Poisson's equation Puns

I tried asking everywhere, and I can't find extremely similar solutions online, where I actually understand what's going on.

I'm trying to solve the Child-Langmuir Law, an important law in Plasma Physics where the Current, I is proportional to 3/2's the Voltage, V.

Since it's been a while since I did second-order homogenous differential equations, I'm having a tough time trying to solve a non-linear differential equation to solve for the potential ϕ, after combining the Poisson equation, charge densities (p) and macroscopic velocity (v), and the current density J = pv.

Here's the question/assignment below with its initial conditions set in the question such as voltage at the cathode being zero and what I did to try to solve the equation:

https://imgur.com/a/pWOMev6

https://imgur.com/gallery/pWOMev6

Hope you can access the link above, I uploaded on Imgur since i'm not sure how to upload it on here.

So if anyone can help, please do.

I hit a break with not sure where to go from here, as I'm stuck solving this second-order diff equation to solve for potential ϕ and then the current density, and then obtain a proportionality expression for current, I, against Voltage, V.

Everything you need to understand what I did is also in those links above - question, what i did to try solve it, and the expected result (what I need to get) as the last image.

Please help if you can!

Thank you. :))

For a computational class, we have been asked to solve the Poisson equation via the relaxation method.

Eq.1: Poisson eqasion

An N by N grid is create for solving Eq.1 in 2D. The grid cell is denoted by i and j in Eq.2 for x and y respectivly.

Eq.2: relaxation equation for Eq.1

In the question, we are told that a charge Q resides in the centre and there is a boundry condition of 10V at each of the edges. What I can't work out is how we determine the value for f at each point? Any help is apprecaitied.

Hi, I'm trying to solve the poisson equation in 2D and I'm wondering if there's any already existing finite difference solvers for this problem. I have a large 2D numpy array containing the source terms and I need to apply neumann and dirichlet boundaries on different regions. I thought there would already be many solvers for this because it's such a common equation but the only straightforward one I found was FENICS project, but it uses Finite Elements which seems a bit overkill in this case.

Coding this from scratch myself seems like I'm reinventing the wheel especially because I will need the code to scale, so I came here to be pushed in the right direction and if I've missed any libraries that do this.

I don't know how to use LaTeX, so I'm just going to write this out. I have a two dimensional form of Poisson's equation:

div^2(T) + P/k = 0

d^2T/dx^2 + d^2T/dy^2 = -P/k

My goal is to indefinitely integrate it twice to get an expression for T, I have known boundary conditions I can use to find the constants of integration, so I'm not worried about those. It's been a few years since I've had to analytically integrate partial differentials, so I'm unsure of how to proceed despite spending several hours searching my past notes and the internet. Any help is appreciated.

Hey guys,

I am trying to solve the two dimensional Poisson equation by using the Greens function as stated below

https://imgur.com/0p1CNoE

However I hve a lot of difficulties solving this equaton, any ideas?

Hi guys

I'm a graduate student. and I have a project in electromagnet theory with the title (solving Poisson's equation by using the finite element method). and make computation by use python.

anyone can help me to find some sources for this topic!

If anyone has a minute to step away from the ~stonks~ and help me with this homework question I would be super grateful! The question is as follows:

Write a matlab routine that solves the two-dimensional Poisson's equation, (del)u = f(x; y); in the domain x = [0; Lx] and y = [0; Ly] subject to the following Dirichlet boundary conditions on the boundaries:

u(x, 0) = sin(x); u(x, Lx) = sin(x)*cos(Ly); u(0, y) = 0; u(Lx, y) = sin(Lx)*cos(y)

Assume a uniform grid with Nx and Ny equally spaced points in the x and y directions, respectively. Build the full matrix (i.e. don't use a sparse matrix), and show that you get second-order accuracy using the test solution:

u(x, y) = sin(x)*cos(y)

for Lx = Ly = 8: Use an equal number of points in each direction, Nx = Ny = N: Note that you will need to build the required forcing term f(x, y).

I have a code that is working, but it does not match up with the analytical solution and I have no idea why. I've been working on it all day and I just can't look at it anymore. I appreciate any help, thank you in advance! I have pasted what I have below.

clear all; close all; 

% length of the area to be solved
Lx = 8; Ly = 8;

% number of grid points
Nx = 100; Ny = 100;

% grid points in the x-direction
dx = Lx/(Nx-1);

for i = 1:Nx
    xx(i,1) = (i-1)*dx;
end

% grid points in the y-direction
dy = Ly/(Ny-1);

for i = 1:Ny
    yy(i,1) = (i-1)*dy;
end

% initialize the size of matrix AA and vector cc
AA = zeros(Nx*Ny,Nx*Ny);
cc = zeros(Nx*Ny,1);

