I want to solve the transient heat equation on a ball. The boundary condition is the same over the hole outer surface. So this should reduce to a one-dimensional problem in radial direction. However I cannot use the one-dim heat equation, since the surface through which the heat flows goes quadratic with the radius.
Is the following approach correct: Take the heat equation, transform it into sperical coordinates and eliminate the derivatives in angular directions. I think the last part is justified, because of the rotational symmetry of the boundary condition. So I end up with this equation:
$$ \frac{\partial T}{\partial t} = \alpha \frac{1}{r^2}\frac{\partial}{\partial r} \left( r^2\frac{\partial T}{\partial r}\right) $$ with thermal diffusivity $\alpha$. Is this the correct reformulation of the heat equation on a ball to a one-dimensional problem?
EDIT: When I solve this with the finite element method, will there be a problem in the centre of the ball? The $1/r^2$ factor might cause a problem.
Answer
That sounds about right to me. Although I've never done this for the heat equation specifically, that's much the same approach we use to solve the Schrödinger equation in spherical coordinates, e.g. for the hydrogen atom. Your full solution will be products of the radial functions $T(r,t)$ with appropriate spherical harmonics.
When determining the radial part of the solution, assuming this goes like the Schrödinger equation, one of the boundary conditions you'll need to use is consistency at the origin: basically you need your solution for $T(r)$ to be finite and differentiable at $r=0$. I can come back and expand on this later, or perhaps someone else will give you a more detailed explanation before I get to it.
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