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Numerical solution of Fisher's equation

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

    Solve Fisher's equation
    u_t = u_{xx} + u(1 - u)
    numerically, with the initial condition (step function)
    u(x,0) = heaviside(-x)

    2. Relevant equations

    One can assume a travelling wave solution:
    u(x,t) = u(\xi)
    where \xi = x-vt
    such that
    u_{\xi \xi} = -vu_{\xi} - u(1 - u)
    which is a second order nonlinear ode (right?)

    3. The attempt at a solution

    I have tried to implement the split-step (pseudospectral) method, but I ran into trouble when I tried to deal with the nonlinear term u^2. I tried the (implicit) Crank-Nicolson method - had difficulty there too. I am now attempting the (explicit) Newton-Kantorovich method - not making much progress.

    The travelling wave solution should be a helpful simplification, though I'm not sure what the next step is: maybe a substitution of the form w = u_{\xi}, to reduce to a first order nonlinear ode (?).
    Would a straightforward explicit finite differences method work or will I need something more sophisticated?

    Any help will be fantastic!
     
  2. jcsd
  3. Oct 17, 2011 #2
    I'm interested in this too... despite it being posted a long time ago.
     
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