% apply BCs on lower/upper boundaries
for j = 1:Nx
    % lower wall (y = 0)
    row = j;
    AA(row,row) = 1.0;
    cc(row) = sin(xx(j))*cos(yy(1));
    % upper wall (y = Ly)
    row = Nx*(Ny-1)+j;
    AA(row,row) = 1.0;
    cc(row) = sin(xx(j))*cos(yy(Ny));
end

% apply BCs on left/right boundaries
for i = 2:Ny-1
    % left wall (x = 0)
    row = i;
    AA(row,row) = 1.0;
    cc(row) = sin(xx(1))*cos(yy(i));
    % right wall (x = Lx)
    row = Nx*(Ny-1)+i;
    AA(row,row) = 1.0;
    cc(row) = sin(xx(Nx))*cos(yy(i));
end

% interior grid points
for i = 2:Ny-1
    for j = 2:Nx-1
        row = (i-1)*Nx+j;
        AA(row,row) = -2.0*(1.0/(dx*dx) + 1.0/(dy*dy));
        AA(row-1,row) = 1.0/(dx*dx) + 1.0/(dy*dy); % west stencil point, x(j-1)
        AA(row+

Hi, I would really appreciate it if someone can explain it to me step by step like I'm a 10 year old or something because I've been really struggling with this class and spent lots of hours trying to understand the concepts. I've reached out to my professor and he's not much help. (He just reads off of the powerpoints given by the publisher for every lecture)

https://preview.redd.it/wzdq7wa3qoj61.png?width=474&format=png&auto=webp&s=0529a9de1561b34b4f1e05ef10248151f3ccf39f

First of all: This is not a homework, but more of a challenge to myself.

Some physical context: Let's say you have a square plane of the size L*L. The outer edges are kept at a temperature T_{B}. A heating source heats a small square of the size l*l in the middle of the larger one with a (constant in time) power, so that the temperature of the center point stays at T_{C}. A certain time has passed so that the system is in an equilibrium.

The beginning heat equation can therefore be written as (with ∂/∂t u = 0)

0 = D * ∆u(x,y) + θ * R(x,y), R = { R_{0}, x,y∈[-l/2,l/2]

u(0,0) = T_{C}, u(±L/2,y) = u(x,±L/2) = T_{B}

which is a beautiful Poisson equation with Dirichlet boundary conditions and an initial value. The variables are seperable and the function is constant in time, so you first get to

u(x,y) = X(x) * Y(y) and X''(x) = -p * X(x) (with p = (θ-b)/D and b = D * Y''/Y )

It is obvious, that X and Y will be the same functions.

So you solve this to

X(x) = A_{1}* sin(\sqrt{p}* x) + A_{2}* cos(\sqrt{p}* x) and
Y(y) = B_{1}* sin(\sqrt{q}* x) + B_{2}* cos(\sqrt{q}* x)

and from the initial value condition you'll get that A_{2}*B_{2} = T_{C}

But the next step would be to apply the boundary conditions and this is where it becomes complicated, as you somehow need to manage

u(x,±L/2) = u(±L/2,y) = u(±L/2,±L/2)

and that's where my skills leave me. Any suggestions? Or have I already done a mistake?

(This post might be edited, if the markdown doesn't look as it should)

https://prajwalsouza.github.io/Experiments/Electrostatics.html

An attempt to solve Poisson's equation for Electrostatics using Finite difference method (generate difference equations) and then Gauss-Seidel Method to solve the difference equations.

The simulation may not be mobile friendly.

Simulation: https://prajwalsouza.github.io/Experiments/Electrostatics.html

It's original a problem from calculus of variation but it boils down to (i believe so at least) solve the Poisson's equation, and i can't solve it.

Edit: Here is my solution. Thanks everyone that took the time to help.

What could be the reason that the solution of a Poisson equation is smooth when obtained by an iterative solver, only if the maximum residual is set to a high value (e.g. 0.1)? When the maximum residual is set as a typical lower value (e.g. 1e-6), then the solution oscillates - often (but not exclusive) near Neumann boundary conditions. The solution with the high residual is underpredicted, and with the low residual oscillates around expected values.

The matrix is non-symmetric sparse, weakly diagonally-dominant (a diagonal entry equals the negative sum of other row entries). The system includes both Dirichlet and Neumann boundary conditions. Similar outcomes happen using various iterative solvers: BiCGStab, CGS, QMR and GMRES. The preconditioner is always a diagonal preconditioner, since I have to use matrix-free solving on the GPU. Using no preconditioner yields similar (tho a bit smoother) results.

• DOI 10.1112/plms/s3-24.2.348

• [URL] http://onlinelibrary.wiley.com/doi/10.1112/plms/s3-24.2.348/abstract

Any help much appreciated!

How do i plot this in matlab? Please help :)

Hi /r/matlab, I need some help on how to build this matrix A shown in this wikipage.

