Laplace equation Definition and 154 Threads

Homework Statement Since the potential field is only a function of position, not velocity, Lagrange's equations are as follows: (Wikipedia, image 1) Homework Equations (Wikipedia, image 2) The Attempt at a Solution Now, -\frac{\partial{V}}{\partial{q_{j}}} How is speed...

Homework Statement Determine the stationary temperature distribution in the hollow sphere a<r<b where r=a is kept at T1 and r=b is kept at T2. Homework Equations \triangle u =0. But with the Laplacian in spherical coordinates. The Attempt at a Solution I think I must solve the given equation...

Homework Statement Consider a circle of radius a whose center is in (0,0). Let (r, \phi) be the polar coordinates and (x,y) the corresponding rectangular coordinates of the plane. Calculate the solution to Dirichlet problem (interior) for Laplace equation \nabla ^2 u =0 with the following...

Solving the Laplace equation in Cartesian Coordinates leads to the 2nd order ODEs: \frac{X''}{X}=k_1, \qquad \frac{Y''}{Y}=k_2 \qquad \frac{Z''}{Z}=k_3 In each case the sign of k_i will determine if the solution (to the particular ODE) is harmonic or not. Hence, if two people solve the...

[/itex][/itex]Homework Statement A battery consists of a cube of side L filled with fluid of conductivity s. The electrodes in the battery consist of two plates on the base at y = 0, one grounded and one at potential V = 12 Volts. The other sides of the battery casing are not conductive...

I was trying to solve this partial differential equation which arose because I wanted to find a general solution to the Laplace equation in the case f=f(x,y). \frac{{\partial}^{2}f}{{\partial x}^{2}}+\frac{{\partial}^{2}f}{{\partial y}^{2}}=0 Thanks in advance.

Is it possible to obtain a general expression for the solution of laplace equation that is valid in an euclidean space with an arbitrary dimension ?

Verify that the function U = (x^2 + y^2 + z^2)^(-1/2) is a solution of the three-dimensional Laplace equation Uxx + Uyy + Uzz = 0. First I solved for the partial derivative Uxx, Ux = 2x(-1/2)(x^2 + y^2 + z^2)^(-3/2) = -x(x^2 + y^2 + z^2)^(-3/2) Uxx = -(x^2 + y^2 + z^2)^(-3/2) +...

Assumed solution for the Laplace EQUATION?? The book I'm using says that the method of separation of variables is a must when solving the Laplace equation. OK, well they ASSUME that the solution looks like V(x,y,z) = X(x)*Y(y)*Z(z) but why can't they assume a solution of V(x,y,z) =...

Just want to verify Laplace equation in rectangular coordinates that: \nabla ^2 \vec E = 0 \Rightarrow\; \nabla^2 \vec E = \left ( \frac {\partial^2}{\partial x^2} +\frac {\partial^2}{\partial y^2} +\frac {\partial^2}{\partial z^2} \right ) ( \hat x E_x +\hat y E_y + \hat z E_z) = 0 \hbox...

Homework Statement water potential=-KH=-x^3y+xy^3+5. Is this a solution to the Laplace equation and is yes, show how Homework Equations LaPlace equation= Partial derivitive^2 of H/partial deriv x^2+partial deriv^2H/partial deriv y^2=0 The Attempt at a Solution I have absolutely no...

Hi everybody, I just want to know, anybody has any information or sources about the method of Finite-Integral-Transform method in order to solve the Laplace Equations. I couldn't find this topic in any texts, mostly they just introduce the method of SOV or Fourier Integral Transform. I...

Hello, I'm trying to solve the following problem: \nabla^2 p = 0 \frac{\partial p}{\partial y}(x, y_{bot}) = \frac{\partial p}{\partial y}(x, y_{top}) = 0 \frac{\partial p}{\partial x}(x_{left}, y) = \frac{\partial p}{\partial x}(x_{right}, y) = C_0 which is the laplace equation...

Homework Statement I'm considering Laplace's equation in 2D, written in circular polar coordinates (so that's u_rr + 1/r*u_r + (1/r^2)*u_theta,theta). I've worked out what all the seperable solutions are. My question is: is this set of seperable solutions complete. (That is, can all...

Homework Statement "The potential in the x-z plane is independent of z and given by a repeating step-function of magnitude 2(phi_0) and period 2a. The plane at y = y_0 is held at ground potential. Find the potential in teh region 0 < y < y_0." - Marion and Heald Classical Electromagnetic...

Homework Statement u_{tt}=u_{xx} The Attempt at a Solution Where do I start? I have this wonderful Schaums outline at hand, and by looking at similar (unfortunately unsolved problems) I can guess that the answer will be in the range of: u=F(x+iy)+G(x-iy) I'm saying something in...

