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10.7: Our Initial Problem and Bessel Functions

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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

ut=k[2uρ2+1ρuρ],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}.

BesselZero.png
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.


This page titled 10.7: Our Initial Problem and Bessel Functions is shared under a CC BY-NC-SA 2.0 license and was authored, remixed, and/or curated by Niels Walet via source content that was edited to the style and standards of the LibreTexts platform.

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