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Spherical coordinates of Partial Differential Equation

  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:

    $$\frac{\partial F(r,t)}{\partial t}=\alpha \frac{1}{r^2} \frac{\partial}{\partial r} r^2 \frac{\partial F(r,t)}{\partial r} +g(r,t) $$

    I have BC's:

    $$ \frac{\partial F}{\partial dr} = 0, r =0$$
    $$ \frac{\partial F}{\partial r} = 0, r =R$$

    2. Relevant equations

    So, in spherical coord.

    First, I know that:

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

    $$\frac{\partial w}{\partial t} =\alpha \frac{\partial^2 w}{\partial r^2}+r*g(r,t) $$

    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.
     
    Last edited by a moderator: Dec 6, 2014
  2. jcsd
  3. Dec 6, 2014 #2

    Mark44

    Staff: Mentor

    I changed your derivatives to partial derivatives. In LaTeX it's \partial F instead of dF, and so on.
     
  4. Dec 6, 2014 #3
    Ok, thanks, do you know how to solve this question? Or at least maybe the next step? I am very confused on what to do next.
     
  5. Dec 6, 2014 #4
    What's the initial condition?

    Chet
     
  6. Dec 6, 2014 #5
    All i know I think I already posted. We start with the initial PDE. I know that dF/dr =0 in the center and on the surface of the sphere and g(r,t) is a known source function. I am told to substitute F=W/r to simplify the 1st term on right-hand side (which I think I did correctly but not sure). Then I need to use eigenfunctions to express g(r,t) and then the solution. That's basically all the info I have. Do you see a way in which to proceed?
     
  7. Dec 6, 2014 #6
    I see ways to proceed, but the problem is underspecified without an initial condition. Is F equal to zero for all r at time zero?

    Chet
     
  8. Dec 6, 2014 #7
    All I know is that F=W/r.

    Would I start by separation of variables? I am very confused. I just know what the derivative of F with respect to r is at r=0 and r=R. other than that, your guess is as good as mine. Thanks for taking the time to help by the way, I really appreciate it.
     
  9. Dec 6, 2014 #8
    Here I attached the full question
     

    Attached Files:

  10. Dec 6, 2014 #9
    How do the boundary conditions transform when you make your substitution W/r? Also, how does W behave near r = 0?

    Chet
     
  11. Dec 6, 2014 #10
    Did you see the original question? I attached file in previous response
     
  12. Dec 6, 2014 #11
    Yes. I see the question, and the problem is underspecified. You know that because you can add an arbitrary constant to F, and still satisfy the differential equation and boundary conditions.

    Chet
     
  13. Dec 6, 2014 #12
    This might help. We are suppose to state all our assumptions. So, we could assume something here in order to solve the problem.
     
  14. Dec 6, 2014 #13
    I think that taking F = 0 at time t = 0 would be reasonable.

    Substituting F = W/r is not how I ordinarily would attack this problem. I would like to think about this a little more before giving you advice.

    Chet
     
  15. Dec 6, 2014 #14
    Ok, fair enough. I think I might have to either do separation of variable or come up with a general solution. I'm not fully sure though.
     
  16. Dec 6, 2014 #15
    I'm guessing the eigenfunctions will have a sin or cosine form.
     
  17. Dec 7, 2014 #16
    I think that, with the transformation F=W/r, the transformed boundary condition at r = 0 becomes problematic mathematically. See what you get. What to other members think? Mark?

    Chet
     
  18. Dec 8, 2014 #17
    This problem is the same as transient heat conduction with heat generation inside an insulated sphere. If we multiply both side of the differential equation by r2 and integrate between r = 0 and r = R, we obtain:
    [tex]\frac{d(\int_0^R{r^2F(r,t)dr})}{dt}=\int_0^R{r^2g(r,t)dr}[/tex]
    This equation can be integrated between t = 0 and arbitrary t to yield:
    [tex]\int_0^R{r^2F(r,t)dr}=\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ[/tex]
    This suggests that it might be advantageous to express F(r,t) as:
    [tex]F(r,t)=H(r,t)+\frac{[3\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ]}{R^3}[/tex]
    The second term on the right here represents the temperature F averaged over the volume of the sphere at any time (assuming F = 0 at time zero). If we substitute this relationship into the original differential equation, we obtain:

    [tex]\frac{\partial H}{\partial t}=\alpha \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial H}{\partial r}\right)+\left[g(r,t)-\frac{3\int_0^R{r^2g(r,t)dr}}{R^3}\right][/tex]
    The term in brackets is the local rate of heat generation within the sphere minus the average rate of heat generation (averaged over the volume of the sphere). In addition, the function H satisfies the equation:
    [tex]\int_0^R{r^2H(r,t)dr}=0[/tex]
    This equation means that the average value of H (averaged over the volume of the sphere) does not change with time.

    If the heat generation rate g were constant, the term in brackets would vanish, which would make this problem very easy to solve. If the heat generation g were a function of r but not t, the problem would also be very easy to solve (since there would be a long time steady state solution). The time-dependence of the term in brackets makes the problem more difficult to solve (even though the average value of the term in brackets is equal to zero over the volume of the sphere).

    Anyone else out there working on this problem. Thoughts?

    I have another idea on how to approach this problem, but I'm not going to present it unless someone has more interest.

    Chet
     
    Last edited: Dec 9, 2014
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