Hello, I'm learning how to solve Laplace's equation to find Potentials in Cartesian, Cylindrical, and Spherical Coordinates and let's just say it's not going as smoothly as I'd like. In particular, I'm having difficulty with the Spherical case which involves Legendre Polynomials, Method of...

Question on Solving Laplace Equation Potentials. Hello, I'm learning how to solve Laplace's equation to find Potentials in Cartesian, Cylindrical, and Spherical Coordinates and let's just say it's not going as smoothly as I'd like. In particular, I'm having difficulty with the Spherical case...

Usually, we use the technique of "separation of variables" as follows: In a "separable coordinate system", we assume a separable solution \Phi=A(a)B(b)C(c) Then we obtain 3 ODEs for A(a), B(b), C(c) We note that there are actually entire families of solutions to each ODE, that happen to be...

Harmonic function satisfies Laplace equation and have continuous 1st and 2nd partial derivatives. Laplace equation is \nabla^2 u=0. Using Green's 1st identity: \int_{\Omega} v \nabla^2 u \;+\; \nabla u \;\cdot \; \nabla v \; dx\;dy \;=\; \int_{\Gamma} v\frac{\partial u}{\partial n} \; ds...

Homework Statement Could someone check my work plaese. \frac{\partial^2u}{\partial x^2}(x,y)+\frac{\partial^2u}{\partial y^2}(x,y)=0 (0<x<1, 0<y) \frac{\partial u}{\partial x}(0,y)=\frac{\partial u}{\partial y}(1,y)=0 u(x,y)\rightarrow k as y\rightarrow\infty u(x,0)=f(x) (0\leqx\leq1)...

Homework Statement I need to solve the Laplace equation, u_rr + (1/r)u_r + (1/r^2)u_{theta}{theta} = 0, on a circular wedge with radius R, angle {alpha}, where u(r,0) = 0, u(R,{theta}) = 0, and u(r,{alpha}) = 50. Homework Equations The Attempt at a Solution Separate variables -...

Homework Statement Solve the Laplace equation: delta u = d2u/dx2+d2u/dy2 inside the half disk 0<r<R, 0<phi<pi Temperature on the bottom side of the disk is zero, u(x,y=0)=0. Temperature on the upper side of the disk is u(r=R, theta) = u0(phi), 0<phi<pi Homework Equations I'm...

In relativity, the scalar wave equation in the coordinate system (x,y,z,ict) is \frac{\partial^2\phi}{\partial x^2}+\frac{\partial^2\phi}{\partial y^2}+\frac{\partial^2\phi}{\partial z^2}+\frac{\partial^2\phi}{\partial (ict)^2}=0 In 3D classical mechanics, the Laplace equation is:{when...

Homework Statement Find the potential outside of a long grounded conducting cylindrical rod of radius R placed perpendicular to a uniform electric field E0. Homework Equations V(s,\phi) = a_{0}+b_0{}ln(s) + \sum(A_n{}cos(n\phi)+B_n{}sin(n\phi))*(C_n{}s^n{}+D_n{}s^{-n}) The sum being...

Homework Statement The capacitor is assumed to consist of two parallel circular disc electrodes of radius R. The electrodes are of infinite small thickness, placed a distance 2H apart, and are equally and oppositely charged to potentials +U and -U. A metal cylinder is placed near the two...

Can anyone help with the solution of the Laplace equation in cylindrical coordinates \frac{\partial^{2} p}{\partial r^{2}} + \frac{1}{r} \frac{\partial p}{\partial r} + \frac{\partial^{2} p}{\partial z^{2}} = 0 with Neumann no-flux boundaries: \frac{\partial p}{\partial r}...

Homework Statement I have problems solving the related Laplace equations in the problem Homework Equations \frac{1}{\rho}\frac{\partial}{\partial\rho}\rho\frac{\partial g_m(\rho,\rho^')}{\partial\rho}-m^2g_m(\rho,\rho^')}=-4\pi\frac{\delta(\rho-\rho^')}{\rho} The Attempt at a...

Sorry but i have a question regarding Laplace Equation, say if a potential function P represents the inverse square propotional field, then how am i going to visualize taking twice partial derivative of P is equal to zero? Because since grad of P is pointing inward (which looks to me is a sink...

Homework Statement A uniform line charge \lambdais placed on an infinite straight wire , a distance above the conducting plane . (Lets say the wire runs parallel to the x-axis and directly above it, and the conducting plane is in the xyregion) a) Find the potential in the region above the plane...

Homework Statement The capacitor is assumed to consist of two circular disc electrodes of radius \alpha . The electrodes are of infinitesimal thickness, placed a distance 2L apart, and are equally and oppositely charged to potentials +V and -V. To solve the potential distribution in and...

Homework Statement This is a try for the solution of Laplace Equation. We have to calculate the potential distribution in a cylinder coordinate. However, there is a step really bring us trouble. Please go to the detail. You can either read it in the related URL, or in my PDF attachment...

