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Math Help - heat equation

  1. #1
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    heat equation

    1. Show that the heat equation has complex-valued solutions of the form
    F(x) exp(iwt) provided that kF′′ = iwF.

    Done This

    2. Find F if F(x) -> 0 as
    x -> infinity and F(0) = A, where A is a real constant. Let T(x, t) be the
    real part of F(x) exp(iwt). Verify that

    T(x, t) = Aexp(−((w/2k)^1/2)x)cos(wt − ((w/2k)^1/2)x)

    and that T is a solution of the heat equation for which T(x, t) --> 0 as
    x --> inifinity and T(0, t) = Acoswt. Can you think of a physical situation
    to which T corresponds?

    I am not sure how to start this second bit, any hints would be much appreciated

    many thanks
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  2. #2
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    The equation kF'' = i\omega F is a standard SHM equation, with solutions F(x) = Ae^{\gamma x} + Be^{-\gamma x}, where \gamma^2 = i\omega/k. Notice that to find \gamma you need to find the square root of i\omega/k. For that, remember that i = e^{i\pi/2}, which has a square root e^{i\pi/4}.
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