# Verify 2D Laplace Solution

• Jan 11th 2013, 11:37 AM
helpthanks
Verify 2D Laplace Solution
• Jan 11th 2013, 12:47 PM
vincisonfire
Re: Verify 2D Laplace Solution
• Jan 11th 2013, 12:59 PM
helpthanks
Re: Verify 2D Laplace Solution
In part A I am not sure how to verify that the analytic solution is of the form described? Additionally I have done Laplace's transforms before but i have no idea how to do part B as well. Thanks for all your help!
• Jan 11th 2013, 01:18 PM
vincisonfire
Re: Verify 2D Laplace Solution
Quote:

Originally Posted by helpthanks
In part A I am not sure how to verify that the analytic solution is of the form described?

To verify the solution, just apply the Laplace operator to it and see that it gives 0 indeed. Note that your solution is separable $u(x,y) = X(x)Y(y)$, which should ease the procedure.
Quote:

Originally Posted by helpthanks
Additionally I have done Laplace's transforms before but i have no idea how to do part B as well.

No need for Laplace transforms. This is the Laplace equation.
• Jan 11th 2013, 02:08 PM
helpthanks
Re: Verify 2D Laplace Solution
Would you mind doing or starting this problem for me, i have quite a few like this i have to do but am having trouble going through the whole process of one problem. Thanks again!
• Jan 11th 2013, 05:01 PM
helpthanks
Re: Verify 2D Laplace Solution
Attached is my work so far.
• Jan 11th 2013, 06:49 PM
vincisonfire
Re: Verify 2D Laplace Solution
You fixed $a$ and $b$ such that the boundary conditions are satisfied. Good. But you need to check that the solution satisfies the Laplace equation, i.e. $\nabla^2 u = {\partial^2 u \over \partial x^2} + {\partial^2 u \over \partial y^2} = 0$.
• Jan 11th 2013, 07:09 PM
helpthanks
Re: Verify 2D Laplace Solution
The first page is where i verified that it is equal to zero and the second page I solved for A & B. What should i do differently on the first page that i did not do to show it is equal to zero?
• Jan 12th 2013, 04:23 AM
vincisonfire
Re: Verify 2D Laplace Solution
On the first page you checked that the Dirichlet boundary conditions $u(x,0) = 0$ and $u(x,L_y ) = 0$. This shows that 2 boundary conditions are satisfied. You need to compute the Laplacian of the function to show that the equation is true everywhere in the domain.