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Solving Dirichlet problem using Green's identities.

  1. Jul 27, 2010 #1
    This is to solve Dirichlet problem using Green's identities. The book gave some examples.

    My question is: Why the book keep talking [itex]v[/itex] is harmonic(periodic) function. What is the difference whether [itex]v[/itex] is harmonic function or not as long as [itex]v[/itex] has continuous first and second derivatives.?

    Green's identity:

    [tex]\int\int_{\Omega}(u\nabla^2v + \nabla u \cdot \nabla v)dx dy = \int_{\Gamma} u\frac{\partial v}{\partial n} ds[/tex]

    If we let u=1:

    [tex]\int\int_{\Omega} \nabla^2 v dx dy = \int_{\Gamma} \frac{\partial v}{\partial n} ds[/tex]

    For Dirichlet problem, [itex]\nabla^2 v = 0 [/itex], Therefore:

    [tex] \int_{\Gamma} \frac{\partial v}{\partial n} ds = 0 [/tex]

    I have no issue with the math portion. It will be the same even though [itex]v[/itex] is not harmonic as long as [itex]v[/itex] has continuous first and second derivatives.

    As long as [itex]\nabla^2 v = 0 [/itex], the result is the same. Why the book keep mentioning [itex] v[/itex] being harmonic function in a few example.
     
    Last edited: Jul 27, 2010
  2. jcsd
  3. Jul 27, 2010 #2

    Mute

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  4. Jul 28, 2010 #3
    Thanks

    Alan
     
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