10.7: Our Initial Problem and Bessel Functions
( \newcommand{\kernel}{\mathrm{null}\,}\)
We started the discussion from the problem of the temperature on a circular disk, solved in polar coordinates, Since the initial conditions do not depend on ϕ, we expect the solution to be radially symmetric as well, u(ρ,t), which satisfies the equation
∂u∂t=k[∂2u∂ρ2+1ρ∂u∂ρ],u(c,t)=0,u(ρ,0)=f(ρ).
With u(ρ,t)=R(ρ)T(t) we found the equations
ρ2R″
The equation for R is clearly self-adjoint, it can be written as
[\rho R']' + \lambda \rho R = 0 \nonumber
So how does the equation for R relate to Bessel’s equation? Let us make the change of variables x= \sqrt{\lambda} \rho. We find
\frac{d}{d\rho} = \sqrt{\lambda} \frac{d}{dx}, \nonumber
and we can remove a common factor \sqrt{\lambda} to obtain (X(x)=R(\rho))
[x X']' + x X = 0, \nonumber
which is Bessel’s equation of order 0, i.e.,
R(\rho) = J_0(\rho \sqrt{\lambda}). \nonumber
The boundary condition R(c)=0 shows that c \sqrt{\lambda_n} = x_n, \nonumber
where x_n are the points where J_0(x)=0. We thus conclude R_n(\rho) = J_0(\rho \sqrt{\lambda_n}). \nonumber the first five solutions R_n (for c=1) are shown in Fig. \PageIndex{1}.

From Sturm-Liouville theory we conclude that
\int_0^\infty \rho d\rho \,R_n(\rho)R_m(\rho)=0\;\;{\rm if\ }n \neq m. \nonumber
Together with the solution for the T equation,
T_n(t)= \exp(-\lambda_n k t) \nonumber
we find a Fourier-Bessel series type solution
u(\rho,t) = \sum_{n=1}^\infty A_n J_0(\rho\sqrt{\lambda_n})\exp(-\lambda_n k t), \nonumber
with \lambda_n= (x_n/c)^2.
In order to understand how to determine the coefficients A_n from the initial condition u(\rho,0)=f(\rho) we need to study Fourier-Bessel series in a little more detail.