Laplace's equation Definition and 116 Threads

Homework Statement Solve Laplace's equation u_{xx} + u_{yy} = 0 on the semi-infinite domain -∞ < x < ∞, y > 0, subject to the boundary condition that u_y = (1/2)x u on y=0, with u(0,0) = 1. Note that separation of variables will not work, but a suitable transform can be applied...

Homework Statement Find the solution of Laplace's equation for \phi(r,\theta) in the circular sector 0 < r < 1; 0 < \theta < \alpha with the boundary conditions \phi(r,0) = f(r), \phi(r,\alpha) = 0, \phi(1,\theta) = 0. (also, implicitly, the solution is bounded at r = 0). Use two different...

The Poisson Integral Formula is a representation of the bounded solution of the Dirichlet problem for Laplace's equation in the interior of the disc. Derive the corresponding formula for the Dirichlet problem in the exterior of the disc, again assuming that the solution is bounded.So we derived...

I'm struggling here so please excuse if I'm writing nonsense. I'm trying to understand how, for a gravitational field, Laplace's equation (I think that's the right name) equals zero in empty space. I understand that the gravitational potential field, a scalar field, is given by...

\nabla^2(Z)=0 Z= 0 for x=0, y=0 Z= x(1-x) for y=0 Z=0 for y=infinity Range 0<x<1 and y>0 (suppose strictly speaking should be x=1 and x=0 too) So all I want to do is solve this Use separation of variables: X''/X = a^2 = -Y''/Y Gives X = Aexp(ax) + Bexp(-ax) and Y=Ccos(ay) +...

Hi all, I am trying to construct a numerical solution to the following linear harmonic problem posed in a wedge of interior angle 0<\alpha<pi/2 \bigtriangledown^2\phi(r,\theta), \ r>0, \ -\alpha<\theta<0 \bigtriangledown\phi\cdot\mathbf{n}=0, r>0,\ \theta=-\alpha...

Homework Statement Solve Laplace's equation on a ring in the plane with r_1<r<r_2. And arbitrary functions on the edges of the ring.The Attempt at a Solution After separation of variables the solution to the radial factor is an often-seen problem. It's eigenvalues are λ=n² the eigenfunctions...

Homework Statement f = r^-n-1 * cos(n+1) θ satisfies laplace's equation. r^2 \partial^2 f / \partial r^2 + r \partial f / \partial r + \partial^2 f / \partial θ^2 = 0 Homework Equations P.D.E The Attempt at a Solution \partial f / \partial r = nr^n-1 * sin (nθ) \partial f/ \partial θ =...

As the title suggests, the general solution of Laplace's equation has 4 arbitrary constants. One would imagine that if you e.g. have the potential at 4 points in a domain, you could get the specific solution by replacing: V(x1,y1)=V1, V(x2,y2)=V2, V(x3,y3)=V3, V(x4,y4)=V4, and solving the...

The question involved solving the "T" on a plate sized from (0,0) to (1,2). Any ways, I will spare you of the details and get to the line in the solution I was confused with: Bottom: T(x,0) = 1-x = Bn (sinh 2n∏) (sin n∏x) = bn (sin n∏x) The next line confused me: bn = 2 0∫1 (1-x) sin n∏x dx...

Hi all. Suppose that U1 is the solution of the Laplace's equation for a given set of boundary conditions and U2 is the the solution for the same set plus one extra boundary condition. Thus U2 satisfies the Laplace's equation and the boundary conditions of the first problem, so it's a solution...

We can easily derive Poisson's and Laplace's equations in electrostatics by using Gauss's law. However, my question is what are the importance of these equations in Electrostatics ?

Homework Statement Two long metal plates of length L>>H (their height) intersect each other at right angles. Their cross section is a cross with each line of length H. This configuration is held at a potential V=Vo and the total charge up to a distance d (d<<H) from the center of the cross is...

Homework Statement Suppose that u(x,y) is a solution of Laplace's equation. If \theta is a fixed real number, define the function v(x,y) = u(xcos\theta - ysin\theta, xsin\theta + ycos\theta). Show that v(x,y) is a solution of Laplace's equation. Homework Equations Laplace's equation...

Homework Statement http://workspace.imperial.ac.uk/mathematics/public/students/ug/exampapers/2010/M2AA2-2010.PDF Questions 3(iii) and 3(iv) Homework Equations The Attempt at a Solution 3(iii) So here we "guess" that the solution is f = f(r) (f for phi). Then we just have...

- exp(f) = Laplacian(f) where f is a real valued function of two variables in an open domain.

