Dirichlet Problem

1. The problem statement, all variables and given/known data

We wanna solve the equation

\(u_{xx}(x,y) + u_{yy}(x,y) = xu(x,y) \)

on the square [0,1]x[0,1] with boundary conditions

u(x,0) = x
u(x,1) = x
u(0,y) = 0
u(1,y) = 1

Let h = 1/3 be the mesh. Set up a system of equations for \(u_{1,1}, u_{2,1}, u_{1,2}\) and \(u_{2,2}\)

2. The attempt at a solution

First point (1/3, 1/3):

-4\(u_{1,1}\) + \(u_{2,1}\) + \(u_{1,2}\) = -x + x\(u_{1,1}\)

This can be written

-13/3\(u_{1,1}\) + \(u_{2,1}\) + \(u_{1,2}\) = -1/3

The correct equation for this point is

-109\(u_{1,1}\) + 27\(u_{2,1}\) + 27\(u_{1,2}\) = -9

What is wrong?

First of all man, I think that some of these mathematical concepts are a bit advanced for most people in this forum. You might want to consider trying on a forum that specializes more in advanced mathematical concepts rather than a forum on all circuit-related information.

That being said, how do you possibly arrive at either of these two equations:

"-4\(u_{1,1}\) + \(u_{2,1}\) + \(u_{1,2}\) = -x + x\(u_{1,1}\)"

"-109\(u_{1,1}\) + 27\(u_{2,1}\) + 27\(u_{1,2}\) = -9"
?

I personally can't help you because I just don't see what you're doing... at all...

Sorry.

-blazed

Here's how

http://www.geocities.com/dharwadker/lecture.html

Alright, I get it.

First of all though, shouldn't you be solving for u sub (1/3,1/3), (1/3,2/3), (2/3,1/3), and (2/3,2/3)?

I'm just gonna pretend that's what you were solving for and in that case...

Your first point (1/3,1/3) should yield this equation:

4*\(u_{1/3,1/3}\) = 1/3 + 1/3 + \(u_{2/3,1/3}\) + \(u_{1/3,2/3}\)

Hope that helps, good luck

-blazed

First of all man, I think that some of these mathematical concepts are a bit advanced for most people in this forum.

Click to expand...

I'm willing to admit that I balk at mention of the the Dirichlet Problem..

triggernum5 said:

I'm willing to admit that I balk at mention of the the Dirichlet Problem..

Click to expand...

After reading it through, while it seems intimidating, it's actually quite simple.

Think about it this way: if you had two points and you wanted to find the one right in between both of them, you would take the average of the two...

That makes sense right?

This is just adding another dimension to the picture... on a two-dimensional object, if you wanted to find the point in between four others, you would take the average of all four points...

That's it. All the rest of it is mathematical jargon, which while it's all well and good to understand, is not completely necessary.

-blazed