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Math Help - Harmonic Functions

  1. #1
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    Mar 2009
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    Harmonic Functions

    Hi, I am having a bit of trouble with a problem so I thought I would post here to see if someone can put me on the right track.

    It is a three part problem; the first two parts involve showing that a*v(x,y) + b (a,b consts) is harmonic in the first quadrant, as well as showing that Arg(z) = Arctan(y/x) and Arg(z) is harmonic in the first quadrant. Both are easily proved.
    The problem arises with the third part, it asks: a large sheet is kept at constant temperature alone the bottom edge (20 degrees) and constant along the left edge (10 degrees). It asks to find the steady state temp distribution u(x,y) that satisfies laplace's equation del^2(u)=0.
    I assume that the solution relies on the information gathered in parts 1 and 2. Intuitively I am expecting an answer like u(x,y)=20-20/pi arctan(y/x) as it would satisfy the boundaries and has been shown to be harmonic already.

    My attempt at the problem is as follows; del^2(u) = d^2/dx^2 (u) + d^2/dy^2 (u) = 0, Assuming u(x,y)=Y(y)X(x) we can separate the problem into 2 ODE's 1/X d^2/dx^2 (x) = -k^2 and 1/Y d^2/dy^2 (y) = k^2. Solving each equation gives us u(x,y)=(c1cos(kx)+c2sin(kx))(c3e^ky+c4e^-ky). I am unable to find the coefficients with the 2 boundary values u(0,y)=10 and u(x,0)=20.

    I am lost as to how to proceed from this point (or whether the ODE approach is right). Any help would be greatly appreciated. Sorry for the lack of latex (I know its difficult to read). Thanks
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  2. #2
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    Re: Harmonic Functions

    I managed to solve the problem.
    Thanks, AJ
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  3. #3
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    Mar 2012
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    Re: Harmonic Functions

    Hi Josh, is it possible to explain your solution to the third part? I am working on a similar problem.
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