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Diffusion equation PDE

  1. Sep 12, 2010 #1
    1. The problem statement, all variables and given/known data
    Solve
    [tex]u_{tt} - 4u_{xx} = 0[/tex], [tex] x \in \mathbb{R}, t > 0 [/tex]
    [tex]u(x, 0) = e^{-x^2} [/tex], [tex] x \in \mathbb{R} [/tex]

    2. Relevant equations
    General solution to the diffusion equation:
    [tex]u(x, t) = \frac{1}{\sqrt{4\pi kt}} \int\limits_{-\infty}^{\infty} e^\frac{{-(x - y)^2}}{4kt} \varphi(y) \, dy[/tex]


    3. The attempt at a solution
    [tex] u(x, t) = \frac{1}{\sqrt{4\pi kt}} \int\limits_{-\infty}^{\infty} e^\frac{{-(x - y)^2}}{4kt} e^{-y^2} [/tex]

    That's about as good as I've got. Integration by parts gets me no further. I've tried to combine the exponents in the integrand, but that leaves me with
    [tex] - \frac{x^2 + y^2 - 2xy + 4kty^2}{4kt} [/tex]
    I have an example in a textbook where they do similar, then complete the square so that they can substitute [tex]p[/tex], then integrate [tex]\int\limits_{-\infty}^{\infty}e^{-p^2} \, dp[/tex] as [tex]\sqrt{\pi}[/tex]... but I can't complete the square in this case.
     
  2. jcsd
  3. Sep 12, 2010 #2

    hunt_mat

    User Avatar
    Homework Helper

    Start from scratch, it'll be easier. Take Fourier transforms w.r.t x of the PDE and the initial condition, then look for the inverse transform.
     
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