I used diag to get the diagonal matrix to be all matrix D, but I dont know how to input the matrix -I.

here is my code for the matrix D with size m*m:

u = repelem(4,m);
D = diag(u);
u_1 = repelem(-1,m-1);
D_1 = diag(u_1,1);
D_2 = diag(u_1,-1);
D = D + D_1 + D_2;

Sorry if my question seems naive, I am new to MATLAB and I am having fun coding in my spare time. Any hints or help are appreciated :).

In which classes are poisson boltzman equation and statistical mechanics taught. The class that I am taking went over these concepts rather quickly. I want to discuss these topics to my professor however, prior to talking to him, I want grapple with the topics before hand rather than being spoon fed information.

Also, if any of you have any textbook recommendations I will gladly read those too.

Hi!

I was wondering if there is a solution to the following equation:

nabla^2 phi = = -rho/eps

in the half-sphere. Ideally spherical solution is quite easy after applying polar coordinates and angular symmetry, but is there some clever way to solve it for a half-sphere?

I suspect that numerical solution is needed due to the edge effects, but I am a physicist and want to ask more math-oriented people just in case.

I was reading some E&M and I was wondering, for most formulas, you can interchange line charge density, surface charge density and volume charge density depending on the charge distribution you're working with. Is this also the case for the Poisson equation? I can't find any sources that reference it for line or surface charges.

Can anyone point out to a reference or a step-by-step tutorial? Im having a hard time figuring out how to do this in SPSS and how to interpret the output. I still keep on getting errors such as "The Hessian matrix is singular. Some convergence criteria are not satisfied." and "The maximum number of iterations was reached but convergence was not achieved" even after following the information in the IBM SPSS website. I will appreciate any guidance. Thank you!

I tried to post this here a few days ago but got no reply so I'll give it another try:

I'm given:

• Δu(x,y)=u_xx(x,y)+u(x,y) on 0<x<1, 0<y<1 (Δ=∇^(2) is the laplacian)
• with the BC: u(x,0)=u(x,1)=u(0,y)=u(1,y)=0.

I'm told to solve this by the method of eigen function expansion, so the e-functions are of the form:

• φ_mn(x,y)=sin(mπx)sin(nπy)

corresponding to eigen values:

• λ_mn=(mπ)^(2)+(nπ)^(2)

and therefore we find solutions of the form:

• u(x,y)=Σ(m=1..inf)Σ(n=1..inf)E_mn*sin(mπx)sin(nπy)

No matter how I go about solving this problem I get that E_mn=0 and I'm fairly certain this is incorrect. I was told by someone this isn't homogenous but I can't see how. I though homogenous PDEs were PDEs where every term has u or one of it's derivatives, which is the case here. When you take derivatives of u and combine like terms, you get a fourier series on one side and 0 on the other, so the combined constant is the fourier coefficients of 0, which are 0. All the examples we've done in class on this had a non-homogenous term f(x,y) and so we get non zero fourier coefficients but I don't know how to go about this if f(x,y)=0.

If you feel so inclined, please point out why I'm wrong and how this PDE can have a non-zero solution. Thanks in advance.

Hey guys,

I am trying to solve the two dimensional Poisson equation by using the Greens function as stated below

https://imgur.com/0p1CNoE

However I hve a lot of difficulties solving this equaton, any ideas?

This post was deleted on the big math-thread and I've been sent here. I'm sorry if this is not the right thread.

It's a personal challenge to solve this one:

Some physical context first: Let's say you have a square plane of the size L*L. The outer edges are kept at a temperature T_{B}. A heating source heats a small square of the size l*l in the middle of the larger one with a (constant in time) power, so that the temperature of the center point stays at T_{C}. A certain time has passed so that the system is in an equilibrium.

The beginning heat equation can therefore be written as (with ∂/∂t u = 0)

0 = D * ∆u(x,y) + θ * R(x,y), R = { R_{0}, x,y∈[-l/2,l/2]; 0, else
u(0,0) = T_{C}, u(±L/2,y) = u(x,±L/2) = T_{B}

which is a beautiful Poisson equation with Dirichlet boundary conditions and an initial value. The variables are seperable and the function is constant in time, so you first get to

u(x,y) = X(x) * Y(y) and X''(x) = -p * X(x) (with p = (θ-b)/D and b = D * Y''/Y )

It is obvious, that X and Y will be the same functions.

So you solve this to

X(x) = A_{1}* sin(√(p)* x) + A_{2}* cos(√(p)* x) and
Y(y) = B_{1}* sin(√(q)* x) + B_{2}* cos(√(q)* x)

and from the initial value condition you'll get that A_{2}*B_{2} = T_{C}

But the next step would be to apply the boundary conditions and this is where it becomes complicated, as you somehow need to manage

u(x,±L/2) = u(±L/2,y) = u(±L/2,±L/2)

and that's where my skills leave me. Any suggestions? Or have I already done a mistake?