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Math Help - Boundary conditions to solve Laplace Equation on a Hexagon

  1. #1
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    Boundary conditions to solve Laplace Equation on a Hexagon

    Hello All,

    In order to solve a 2D Laplace Equation in x-y coordinates, we need to prescribe 4 boundary conditions (2 in x and 2 in y). For example, if the domain of interest is a 1x1 rectangle, we can specify BCs at x=0, x=1, y=0, and y=1.

    But, what if the domain is arbitrary, say a hexagon and there are values on all 6 sides that I'm interested to prescribe ? It seems like I will end up with more than 4 BCs in such case ? But, BCs have to be specified all along the boundary for the PDE to be well-posed.

    Perhaps, I'm missing something very basic here. Could one of you help me understand this ? Your help will be much appreciated.

    Thanks in advance,
    aeroaero
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  2. #2
    MHF Contributor

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    Re: Boundary conditions to solve Laplace Equation on a Hexagon

    The first thing you would have to do is change the Laplace eqution to fit the geometry of the situation. This is going to be much more than just an xy- coordinate system.
    Thanks from topsquark
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  3. #3
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    Rogers, Arkansas
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    Re: Boundary conditions to solve Laplace Equation on a Hexagon

    Hello HallsofIvy,

    Thanks for your response. I'm not sure if I understood what you said completely though. Do you think you can throw some more light or may be even point me to a reference (a textbook or something) that can help me understand this ? Most references I know of only discuss solution of PDEs on rectangular domains which is fairly straightforward w.r.t. BCs.

    In fact, I'm not particular that the domain be hexagon since this is not a real problem I'm trying to solve. I'm only trying to understand how the requirement of 2 BCs per direction for a 2nd order PDE such as the Laplace equation is satisfied for arbitrary domains. For instance, we can even consider a 5-sided geometry like the one attached. How can specify BCs for it ? Notice that this is very close to being a rectangle but there is an extra side AB.

    Boundary conditions to solve Laplace Equation on a Hexagon-geometry.png

    Thanks in advance,
    aeroaero
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