How Do You Solve the Cylindrical Heat Equation with Non-Constant Coefficients?

[Yoni]

I have tried to solve the cylindrical case of the heat equation and reached the second order differential equation for the function R(r):

R'' + (1/r)*R' + (alfa/k)*R = 0

(alfa, k are constants)

I couldn't find material on the web for non-constant coefficients, does anyone know how to solve this?

thanks

 

[phyzguy]

Try re-scaling your r-coordinate to a new coordinate:
$$r' = r\sqrt{\alpha/k}$$
This will get rid of the constants. Then this becomes Bessels' equation. Go here:

http://en.wikipedia.org/wiki/Bessel_function