Solving a PDE in spherical with source term

[jhartc90]

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

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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.

 

[Orodruin]

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.

 

[jhartc90]

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(...).