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Math Help - Number of boundary conditions to solve a 2D PDE ?

  1. #1
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    Number of boundary conditions to solve a 2D PDE ?

    Hi All,

    I've already asked this question in a different thread but that fetched only luke warm response. Perhaps, my question was not very clear. I'm now starting this new thread after rephrasing my question so as to make my question clearer.

    How many boundary conditions are needed to solve a 2nd order PDE in x and y ? For example, take the case of a 2D Laplace equation in x-y. It seems obvious that we need a total of 4 BC, 2 in x and 2 in y. But, what is confusing to me is the relation between the number of BC and the shape of the domain on which the PDE is to solved ? Text books always show the example of a rectangle domain which is pretty straight-forward. But, what if the domain is anything else arbitrary ? Will the number of BC be still 4 ?

    Can you guys help me understand this ?

    Thanks a lot in advance,
    aeroaero
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  2. #2
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    Re: Number of boundary conditions to solve a 2D PDE ?

    The number of boundary conditions for a second order partial differential equation is four. That does NOT have to be the four edges of a rectangle. For example, if you have a pde on a disk then the four conditions will typically be:
    1) The value of the function on the boundary of the disk.
    2) Whether the value of the function as you go to the center of the disk is finite or not (Solutions to pde's involving the Laplacian, on a disk, for example, typically involves the Bessel functions- and some of the Bessel functions are finite at 0, others not).
    3) The fact that the function is periodic with period [itex]2\pi[/itex] in the angle.
    4) The fact that the derivative of the function, with respect to the angle, is also periodic with period [itex]2\pi[/itex].
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  3. #3
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    Re: Number of boundary conditions to solve a 2D PDE ?

    I appreciate your quick response. Here is where I am confused: In the case of a rectangle domain, I am able to identify two boundaries at constant x and y. Similarly, for a circle in polar coordinates, I can identify two boundaries at constant r and constant theta as you mentioned. The problem is when you have a domain for which I cannot identify boundaries at constant values of my independent variables. An example of such a domain is a pentagon.
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