Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Age Diffusion Theory and Fourier Transforms

  1. Jan 21, 2009 #1
    I am attempting to solve a second order differential, but I am have never done anything like this. I was told that it was a good Idea to think about fourier transforms.


    Boundary Conditions:

    Apparently the final solution is:


    If you were wondering the problem statement:
    Determine the slowing down density established by a monoenergetic plane source at the origin of an infinite moderating medium as given by age-diffusion theory.
  2. jcsd
  3. Jan 29, 2009 #2
    This is one of the simplest PDE you can solve via Fourier Transform which accounts for two physical problem of interest, i mean heat equation and Schroedinger equation.
    We start with PDE

    \partial^2_x u(x,t) = \partial_t u(x,t)

    First start with projecting the equation on Fourier space (with respect to x), we get

    - k^2 \hat{u}(k,t) = \partial_t \hat{u}(k,t)

    This is a first order ODE in t for unknown [tex] \hat{u}(k,t) [/tex] its solution can be found by separation of variables and reads

    \hat{u}(k,t) = f(k) e^{- t k^2}

    for some unknown f(k) constant in time. To recover [tex] u(x,t) [/tex] we simply use inverse Fourier transform

    u(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} \hat{u}(k,t) e^{ i k x} dk = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} f(k) e^{- t k^2} e^{ i k x} dk



    u(x,0) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} f(k) e^{ i k x} dk

    hence f(k) is just the Fourier transform of the initial condition of the PDE. In our case

    f(k) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} u(x,0) e^{- i k x} dx = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} S_0 \delta(x) e^{- i k x} dx = \frac{S_0}{\sqrt{2 \pi}}

    So we find the solution by evaluating the integral

    u(x,t) = \frac{1}{\sqrt{2 \pi}} \int_{- \infty}^{\infty} \frac{S_0}{\sqrt{2 \pi}} e^{-t k^2} e^{i k x} dk = \frac{S_0}{2 \pi} \int_{- \infty}^{\infty} e^{-t k^2 + i k x} dk = \frac{S_0}{2 \pi} \int_{- \infty}^{\infty} e^{-t ( k^2 + i \frac{k x} {t} + \frac{i^2 x^2} {4 t^2} - \frac{i^2 x^2} {4 t^2} ) } dk = \frac{S_0 e^{- \frac{x^2} {4 t}}} {2 \pi} \int_{- \infty}^{\infty} e^{- t ( k + \frac{i x} {2 t} )^2 } dk =
    = \frac{S_0 e^{- \frac{x^2} {4 t}}} {2 \pi} \int_{- \infty - \frac{i x} {2 t} }^{\infty - \frac{i x} {2 t} } e^{- t p^2 } dp = \frac{S_0 e^{- \frac{x^2} {4 t}}} {2 \pi} \sqrt{\frac{\pi}{t}} = \frac{S_0} {\sqrt{4 \pi t}} e^{- \frac{x^2} {4 t}}

    Hope this can help, ask if some steps are not clear. bye.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?

Similar Discussions: Age Diffusion Theory and Fourier Transforms
  1. Fourier transformation (Replies: 3)

  2. Fourier Transforms (Replies: 3)