Can anyone help me think of the three similar cases I need to examine, I was thinking 0<x<pi/2 0<y<pi/2, 0<x<pi 0<y<pi/2, 0<x<pi/2 0<y<pi, with the same boundaries as those parts of the original square, but it doesn't really work for me, any help would be greatly appreciated...

I don't seem to grasp the meaning of boundary conditions for Laplace's equation. Consider the Lagendre expansion of the potential at x due to a unit charge 1/|x-x'|, where x' is the position of the unit point charge. To do the expansion, we need to consider a volume in space where the...

Homework Statement I'm taking an independent study course at my college in which my professor and I have decided to make a neutron spin flipper. To do this, I've got to solve Laplace's Equation for a cylindrical shell of (obviously) finite length and thickness. Can I assume azimuthal symmetry...

Homework Statement Show that for a solution w of Laplace's equation in a region R with boundary curve C and outer unit normal vector N, \int_{R}\left\| \nabla w\right\| dxdy = \oint_{C}w\frac{\partial w}{\partial N}dsHomework Equations The book goes through the steps to show that the following...

Homework Statement an infinitely long thin conducting cylindrical shell(radius R) of surface charge density \sigma=\sigma_{1}sin(2\Phi)+\sigma_{2}cos(\Phi). what are the four boundary conditions for this problem? using the four boundary conditions and the identification of the coefficients of...

Hello i have a problem with a laplace equation in a box, because the problem says that in the six face the tempeture is constant and given by the function f(x,y). and the other faces the temperature its zero. My teacher says that i must express the f(x,y) as a Fourier series, but i can't...

Homework Statement Consider a box that has a top and bottom at a/2 and –a/2, while the sides are located at –b/2 and +b/2. Also, the top and bottom are at potential V0=100V and the sides have V=0. You will need to use the separation of variables technique. 1) Find the general solution using...

I'm writing a paper, and as a motivation to the forthcoming finite element modeling, I want to state, with some sort of "proof" that Laplace's equation in a heterogeneous volume: \del (sigma \del V) = 0 exhibits linearity. By "linearity", I mean that if a set of initial conditions...

Homework Statement uxx+uyy=0 u(x,0)=u(x,pie)=0 u(0,y)=0 ux(5,y)=3siny-5sin4y Homework Equations The Attempt at a Solution Using separable method I get Y"-kY= 0 and X"+kX=0 For Case 1 and Case 2 where k>0 and k=0 there are no eigenvalues So Case 3 k<0 gives Y=ccos(sqrk...

Homework Statement Consider Laplace's equation uxx + uyy = 0 on the region -inf <= x <= inf, 0 <= y <= 1 subject to the boundary conditions u(x,0) = 0, u(x,1) = f(x), limit as x tends to inf of u(x,y) = 0. Show that the solution is given by u(x,y) = F-1(sinh(wy)f(hat)/sinh(wy))...

Homework Statement I want to cook a 4" meatball. The meatball is being stored in the fridge at 35 degrees F. The meatball will go into a convection oven at 350 degrees F (surface is maintained at precisely 350 for the duration of cooking). I want to cook the meatball to 130 degrees F (in the...

Homework Statement I have a really dumb question, but I want to make sure this is right... So I have the integral (d2V)/(dф2) = 0. I am solving for the potential function on the bounds, 0 < ф < фo. I will also be solving on range of фo < ф < 2∏. Homework Equations The Laplace equation...

Homework Statement A surface at z = 0 is held at potential V (x, y) = V0 cos(qx) sin(qy). Find the potential in the region z > 0. Homework Equations Laplace's equation in Cartesian coordinates The Attempt at a Solution I wrote at least a page of my past 2 attempts at a solution...

Homework Statement I have to show for a conducting sheet bent along one axis into the shape of a wedge, with a certain angle, that the magnitude of the electric field in the bend is proportional to r^{(\pi/\theta) - 1}, where theta is the opening angle. Homework Equations The...

Hi, Can someone point me in the right direction to a derivation of Poisson's Equation and of Laplace's Equation, (from Maxwell's equations I think) both in a vacuum and in material media? How does one get from Maxwell's equations to Poisson's and Laplace's?

Homework Statement Show that arctan(y/x) satisfies Laplace's equation.Homework Equations Laplace's equation: \frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^{2}}=0The Attempt at a Solution We haven't really done this is class too thoroughly, I've looked a...

Is it true that any solution to Laplace's equation, subject to any set of boundary conditions, can be written as a linear superposition of separable solutions? I'm sure there are some vagaries in what I've written above. Feel free to point them out and rectify them.

Homework Statement Use Poisson's equation and Laplace's equation to determine the scalar potential inside and outside a sphere of constant charge density po. Use Coulomb's law to give the limit at very large r, and an argument from symmetry to give the value of E at r=0. Homework...

