Spherical coordinates of Partial Differential Equation

[jhartc90]

Homework Statement

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:

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

Homework Equations

So, in spherical coord.

First, I know that:

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

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

The Attempt at a Solution

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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.

 

[Mark44]

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

 

[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.

 

[Chestermiller]

What's the initial condition?

Chet

 

[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?

 

[Chestermiller]

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?

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

 

[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.

 

[jhartc90]

Here I attached the full question

 

[Chestermiller]

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

Chet

 

[jhartc90]

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

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

Chet

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

 

[Chestermiller]

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

 

[jhartc90]

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

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

 

[Chestermiller]

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

 

[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.

 

[jhartc90]

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

 

[Chestermiller]

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

 

[Chestermiller]

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