1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: PDE: a traveling wave solution to the diffusion equation
  1. Diffusion equation PDE (Replies: 1)

Loading...