Spherical coordinates of Partial Differential Equation

1. Dec 6, 2014

jhartc90

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. Dec 6, 2014

Staff: Mentor

I changed your derivatives to partial derivatives. In LaTeX it's \partial F instead of dF, and so on.

3. Dec 6, 2014

jhartc90

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.

4. Dec 6, 2014

Staff: Mentor

What's the initial condition?

Chet

5. Dec 6, 2014

jhartc90

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?

6. Dec 6, 2014

Staff: Mentor

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

7. Dec 6, 2014

jhartc90

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.

8. Dec 6, 2014

jhartc90

Here I attached the full question

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9. Dec 6, 2014

Staff: Mentor

How do the boundary conditions transform when you make your substitution W/r? Also, how does W behave near r = 0?

Chet

10. Dec 6, 2014

jhartc90

Did you see the original question? I attached file in previous response

11. Dec 6, 2014

Staff: Mentor

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

12. Dec 6, 2014

jhartc90

This might help. We are suppose to state all our assumptions. So, we could assume something here in order to solve the problem.

13. Dec 6, 2014

Staff: Mentor

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

14. Dec 6, 2014

jhartc90

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.

15. Dec 6, 2014

jhartc90

I'm guessing the eigenfunctions will have a sin or cosine form.

16. Dec 7, 2014

Staff: Mentor

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

17. Dec 8, 2014

Staff: Mentor

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:
$$\frac{d(\int_0^R{r^2F(r,t)dr})}{dt}=\int_0^R{r^2g(r,t)dr}$$
This equation can be integrated between t = 0 and arbitrary t to yield:
$$\int_0^R{r^2F(r,t)dr}=\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ$$
This suggests that it might be advantageous to express F(r,t) as:
$$F(r,t)=H(r,t)+\frac{[3\int_0^t{\int_0^R{r^2g(r,ξ)dr}}dξ]}{R^3}$$
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:

$$\frac{\partial H}{\partial t}=\alpha \frac{1}{r^2}\frac{\partial}{\partial r}\left(r^2\frac{\partial H}{\partial r}\right)+\left[g(r,t)-\frac{3\int_0^R{r^2g(r,t)dr}}{R^3}\right]$$
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:
$$\int_0^R{r^2H(r,t)dr}=0$$
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