Spherical coordinates of Partial Differential Equation

In summary, the conversation revolved around a PDE in a spherical domain with known boundary conditions and a known source function. The main question was how to proceed with solving the problem, with suggestions such as using separation of variables or finding a general solution. One member also suggested using a transformation to simplify the problem, while another member proposed a different approach involving heat conduction. The conversation ended with the suggestion of waiting for more input before presenting another idea.
  • #1
jhartc90
43
0

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


[/B]
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:
Physics news on Phys.org
  • #2
I changed your derivatives to partial derivatives. In LaTeX it's \partial F instead of dF, and so on.
 
  • #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.
 
  • #4
What's the initial condition?

Chet
 
  • #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?
 
  • #6
jhartc90 said:
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
 
  • #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.
 
  • #8
Here I attached the full question
 

Attachments

  • Capture.PNG
    Capture.PNG
    56.9 KB · Views: 498
  • #9
How do the boundary conditions transform when you make your substitution W/r? Also, how does W behave near r = 0?

Chet
 
  • #10
Chestermiller said:
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
Chestermiller said:
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
 
  • #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
 
  • #12
Chestermiller said:
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.
 
  • #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
 
  • #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.
 
  • #15
I'm guessing the eigenfunctions will have a sin or cosine form.
 
  • #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
 
  • #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:

1. What are spherical coordinates?

Spherical coordinates are a system of coordinates used to describe the location of a point in three-dimensional space. They consist of a distance from the origin (r), a polar angle (θ), and an azimuthal angle (φ).

2. How are spherical coordinates related to partial differential equations?

Spherical coordinates can be used to solve partial differential equations in three-dimensional space. This is because they allow for the separation of variables, which simplifies the equation and makes it easier to solve.

3. What is the Laplace operator in spherical coordinates?

The Laplace operator in spherical coordinates is given by

Similar threads

Replies
6
Views
859
Replies
8
Views
155
  • Differential Equations
Replies
3
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
878
  • Advanced Physics Homework Help
Replies
1
Views
766
  • Advanced Physics Homework Help
Replies
10
Views
962
  • Advanced Physics Homework Help
Replies
1
Views
875
  • Advanced Physics Homework Help
Replies
3
Views
286
  • Calculus and Beyond Homework Help
Replies
4
Views
436
  • Advanced Physics Homework Help
Replies
3
Views
627
Back
Top