Let $\rho$ be a given function $\mathbb R^n \to \mathbb R^{*+}$, which is as regular as we would like. We consider the following equation, on a smooth open domain $\Omega \subset \mathbb R^n$: $$ \...
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Finite Element Method Poisson Equation in 1D with Inhomogenous Boundary Conditions

Im trying to solve the Poisson equation in 1D: $$-u_{xx} = f(x), \hspace{6mm} u(a) = d1, \hspace{2mm} u(b) = d2$$Assuming a uniform partition such that $x_n = a + nh$, where $h = (b-a)/N$ and $n \in [...
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weak variational formulation of Poisson equation with Dirichlet boundary conditions

I have given the Poisson equation with Dirichlet boundary conditions \begin{cases} -\Delta u & = f & \text{in} &\Omega \\ \quad u & = g & \text{on} & \partial\Omega \end{cases}...
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A problem about gradient estimates for Poisson's equation

During studying Gilbarg-Trudinger's Elliptic PDEs of $2^{nd}$ order, I've been confused about an derivation of the estimates inequality for a long time. In sec. 3.4 Gradient Estimates for Poisson's ...
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Poisson Model $r=5$ [closed]

Suppose that requests to a web server follow the Poisson model with rate $r=5$ per minute. Find the probability that there will be at least $8$ requests in a $2$ minute period. The answer is: $0.7798$,...
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Non-homogeneous Cauchy problem for vibrations of a string of infinite length

I'm new to differential equations, for the following non-homogeneous Cauchy problem for vibrations of a string of infinite length: \begin{equation} \begin{cases} v_{tt} - c^2v_{xx} = f(x,t) &...
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Questions about Solving Poisson's equation in Evan's PDE

In this postSolving Poisson Equation Proof in Lawrence C. Evans Theorem 1 page 23, someone has asked about the inequalities 12 and 14, which also confused me. I can understand the approximation of ...
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How to solve Poisson's equation with incomplete knowledge of the right-hand side?

Consider the following Poission equation, $$\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2}\right)\varphi(x,y,z) = f(x,y,z).$$ I was wondering ...
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Poisson surface reconstruction: toy example

I am trying to get my head around Poisson surface reconstruction with a toy example. I am trying to get the isosurface (in this case line as I'm coding it in 2D) given $m$ samples (points) lying on a $...
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The gradient of poisson's equation is unbounded at zero on the upper half space

Assume $g\in C\left( \mathbb{R}^{n-1}\right) \cap L^{\infty}\left( \mathbb{R}^{n-1}\right) $ and $g\left( x\right) =\left\vert x\right\vert $ for $\left\vert x\right\vert \leq1$. Let $ u\left( x\...
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Get rid of mixed derivatives in stationary PDE using coordinate transform

In this question the PDE $$u_{xx} + u_{yy} + u_{zz} + u_{zy} = 0,$$ was rewritten to the standard Laplace equation $\Delta u=0$ using a coordinate transform, in some sense similar how you can map an ...
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Do solutions to the forced Stokes equations for zero-Reynolds number flows always exist?

I'm in the process of solving a forced Stokes flow problem in two dimensions, the governing equations of which are given by $$ \nabla p = \eta \Delta \mathbf{v} + \mathbf{f}, \quad \nabla\cdot \mathbf{...
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The fundamental solution to Poisson equation using complex function analysis

Is there any way to attain the fundamental solution of 2D Poisson's equation $\nabla^2u(\mathbf{x})=\delta(\mathbf{x})$ using the theorem of complex functions? I think a candidate solution could be to ...
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2D Poisson's Equation with constant source function and Dirichlet boundary conditions on rectangular boundary

I'm trying to solve the torsion problem for a rectangular cross-section of shape $2a$ by $2b$ in the x-y plane. The coordinate system is set up so that the origin is the centroid of the cross-section ...
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Confusing dot product and inner product in a weak formulation

I have been struggling with this for a while. Here, as you can see, they define the weak formulation of the Poisson equation as: $-\int_{\Omega}\nabla u\cdot\nabla v\,ds = \int_{\Omega}fv\,ds \equiv -\...
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Poisson equation in a cylinder

I need to solve the problem $\nabla^{2} u(r,\theta,z)=Q(r,\theta,z)$ inside a circular cylinder $(0 < r < a, 0 < \theta < 2\pi, 0 < z < H)$ subject to $u = 0$ on the sides. I'm ...
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Solution of 2nd order elliptic PDE in 3 variables

I was deriving Low Reynolds Number Flow around a Sphere of radius R placed in uniform stream U. There I got stuck in following PDEs, which are as under - $\nabla^2 w = \frac{3UR}{2}(\frac{3z^2}{r^5}-\...
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Is the solution to the Poisson equation smooth?

