Dirichlet Problem

Discussion in 'Math' started by boks, Nov 27, 2008.

  1. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
    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):

    -4u_{1,1} + u_{2,1} + u_{1,2} = -x + xu_{1,1}

    This can be written

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

    The correct equation for this point is

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

    What is wrong?
     
  2. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    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:

    "-4u_{1,1} + u_{2,1} + u_{1,2} = -x + xu_{1,1}"

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

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

    Sorry.

    -blazed
     
  3. boks

    Thread Starter Active Member

    Oct 10, 2008
    218
    0
  4. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    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
     
  5. triggernum5

    Active Member

    May 4, 2008
    216
    0
    I'm willing to admit that I balk at mention of the the Dirichlet Problem..:)
     
  6. blazedaces

    Active Member

    Jul 24, 2008
    130
    0
    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
     
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