Hello all! I just finished the following problem: Consider a thin semi-infinite plate of negligible thickness made of an isotropic conductive material. A voltage V0=1V is applied at x=0 on the plate (across the short dimension). At a distance x=d=1cm from the end (x=0) V is measured to be...

Why does laplace's equation only apply in limited regions, while Poisson's equation can apply in unbounded domains ?

When one compares the fundamental solution for Laplace's Equation one might note that in 2 dimensions this solution becomes unbounded as r goes to infinity while in 3 dimensions the solution goes to zero as r goes to infinity. Now I understand both mathematical derivations so my question is...

Homework Statement Hi, I've attached the question as I don't know how to write equations on here without them looking awful.Homework Equations Laplace -> del^2 V=0The Attempt at a Solution I've done the first bit (expression for Q0). For the next bit I tried to solve laplace's equation to find...

Homework Statement Laplace's equation in 2 dimensions may be written, using plane polar coordinates r, θ, as Find all separable solutions of this equation which have the form V(r, θ)=R(r)S(θ), which are single valued for all r, θ. What property of the equation makes any linear...

Urr+(1/r)*Ur+(1/r^2)*Uθθ=0 a<r<b, 0<θ<w with the conditions U(r,0)=U1 U(r,w)=U2 U(a,θ)=0 U(b,θ)=f(θ)

Homework Statement Calculate potential function and the electric field for the region between two concentric cylinders, where V ( inner cylinder) = 0 at r = 0.015 m and V(outer cylinder) = 100 for r = 0.025 Homework Equations \Delta (square ) V = 0 The Attempt at a Solution so...

I wish to approximate the Laplace PDE in a 1/4 plane by truncation of the domain in (x,y) variables: U_xx + U_yy = 0 Now the PDE is approximated in a box [0, xMax] X [0, yMax] and I can solve it using finite differences. But the problems are: 1. How to choose xMAx, yMax appropriately...

Homework Statement \frac{\partial^2f}{\partial x^2}+\frac{\partial ^2f}{\partial y^2}= 0 Homework Equations Show that the equation above is equal to: \frac{\partial^2f}{\partial r^2}+\frac{1}{r^2} \frac{\partial ^2f}{\partial \theta^2} + \frac{1}{ r} \frac{\partial f}{\partial r}= 0...

Hi, My 2D L.e.: Uxx + Uyy = 0 with boundary conditions: U (x,0) =x U (0,y) =0 U (1,y) =1 U (x,1) =x Please, need help with analytical sollution! i'm trying to do smth like it is described in attachment pic, but i it's not working.. Thanks!

Hi everyone, I read somewhere that solutions to Laplace's Equation must agree at a shared boundary. So for example if \Phi_{1} and \Phi_{2} are two solutions to the Laplace equation in two different regions which share a boundary, then on the boundary \Phi_{1} = \Phi_{2} Is this true...

The code should solve laplace equation through an iterative technique until values change less than the specified tollerence, in this case maxdiff. I've used 3 arrays. one to store all values including initial and boundary conditions, and 2 more to store the new values and differences between...

hello all. my question isn't about the solution, but more how the solution was obtained. i have a circuit from which i obtained the following equation Vo { 1/4000 + 1/(0.08S) + 1/(21000 + 10^9/(5S) } = 300/S however the problem is, that i can't arrange it so that it becomes in the following...

For an elliptic PDE Uxx + Uyy + Ux + Uy = -1 in D = {x^2 + y^2 = 1} and U = 0 on the boundary of D = {x^2 + y^2 = 1} is it possible for me to make a change of variables and eliminate the Ux and Uy and get the Laplace equation Uaa + Ubb = 0? I tried converting into polar coordinates, but the...

Homework Statement Verify that the function u=1/(x^2 + y^2 + z^2)^2 is a solution of the 3-dimensional Laplace equation uxx+uyy+uzz=0 The Attempt at a Solution I know how to solve the partial derivatives, so I know that uxx=uyy=uzz for this problem. How can their sum equal 0?

Homework Statement I need to solve Laplace equation in the domain D= 0 < x,y < pi Neumann boundary conditions are given: du/dx(0,y)=du/dx(pi,y)=0 du/dy(x,pi)=x^2-pi^2/3+1 du/dy(x,0)=1 2. The attempt at a solution first, we check that the integral of directional derivative of u...

[SOLVED] Laplace equation, cylindrical 2D Homework Statement I am given the Laplace eq. in cylindrical coord. (2D), and I am told that we can assume the solution u(rho, Phi) = rho^n * Phi(phi). Find the general solution. The Attempt at a Solution My teacher says that the general...

[SOLVED] solutions to the laplace equation Homework Statement http://mathworld.wolfram.com/LaplacesEquation.html I don't understand why the solutions to the Laplace equation are different in different coordinate system. Obviously, the solutions will look different when you write them out as...