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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+ρR+λρ2R=0R(c)=0T+λkT=0.

The equation for R is clearly self-adjoint, it can be written as

[ρR]+λρR=0

So how does the equation for R relate to Bessel’s equation? Let us make the change of variables x=λρ. We find

ddρ=λddx,

and we can remove a common factor λ to obtain (X(x)=R(ρ))

[xX]+xX=0,

which is Bessel’s equation of order 0, i.e.,

R(ρ)=J0(ρλ).

The boundary condition R(c)=0 shows that cλn=xn,

where xn are the points where J0(x)=0. We thus conclude Rn(ρ)=J0(ρλn). the first five solutions Rn (for c=1) are shown in Fig. 10.7.1.

BesselZero.png
Figure 10.7.1: A graph of the first five functions Rn

From Sturm-Liouville theory we conclude that

0ρdρRn(ρ)Rm(ρ)=0if nm.

Together with the solution for the T equation,

Tn(t)=exp(λnkt)

we find a Fourier-Bessel series type solution

u(ρ,t)=n=1AnJ0(ρλn)exp(λnkt),

with λn=(xn/c)2.

In order to understand how to determine the coefficients An from the initial condition u(ρ,0)=f(ρ) 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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