In my lecture notes this well known theorem is presented in the following way: Let $f\in C_{0}^{\infty}(\mathbb{R}^n)$ and define $$u(x) = (f*Φ)(x) = \int_{\mathbb{R}^n} f (y)Φ(x− y)d y,$$ where Φ is ...
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One-Parameter Group $F:\mathbf{R}\to\text{End}(C^\infty(\mathbf{R}^{2n}))$ Not Generated By Some $\bar F:\mathbf{R}\to\text{End}(\mathbf{R}^{2n})$

Reading this article, and the definition of one-parameter group, I wonder what naturally occuring (in physics or elsewhere) one-parameter groups $\{F_t:C^\infty(\mathbf{R}^{2n})\to C^\infty(\mathbf{R}^...
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Evaluating limits of the type $\left.[e^{ikx}\Phi(x)]\right|_{x=-\infty}^{x=\infty}$

How to evaluate limits of the type: $$ \left.[e^{ikx}\Phi(x)]\right|_{x=-\infty}^{x=\infty} $$ Such terms occur while trying to integrate by parts when taking the Fourier Transform of the Poisson ...
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Show that bilinear form is coercive

I have got the following problem. Let $G$ be bounded and connected. For $u,v \in H^1(G)$ and $\lambda \in \mathbb{C}$ define $B(u,v) := \langle \nabla u, \nabla v \rangle$ and $B_{\lambda}(u,v) := \...
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Solving Poisson's equation on $B_1(0)\subset \mathbb{R}^2$

I am trying to solve a specific Poisson equation on the following set $B_1 =\left \{ (x,y) \in \mathbb{R^2}: x^2 + y^2 \leq 1 \right \}$ \begin{cases} \Delta u = y & \text{in}\quad B_1\\ u = 1 &...
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Resolvent Equation PDE

Let $\lambda>0$. Show that the fundament solution of the resolvent equation $(-\Delta+\lambda^2)u=f$ on $\mathbb{R}^3$ is given by $u(x)=\int_{\mathbb{R}^3} R(x-y)f(y)dy$ where $R(x)=\frac{1}{4\pi|...
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Poisson Equation on an unbounded domain with fast decay

Some text books on PDE (Evans, Gilbarg and Trudinger for instance) give a solution to Poisson's equation $-\Delta u = f$ on $\mathbb{R}^n$ as $u(x) = \int_{\mathbb{R}^n} \Phi(x - y)f(y)\; dy$, where $\...
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How do I workout the Poisson equation?

I have trouble with working out the Poisson equation. I don't know how to go from step $1$ to step $2$. Why is in $(2.)$ the $k$ in front of the nabla sign? How did the nabla sign come in front of ...
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Particular solution to the Non-Homogeneous Poisson

I am stuck with the following Non-Homogeneous Poisson equation: The boundary conditions are: My effort so far is that I got the homogeneous solution (with some BCs applied): How do I find the ...
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Solving Poisson equation on a rectangle with incompressibility condition.

I've come across the following fluid dynamics equation defined on $[0, L_x]\times [0, L_y]$: \begin{equation} \partial^2v_i = \partial_i p + f_i \end{equation} where $v_i$ (velocity) is a 2-component ...
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Series solution to 2D Poisson's equation on a rectangle with Dirichlet boundary conditions.

I've been following this link in order to try to solve Poisson's equation on a rectangle $[L_x, L_y]$: \begin{equation} \left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right)\...
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Poisson's equation with no explicit position dependence

Are there any ways for finding analytic general solutions for Poisson's equation of the form $$ \nabla^{2}\phi=f(\phi) $$ in two or three dimensions, where $f$ is some arbitrary (integrable) function? ...

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