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Math Help - Heat Equation Problem

  1. #1
    Junior Member
    Joined
    Aug 2010
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    53

    Heat Equation Problem

    Hi all,

    Working though this problem, I have reached a proof I am not sure of the method, any opinions much appreciated!

    Q:
    \frac{\partial u}{\partial t} = \frac{\partial^2 u}{\partial x^2} where  u(x,t) = t^{\alpha} \phi (\xi) ; \xi = xt^{-1/2} then \phi(\xi) satisfies :

    \alpha \phi - \frac {1}{2} \xi \phi' = \phi '' where  ' = \frac {\partial}{\partial \xi}

    ------ This part is fine with some partial differentiation, chain rule and substitution, I can get the proof.

    2nd proof:

    show that \int_{-\infty}^{\infty} u(x,t) dx = \int_{-\infty}^{\infty} t^{\alpha} \phi(\xi) dx is independant of t if \alpha = -1/2

    ------ This proof is also fine with some partial differentiation and change of variable

    ending with

    \int_{-\infty}^{\infty} u(x,t) dx = \int_{-\infty}^{\infty}  \phi(\xi) d \xi


    3rd proof:

    further show C - \frac {1}{2} \xi \phi = \phi ' when \alpha = -1/2 where C is an arbitary constant....

    Now given the second proof, is it possible to integrate the entire expression

    \alpha \phi - \frac {1}{2} \xi \phi' = \phi '' where  ' = \frac {\partial}{\partial \xi} w r t : \d \xi?

    reducing the orders of the differential terms? I suspect this is not possible since one of the terms contains a \phi' \xi..... if we cannot directly integrate to reduce the power, is it possible to use a power reduction? I suspect not since \phi is an unknown function- not an explicit solution that allows this method..........

    Thanks for reading!
    (I realise the post is rather long winded, but I didnt want to miss info that the proof relied upon in previous sub part)
    Last edited by padawan; September 2nd 2010 at 01:34 PM.
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