I was solving the heat equation in spherical coordinates with standard boundary conditions: temperature held at 0 at the boundary $r=\alpha$. I was able to find all eigenvalues and eigenfunctions. I'm not going to show all my work because it's unnecessary. I found that the eigenfunctions for the azimuthal problem are the associated Legendre polynomials given by $$ P_{n}^{m}(\cos\phi) $$ and the eigenfunctions for the radial problem are the spherical Bessel functions given by: $$ r^{-1/2} \, P_{n+\frac{1}{2}} \bigl( \tfrac{\lambda}{a} r \bigr) $$
So the solution will involve the product of these eigenfunctions: \begin{gather} \sum_{n=0}^{\infty} \sum_{m=n}^{\infty} A_{nm} \, r^{-1/2} \, P_{n+\frac{1}{2}} \bigl( \tfrac{\lambda}{a} r \bigr) \, P_{n}^{m}(\cos\phi) \cos(n\theta) \, e^{-(\frac{\lambda}{a})^2 \,t} \\ \qquad{}+ \sum_{n=0}^{\infty} \sum_{m=n}^{\infty} B_{nm} \, r^{-1/2} \, P_{n+\frac{1}{2}} \bigl( \tfrac{\lambda}{a} r \bigr) \, P_{n}^{m}(\cos\phi) \sin(n\theta) \, e^{-(\frac{\lambda}{a})^2 \, t} \end{gather}
My problem is I'm not sure how to find the coefficients $A_{mn}$, $B_{mn}$. I read that the associated Legendre polynomials are not always orthogonal so I think I won't be able to use them as an orthogonal system here. Could you please show what the coefficients would be for this solution? Thanks a lot!
EDIT. Please help!
If you are solving the heat equation within a sphere, and if you require the temperature on the boundary of the sphere to be $0$ for all time, then you know one solution is the temperature distribution which is $0$ everywhere in and on the sphere. Uniqueness will dictate that to be the only solution.