Solving a PDE in spherical with source term

jhartc90
Messages
43
Reaction score
0

Homework Statement



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:

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

Homework Equations



So, in spherical coord.

First, I know that:

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

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

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.
 
Physics news on Phys.org
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.
 
Orodruin said:
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.
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(...).
 

Attachments

  • Capture2.PNG
    Capture2.PNG
    17.3 KB · Views: 578
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
Back
Top