A long cylinder (radius $r = b$) initially at $T=f(r)$ is exposed to a cooling medium which extracts heat uniformly from its surface. It was assumed that heat transfer takes place only through radial direction.
From the research I did I could find out that the below is the analytical solution, for unsteady state heat transfer in an infinitely long cylinder along r direction.Using this analytical solution I need to find out the temperature time distribution at each time interval for all nodal points from surface to centre
$$T(r,t)=T_0+\frac{2}{b}(T_i-T_0)\sum_{m=1}^{\infty} \frac{1}{\beta_m}\frac{\mathrm{J_1}(\beta_m\cdot b)\mathrm{J_0}(\beta_m\cdot r))}{\mathrm{J}_0^2(\beta_m\cdot b)+\mathrm{J}_1^2 (\beta_m\cdot b)} \exp\left(-\beta^2_m \cdot\alpha\cdot t\right)$$
$J_0$ is the Bessel function of first kind and $J_1=-J'_0$.
Where the eigenvalues $\beta_m$ are obtained from the roots of transcendental equation
$\beta\cdot b\cdot J_1(\beta\cdot b)-B_i\cdot J_0(\beta\cdot b)=0$ and $B_i,b,T_0,T_i,\alpha,r,t$ are known.
Is there any way I could generate numerical solution to the above equation?