10.7: Our Initial Problem and Bessel Functions

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 $\phi$, we expect the solution to be radially symmetric as well, $u(\rho,t)$, which satisfies the equation

$$\begin{aligned} \frac{\partial u}{\partial t} &= k\left[ \frac{\partial^2 u}{\partial \rho^2} + \frac{1}{\rho} \frac{\partial u}{\partial \rho} \right],\nonumber\\ u(c,t) &=0,\nonumber\\ u(\rho,0) &= f(\rho).\end{aligned} \nonumber$$

With $u(\rho,t)=R(\rho)T(t)$ we found the equations

$$\begin{aligned} \rho^2 R''+\rho R' + \lambda \rho^2 R &= 0\;\;\;\;R(c)=0\nonumber\\ T'+\lambda k T = 0.\end{aligned} \nonumber$$

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. $\PageIndex{1}$.

Figure $\PageIndex{1}$: A graph of the first five functions $R_n$

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.