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Differential Equation with Noise term

  1. Oct 22, 2014 #1
    1. The problem statement, all variables and given/known data
    [tex] (\partial_t - A\nabla^2 + B)f(x) = \eta(x, t) [/tex]
    So I have a homogeneous linear differential equation except for an added noise term ##\eta(t) ##. The noise is uncorrelated between times and has a Gaussian distribution with zero mean. That is we have Gaussian white noise.

    The problem is to calculate the time dependence of f(x,t) and also the time dependence of ##\left| F(k, t)\right|^2 ##, where F(k, t) is the Fourier transform of f(x, t).


    2. Relevant equations


    3. The attempt at a solution

    If I know the specific function for the noise, I can solve using the Green's function - but I do not know ##\eta(t) ##.
    What can I say about the behavior of the solution? If you like, the noise starts at t=0, and we are interested in some kind of transient response I suppose. I began to proceed with solving the Green's function in Fourier space (that is x -> k, but t remained unchanged) but I am not sure what to do next.

    I am wishing I knew or remember more I guess.

     
  2. jcsd
  3. Oct 22, 2014 #2

    Simon Bridge

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  4. Oct 22, 2014 #3
    This is actually for a different course, we haven't been taught at all about random processes. However, years ago when I was getting my bachelors as an engineer I learned a little bit. I'd like to be able to do a nice job with this but I don't have time for a huge amount of extra learning.
     
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