TITLE: Bessel functions and PDE steady state temperature
QUESTION [0 upvotes]: If $u = 0$ at $r = 10$ and $u = 100$ at $z= 0$, how can we find $u$ at $r = 5$, $z = 10$. 
So the situation is for a steady state temperature in a cylinder where coefficients $c_m = 200/(k_m J_1(k_m))$ of the series $u = \sum_{m = 1}^\infty c_m J_0(k_m r) e^{-k_m z}$ where $J_0$ denotes the Bessel function of first kind.
This is from Boas Mathematical tools physical science, not homework, but a practice question

REPLY [1 votes]: For the condition at $r=10$, you have 
$$k_m = \frac{j_{0,m}}{10}$$
where $j_{0,m}$ is the $m$th zero of $J_0$.  Because you have already specified your coefficients, I am not sure of the use of the condition at $z=0$; you should check to see if this condition and the values specified for the coefficients are consistent.
Given the values of the $k_m$ and $c_m$, you need to evaluate that sum numerically for $r=5$ and $z=10$.  You should specify some precision so you know when to stop summing.  You will need a table of zeroes of that Bessel function as well.
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