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PDE: a traveling wave solution to the diffusion equation

  1. Mar 2, 2009 #1
    1. The problem statement, all variables and given/known data

    Consider a traveling wave u(x,t) =f(x - at) where f is a given function of one variable.
    (a) If it is a solution of the wave equation, show that the speed must be [tex]a = \pm c[/tex] (unless f is a linear function).
    (b) If it is a solution of the diffusion equation, find f and show that the speed a is arbitrary.

    2. Relevant equations

    Wave: u(x,t) = f(x+ct) + g(x-ct) and [tex]u_{tt} = c^2u_{xx}[/tex]

    Diffusion: [tex]u_{t} = ku_{xx}[/tex] and [tex]u(x,t) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy[/tex]

    3. The attempt at a solution

    (a) The wave equation is [tex] u_{tt} = c^2u_{xx}[/tex] with the general solutions u(x,t) = f(x + ct) + g(x - ct), so for the solution to be u(x,t) = f(x-at), [tex]a = \pm c[/tex]. If f is linear, then [tex]u_{tt} =u_{xx}=0[/tex], so it doesn't matter what a equals.

    (b) [tex]u(x,t) = f(x-at) = \frac{1}{\sqrt{4kt\pi}}\int^{\infty}_{-\infty}e^{-(x-y)^2/4kt}\phi(y)dy[/tex] I don't know where to go from here.
  2. jcsd
  3. Mar 3, 2009 #2
    Plug f(x-at) into the diffusion equation. You will get a simple ODE for f', from which you can obtain all the possible f so that f(x-at) satisfies the PDE.
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