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Burger's equation

  1. Oct 18, 2011 #1
    1. The problem statement, all variables and given/known data
    a) Show that the function [itex]\phi[/itex](x,t) = 1 + sqrt(a/t)exp(-x^2/(4vt)) with a > 0 satisfies the dispersion equation [itex]\phi[/itex]t = v[itex]\phi[/itex]xx
    b) Use the Cole-Hopf transformation to find the corresponding solution u(x,t) to Burger's equation ut + uux = vuxx.
    c) Show that this solution is anti-symmetric (odd) and that the area under the solution for x > 0 A(t) is given by A(t) = 2v log(1+sqrt(a/t)).

    2. Relevant equations

    3. The attempt at a solution
    a) is easy.
    For b) the Cole-Hopf transformation is [itex]\psi[/itex] = -2vlog([itex]\phi[/itex])
    and to find u we solve u(x,t) = (-2v[itex]\phi[/itex]x(x,t))/[itex]\phi[/itex](x,t)
    so do we just use the [itex]\phi[/itex](x,t) in part a).
    c) For oddness do we show that -u(-x,t) = u(x,t) and to find A(t) do we use A(t) = [itex]\int_0^{∞}[/itex] u(x,t) dx
  2. jcsd
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