When solving Laplace's equation over a triangular domain. Why is it a good idea to take M = N?

Homework Statement \frac{1}{s} \frac{\partial }{\partial s} (\{s} \frac{\partial V}{\partial s}) + \frac{1}{s^2} \cdot \frac{\partial^2 V}{\partial \phi^2} When you do separation of variables what happens to the \frac{1}{s} and the \frac{1}{s^2} after you divide through by \Phi and S to...

I have a question about the general solution to Laplace's equation in spherical co-ordinates, which takes the form of a linear combination of the spherical harmonics. In my problem, I am solving for the potential within two concentric spherical shells, each with its own conductivity. Now...

I have a simple question about the general solution to Laplace's equation in spherical co-ords. The general solution is: u(r, \theta, \phi) = \sum^{\infty}_{l=0}\sum^{l}_{m=-l}\left(a_{lm}r^{l} + \frac{b_{lm}}{r^{l+1}}\right)P_{lm}(cos\theta)e^{im\phi} (where the a_{lm}, b_{lm}...

A scalar "harmonic" function f is one that satisfies \nabla ^2 f = 0 What is the physical meaning or relevance of this?

Where does Laplace's equation in spherical polars come from \frac{\partial^2 u}{\partial r^2} + \frac{1}{r} \frac{\partial u}{\partial r} + \frac{1}{r^2} \frac{\partial^2 u}{\partial \theta^2}=0 ? i can derive from scratch the expression for the laplacian in spherical polars but this...

[solved] solutions to Laplace's equation Homework Statement Find all solutions f(x,y) that satify Laplace's equation that are of the form: ax^3 + bx^2y + cxy^2 + dy^3 Homework Equations Laplace states that fxx + fyy = 0 The Attempt at a Solution fxx = 6ax + 2by fyy = 6dy + 2cx so...

Homework Statement I'm having issues with a deceptively simple Laplace problem. If anybody could point me in the right direction it would be fantastic. It's just Laplace's equation on the square [0.1]x[0,1] (or any rectangle you like) with a mixed boundary. Homework Equations...

Homework Statement Solve Laplace's equation \nabla^2 u(r,\vartheta) =0 in an annulus with inner radius r_1 and outer radius r_2 . (a) For boundary conditions take u(1,\vartheta) = 3 and u(2,\vartheta) = 5. (b) What is the solution using this second set of boundary conditions...

Homework Statement I missed the lecture on this so I just wanted to check if I am doing this correctly? Which of the following functions obey Laplace’s equation? a) Ψ(x, y) = 2xy b) Ψ(x, y) = x^3 - 3y^2 c) Ψ(x, y) = x^4 - 6x^2.y^2 d) Ψ(x, y) = e^x.siny e) Ψ(x, y) = sinxsinhy...

Homework Statement Hi all. I am looking at Laplace's equation on an annulus (just a circle where a < r < b, where "a" and "b" are constants). The boundaries of the annulus at r=a and r=b are kept at a certain temperature, which is theta-independent! Using my physical intuition, of course the...

Homework Statement Find the general solution to Laplace's equation in spherical coordinates, for the case where V depends on on r. Do the same for cylindrical coordinnates assuming V depends only on r. Homework Equations Laplace's Eq (spherical): 1/r^2 (d/dr)(r^2(dV/dr)) +...

I am using separation of variables and superposition to solve: u_{xx}+u_{yy}=0; for (x,y) \in (0,L) X (0,H) u(0,y)=f(y); u(L,y)=0; u(x,0)=g(x); u(x,H)=0 Is it correct to assume that I can write my solution as: u=u_1+u_2 Where: u_1 is the solution with BC u(0,y)=0...

Homework Statement Solve Laplace's equation inside a circular annulus (a<r<b) subject to the boundary conditions \frac{\partial{u}}{\partial{r}}(a,\theta) = f(\theta)\text{, }\frac{\partial{u}}{\partial{r}}(b,\theta) = g(\theta) Homework Equations Assume solutions of the form u(r,\theta)...

Homework Statement Solve Laplace's equation inside a circular annulus (a<r<b) subject to the boundary conditions \frac{\partial{u}}{\partial{r}}(a,\theta) = f(\theta)\text{, }\frac{\partial{u}}{\partial{r}}(b,\theta) = g(\theta) Homework Equations Assume solutions of the form u(r,\theta)...

Homework Statement Solve Laplace's equation inside the rectangle 0 \le x \le L, 0 \le y \le H with the following boundary conditions u(0,y) = g(y)\text{, } u(L,y) = 0\text{, } u_y(x,0) = 0\text{, and } u(x,H) = 0 Homework Equations The Attempt at a Solution I know that with...