Bessel Functions

Dealing with Bessel Functions

I had to deal with some Bessel Functions in dealing with an analytical solution of steady state reaction-diffusion partial differential equation. The diffusion-reaction equation is a simplified, non-dimensionalized equation for a single species and single reaction, and is given by:

$ \nabla^2 u = \phi^2 u $

with the following boundary condition on the boundary of the domain:

$ v (1-u_s) = \dfrac{\partial u}{\partial v} $

For an infinite cylinder (analytical solutions on cylinders typically end up using Bessel Functions), the solution can be written in terms of the non-dimensionalized radial dimension $ \rho $:

$ \dfrac{1}{\rho} \dfrac{d}{d \rho} \left( \rho \dfrac{du}{d \rho} \right) = \phi^2 u $

over the region $ 0 \leq \rho \leq 1 $, and the boundary conditions:

$ \dfrac{du}{d \rho} = 0 $

at $ \rho=0 $, and also

$ v(1-u) = \dfrac{\partial u}{\partial \rho} $

at $ \rho=1 $.