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Solving Dirichlet problem on a Square

  1. Aug 23, 2011 #1
    1. Problem Statement
    Solve by inspection the Dirichlet problem, where [itex]\Omega[/itex] is the unit square 0[itex]\leq[/itex]x[itex]\leq[/itex]1, 0[itex]\leq[/itex] y [itex]\leq[/itex] 1, and where the data is:
    f(x,y) = {
    x for 0[itex]\leq[/itex]x[itex]\leq[/itex]1, y=0,
    1 for x = 1, 0[itex]\leq[/itex] y [itex]\leq[/itex] 1,
    x for 0[itex]\leq[/itex]x[itex]\leq[/itex]1, y=1,
    0 for x=0, 0[itex]\leq[/itex]y[itex]\leq[/itex]1

    2. Relevant equations
    Dirichlet Problem:

    [itex]\Delta[/itex] u = 0 in [itex]\Omega[/itex]
    u = f on d[itex]\Omega[/itex]

    3. The attempt at a solution
    I don't know where to really start with this one. What does it mean "solve by inspection the dirichlet problem?" What is u? How does the "data" help, it's the initial conditions right?
  2. jcsd
  3. Aug 23, 2011 #2


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    It's the boundary conditions. How much PDE theory do you know?
  4. Aug 23, 2011 #3
    Almost nothing, sadly. I have had only one lecture in it so far and that was just going over the three main operators in PDEs.
    Is there a good website that shows how to solve the Dirichlet problem with boundary conditions or just how to solve a problem with boundary conditions?
  5. Aug 23, 2011 #4


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    So if I mention separation of variables, this would draw a blank to you? It sounds like you need to go to a few more lectures before you start looking at the questions.

    Have you tried a google search? It would be the first thing I would do.
  6. Aug 23, 2011 #5
    Unfortunately, mainly yes. I do know that we will be covering three main methods: Separation of Variables, Green's Method, and Variational/Numerical methods, but we haven't even discussed what each means. It isn't even until next week we start discussing the separation of variables though.

    I did a google search for solving differential equations using boundary conditions and it made sense. Specifically, I used http://tutorial.math.lamar.edu/Classes/DE/BoundaryValueProblem.aspx" [Broken]
    A google search of dirichlet problem did no good.

    I suppose I'll just have to keep reading, thank you for the help!
    Last edited by a moderator: May 5, 2017
  7. Aug 23, 2011 #6


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    How about a search on wikipedia?
  8. Aug 23, 2011 #7
    I came across http://en.wikipedia.org/wiki/Dirichlet_boundary_condition" [Broken].
    So, what I'm getting from it is I need to find a solution to a PDE based on the boundary conditions given. Is my PDE [itex]\Delta[/itex]u = 0 in the unit square and u = f where f is our data?
    I'm just confused on where to start.
    Last edited by a moderator: May 5, 2017
  9. Aug 23, 2011 #8


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    Basically yes, your f is your boundary data. Like I said you're probablt jumping into the problems a little too early.
  10. Aug 23, 2011 #9


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    Yet the original question was to solve that Dirichlet problem by inspection. By examining the values on the boundary of the square, it really isn't much to ask to guess a solution, never mind the lack of advanced techniques available.
  11. Aug 23, 2011 #10
    How would you go about that?
  12. Aug 23, 2011 #11


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    Is there a simple function in the square that fits your data on the boundary of the square? Does that function solve the Laplace equation 'by inspection'?
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