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I already know how to apply the separation of variables method to solve transient radial heat equation inside a cylinder. But, when it comes to cylindrical shells, both Bessel J and Y functions appear in the solution and I don't know how to find the coefficients by taking advantage of orthogonality.

Here's the PDE: $$u_t = \frac{k}{r}(ru_r)_r$$

with the following boundary/initial conditions:

\begin{align} &r=r_3 &u_r = 0 \\ &r=r_2 &-ku_r + pu = 0 \\ &t=0 &u = u_0 \end{align}

After applying the separation of variables method, here's what I have:

\begin{align} u(r,t) &= R(r)T(t) \\ T(t) &= A e^{-k\lambda^2t} \\ R(r) &= B J_0(\lambda r) + C Y_0(\lambda r) \end{align}

I appreciate it if you could shed some light on how to proceed.

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There are unique constants $A,B$ so that $R(r)=AJ_0(\lambda r)+BY_0(\lambda r)$ satisfies the endpoint conditions $$ R'(r_3) = 0,\;\; R(r_3)=1. $$ This is because $R(r_3)=0=R'(r_3)$ implies $R\equiv 0$, and this is true for any constants $A,B$. Such a solution is $$ R(r) = \frac{J_0(\lambda r)Y_0'(\lambda r_3)-J_0'(\lambda r_3)Y_0(\lambda r)}{J_0(\lambda r_3)Y_0'(\lambda r_3)-J_0'(\lambda r_3)Y_0(\lambda r_3)} $$ Indeed, the numerator $N$ is a solution of the Bessel equation with eigenvalue $\lambda$ such that $N'(r_3)=0$ and $N(r_3)\ne 0$. So $R$ is a solution of the Bessel equation with $R'(r_3)=0$ and $R(r_3)=1$. The eigenvalue equation for $\lambda$ is then determined by the condition $$ -kR'(r_2)+pR(r_2) = 0. $$

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