Laplace equation Definition and 164 Threads

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...

u(r, θ) satisfies Laplace's equation inside a 90º sector of a circular annulus with a < r < b ; 0 < θ < π/2 . Use separation of variables to find the solution that satisfies the boundary conditions u(r, 0) = 0 u(r, π/2) = f(r) ; a < r < b u(a, θ) = 0 u(b, θ) = 0 ; 0 < θ < π/2 Consider all...

I've recently started studying Laplace's equation and it's solution under various simple circumstances in electrostatics. I tried to solve the equation for a parallel plate condenser system, but I couldn't meet the boundary conditions. I had two plates, one placed on xz plane at y=0 (with...

Does \nabla ^2 u(r,\theta) = 0 with the boundary conditions u(1,\theta) = u(2,\theta) = \sin^2 \theta have any solutions? This was a problem on my exam but someone must have written the conditions wrong, or am I stupid?

Hi, I'm trying to solve the Laplace equatio in oblate and prolate spheroidal coordinates, but it's proving to be too much for me to handle, can anyone help me out? You can see the equations I'm using in: http://mathematica.no.sapo.pt/index.html