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Homework Help: Solving the 'modified diffusion equation' using fourier transform

  1. Apr 27, 2010 #1
    1. The problem statement, all variables and given/known data

    A the density of a gas [tex]\rho[/tex] obeys the modified diffusion equation

    [tex]\frac{\partial \rho(x,t)}{\partial t}-D\frac{\partial^2 \rho(x,t)}{\partial x^2}=K\delta(x)\delta(t)[/tex]

    A) Express [tex]\rho[/tex] in terms of its 2D fourier transform [tex]\widetilde{\rho}(p,\omega)[/tex] and express the right hand side as a Fourier type integral.

    B) Find the function [tex]\widetilde{\rho}(p,\omega)[/tex]
    [Note: It might occur to you that this function could contain a term of the form [tex]f(p)\delta(Dp^2-i\omega )[/tex], where [tex]f(p)[/tex] is an arbitrary function. If so, you should assume that [tex]f(p)=0[/tex]. If this doesn't occur to you, then don't worry about it!]

    3. The attempt at a solution

    A) I'm pretty sure I can do this:

    [tex]\rho(x,t)=\frac{1}{2\pi}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)}\widetilde{\rho}(p,\omega) \right ][/tex]

    and

    [tex]\delta(x)\delta(t)=\frac{1}{4\pi^2}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)} \right][/tex]

    B) Sticking the results from A) into the diffusion equation, taking the derivatives and rearranging:

    [tex]\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)}\widetilde{\rho}(p,\omega) \right]=\frac{K}{2\pi(Dp^2-i\omega)}\iint_{-\infty}^{\infty}dp\; d\omega\; \left [e^{i(px-\omega t)} \right][/tex]

    ...and here i get stuck. how do i solve this? I have noticed that the two sides are conspicuously similar, but I'm not sure what conclusions I can make from that.

    Edit: the choice of exponentials for the transform (negative for the time) is the convention that was recomended in lectures.
     
    Last edited: Apr 27, 2010
  2. jcsd
  3. Apr 28, 2010 #2
    I'm tempted to simply say [tex]\widetilde{\rho}(p,\omega)=1[/tex], and [tex]\frac{K}{2\pi (Dp^2-i\omega)}=1[/tex], but that wouldn't give much diffusion, so I assume that's not the answer...

    I have also considered taking [tex]\frac{\partial^2 }{\partial p \partial \omega}[/tex] to both sides to get a partial differential equation, but i don't think that will work, since it's a definate integral, and you would still have an integral on the r.h.s from the chain rule.
     
  4. Apr 29, 2010 #3
    i'm sorry, this is nought but a shameless bump, can't diguise it as anything else. Any hint would be appreciated.
     
  5. Apr 29, 2010 #4

    marcusl

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    Gold Member

    You can't take the term [tex](Dp^2-i\omega)[/tex] out of the integral on the left since both variables are operated on by the integrals. Then maybe the hint can be used.
     
  6. Apr 29, 2010 #5
    I stared at this for so long, can't believe I didn't realize that...
     
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