I'm having a course on partial diferential equation solving, and we are currently going through cylindrical coordinates problems.
I'm trying to solve the heat equation for a cylinder with fixed surface temperatures like shown here: Infinite semi-cylinder with diferent temperatures fixed in each surface Since there are no heat sources, we have to deal with a Laplace problem such that: $$ \Delta u = 0 $$ With this CC: $$ u(ρ= ρ_0) = T_3\\ u(z=0) = u(z=L) = T_1\\ u(\phi = 0) = u(\phi=\pi) = T_2 $$ So in order to solve this, we have been taught to use variable separation and simplify the operator making Sturm-Lioville problems in the proper directions. Then: $$ u(\rho, \phi, z) = R(\rho)\Phi(\phi)Z(z) $$
And plugging this into the laplace operator and dividing by u:
$$ \frac{1}{R}\Big(\frac{1}{\rho}\frac{dR }{d\rho} + \frac{d^2R }{d\rho^2}\Big) + \frac{1}{\Phi \rho^2}\frac{d^2\Phi }{d\phi^2} + \frac{d^2Z }{dz^2}\frac{1}{Z} = 0 $$
Then we will use the linearity of the operator to find the solution by adding the solution of 2 more simple problems. First we make the change of variables such that the contourn of the chosen "favourite" direction has homogenous condition, like shown here: Separating problem in 2
The thing is that the teacher has stated that we should always try to solve the problems by using orthogonal φ functions, that is, taking for both problems φ with homogenous conditions and orthogonal function:
$$ \frac{d^2\Phi }{d\phi^2} + m^2\phi = 0\\ \Phi(0) = \Phi (\pi) = 0 $$ And aplying the CC for the angular direction we find sinusoidal functions.
However I still don't know why we can't find the solution using in both problems the prefered direction Z. So that, for the problem with $\rho$ also with homogenous conditions we have: The CC: $$ v(ρ= ρ_0) = T_3-T_1\\ v(z=0) = u(z=L) = 0\\ v(\phi = 0) = u(\phi=\pi) = T_2-T_1 $$ We state that: $$ \frac{1}{\rho}\frac{dR }{d\rho} + \frac{d^2R }{d\rho^2} = λR\\ \frac{d^2Z }{dz^2} = μZ $$ The first one is satified by the first order Bessel equation and the second one is a sine function.
The eigenvalues of each are found using the CC: $$ R(\rho_0) = J_0(\sqrt{λ_n}\rho_0)=0\\ Z(L) = \sin(\sqrt{\mu}L) = 0 $$
Substituying in the laplacian: $$ \Big(\frac{1}{\rho^2}\frac{d^2\Phi_n }{d\phi^2} - (λ+μ)\Phi_n\Big)J_0(\sqrt{λ_n}\rho)sin(\sqrt{\mu}z)=0 $$
And if we multiply by $\rho^3J_0(\sqrt{λ_n'}\rho)sin(\sqrt{\mu'})$ and integrate from 0 to $\rho_0$ and from 0 to L in z we get a term proportional to $δ_{nn'}δ_{\mu\mu'}$ (from orthogonality of $J_0$ and sine function) plus the integral I'm concerned about:
$$ \int_{0}^{ρ_{0}}ρ^3J_0(xλ_n)J_0(xλ_{n'})dρ $$
Is this integral proportional to $\delta_{nn'}$? If not why. I cannot find a way to prove it. The reason this is so important, is that, if it is not proportional to $\delta_{nn'}$, then the equation for $\phi$ won't be resoluble since we won't have one equation for each eigenvalue $λ_n$.
Best regards!