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Diffusion eq. with periodic BC using method of images

  1. Oct 9, 2008 #1
    1. The problem statement, all variables and given/known data
    Considering the periodic boundary conditions (given below) I am supposed to find the solution T(x,t) with the initial condition T(x,0)=[tex]\delta[/tex](x) Also I am limited to use method of images so I can't use separation of variables unfortunately.

    2. Relevant equations
    The boundary conditions are give by:

    [tex]\frac{\partial T}{\partial x}(x=-L/2)=\frac{\partial T}{\partial x}(x=L/2)[/tex]

    3. The attempt at a solution
    I've only started and for the initial condition using method of images I get:

    [tex]T(x,t)=\sum{(-1)^{n}\ T_{g}(x+n*L,t)}[/tex]

    where the sum goes from -infinity to infinity.

    My problem is how to implement the periodic boundary conditions into the problem.
    In my textbook it says that using theese kind of boundary conditions in 1-D is equivalent to transforming the coordinates from a line to a circle. What does that mean?

    I'd much appriciate it if you gave me a hint on how to solve this

  2. jcsd
  3. Oct 9, 2008 #2


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    Welcome to PF!

    Hi Simon! Welcome to PF! :smile:

    It just means that under those boundary conditions, the function repeats itself whenever x increases by L.

    So it's the same as a function on a circle with perimeter L. :smile:
  4. Oct 9, 2008 #3
    Thanks for the welcome =)

    Ok so the function repeats itself when x increases by L. How do I use this when "mirroring"?

    Since the delta-function has alternating signs (regarding the initial condition) for every other mirror image. Does this goes for the BC as well?

    A lot of confusion here since I don't know the exact properties of the method of images. If anyone has a link where it is explained I'd appriciate it :P
  5. Oct 10, 2008 #4


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    What exactly is [itex]T_{g}(x+n*L,t)[/itex]? I assume you are summing over n?

    What x interval are you trying to find the solution on? [-L/2,L/2] perhaps? The method of images entails adding additional "image sources" outside of the region that you are looking for a solution on. These extra sources are placed such that the solution T(x,t) due to all of the sources will satisfy the boundary conditions.
  6. Oct 10, 2008 #5
    Yea sorry I forgot to write that I sum over n. Tg is just the gaussian solution to the diffusion eq. And [tex]\int^{-\infty}_{\infty} Tg(x,t)dx=1[/tex]
  7. Oct 10, 2008 #6


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    Okay, so

    [tex]T_g(x,t)=(4 \pi kt )^{-\frac{1}{2}}e^{\frac{-x^2}{4kt}}[/tex]

    where [itex]k[/itex] is the diffusion constant?

    You know that using the method of images is going to involve adding image sources, so say you place one at [itex]x=x_0[/itex] such that [itex]T(x,0;x_0)=\delta (x-x_0)[/itex] what then would [itex]T(x,t;x_0)[/itex] due to just that source be?
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