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Math Help - Laplace's Equation using similarity variable and others

  1. #1
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    Laplace's Equation using similarity variable and others

    I've really spent a long time struggling with these.



    The first one is finding the similarity solutions of

    \frac{\partial u}{\partial t} = x\frac{\partial^2 u}{\partial x^2}

    using the similarity variable n = xt^a, where a is a constant I have to determine all possible values of

    The other problem is solving Laplace's equation

    \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}=0

    using the similarity variable n=xy^a, where a is a constant I have to determine all possible values of
    Last edited by ricka; November 9th 2010 at 01:45 PM.
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  2. #2
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    Show us what you've tried so that we can help.
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  3. #3
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    my attempts at answering the questions

    for the second one I managed to solve by the normal method of writing u as a product of a function of x only and a function of y only and working through to get:

    u = (c_1e^{kx} + c_2e^{-kx})(c_3cos(ky) + c_4sin(ky))

    for constants k and c subscript 1-4

    but i'm not sure how to write up a solution using their similarity variable, or how to be sure how i've found all the possible constants a
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    Happen to go to Edinburgh university?
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  5. #5
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    I think the idea here is to assume a solution of the form

    u = F\left(xy^a\right)

    substitute into the PDE and pick a such that the PDE becomes an ODE in the variable n = xy^a.
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  6. #6
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    Yeah you should find the partial derivatives in terms if the similarity varibles and the other varibles then (i suggest combining to find one co-efficient in front of f' ) attempt to choose alpha such that you have an ode with constant co-efficients or in terms of the similarity vairable, I think you get 2 values forth diffusion equation and three for laplace
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