I have a problem understanding when and how to use the bessel function, I'm completely confused what should I do with boundary conditions. Could you please advise me and if it's possible give me examples or exercises where i should use Bessel function and where i should not
$\Delta u=0$ on a circle with radius $r\in[1,2]$, boundary conditions are
$u|_{r=1}=sin(2\phi)sin(3\phi),$
$\frac{\partial u}{\partial n}|_{r=2} = cos(2\phi)cos(3\phi)$
I'm trying $u(r,\phi) = R(r)\cdot \Phi(\phi)$
$r^2\frac{R''}{R}+\frac{R'}{R}+\frac{\Phi''}{\Phi}=0,$
$\left\{
\begin{aligned}
\Phi''+\lambda_1\Phi=0\\
\Phi(0)=\Phi(2\pi) =0\\
\end{aligned}
\right.$, then
$\Phi_0=1,$
$\Phi_{k,1}=cos\sqrt\lambda_1\phi$
$\Phi_{k,2}=sin\sqrt\lambda_1\phi$
$r^2R''+rR'+R(\lambda_1+\lambda_2)=0,$ $t = \alpha r$
$R_r'= \alpha R_t'$
$R_r''= \alpha^2 R_t''$
$\alpha = \sqrt\lambda$
$R(t)=J_k(\sqrt\lambda r)$
$J_k(\sqrt\lambda R) =0, $ $\mu_n^k$ - the n-th positive root $J_k$
$\lambda_{k,n} = (\frac{\mu_n^k}{R})^2$
$R_{k,n}=J_k(\frac{\mu_n^k}{R}r)$
$u=\sum\limits_{\forall k,n}\Phi_{k,n}R_{k,n}$