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Solving a PDE in spherical with source term

  1. Dec 6, 2014 #1
    1. The problem statement, all variables and given/known data

    I have a PDE and I need to solve it in spherical domain:

    $$\frac{dF(r,t)}{dt}=\alpha \frac{1}{r^2} \frac{d}{dr} r^2 \frac{dF(r,t)}{dr} +g(r,t) $$

    I have BC's:

    $$ \frac{dF}{dr} = 0, r =0$$
    $$ \frac{dF}{dr} = 0, r =R$$

    2. Relevant equations

    So, in spherical coord.

    First, I know that:

    $$F=w/r$$
    Reducing, I get:

    $$\frac{dw}{dt} =\alpha \frac{d^2w}{dr^2}+r*g(r,t) $$

    3. The attempt at a solution

    After I Get this, I need to find eigenfunction expansions to express the source term and
    then, finally, the solution Do I need to do separation of variables? I am confused at this point and not sure how to proceed.
     
  2. jcsd
  3. Dec 7, 2014 #2

    Orodruin

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    Why do you say you are confused when you have essentially described a proper way of attacking the problem? Why don't you simply try doing it?

    Side note: Were you given these boundary conditions or did you implement them based on problem formulation. It seems strange to me to have a boundary condition of that form at r=0. If r=0 is part of your domain, it is not a boundary.
     
  4. Dec 7, 2014 #3

    I should be more specific. The problem is attached for complete clarity, noting that I need to state any assumptions. The reason I haven't started is because I am not fully sure how to start. Should I Start with separation of variables? Should I start with identifying a proper eigenfunction? Would it be of the form sin(...).
     

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