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I need help to do a Fourier Sine Transform!

  1. Feb 27, 2010 #1
    1. The problem statement, all variables and given/known data
    A function u(x, t) satisfies the heat equation
    [tex]K[/tex][tex]\frac{\delta^{2}u}{\delta x^{2}}[/tex] = [tex]\frac{\delta u}{\delta t}[/tex]
    on the half line x [tex]\geq[/tex] 0 for t > 0, where [tex]K[/tex] is a positive constant. The initial
    condition is
    u(x, 0) = cxe[tex]^{\frac{-x^{2}}{4a^{2}}}[/tex]
    with c and a being constants, and the boundary conditions are
    u(0, t) = 0
    u(x, t) [tex]\rightarrow[/tex] 0 as x [tex]\rightarrow[/tex] [tex]\infty[/tex]
    Prove that the Fourier sine transform of u with respect to x is given by
    [tex]\hat{u}[/tex](s,t) = [tex]\hat{u}[/tex](s, 0)e[tex]^{-Kts^{2}}[/tex]
    Hence find the solution of the heat equation.

    I have absolutely no idea what to do, and my books aren't helping at all. Anyone able to help me?
  2. jcsd
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