Mathematical Methods for Physics and Engineering : A Comprehensive Guide

PDES: SEPARATION OF VARIABLES AND OTHER METHODS

z

x

a y

u=0 u=0

u=T 0

Figure 21.5 A uniform metal cylinder whose curved surface is kept at 0◦C

and whose base is held at a temperatureT 0.

By imposing the remaining boundary conditionu(ρ, φ,0) =T 0 , the coefficientsAncan be
found in a similar way to Fourier coefficients but this time by exploiting the orthogonality
of the Bessel functions, as discussed in chapter 16. From this boundary condition we
require

u(ρ, φ,0) =

∑∞

n=1

AnJ 0 (knρ)=T 0.

If we multiply this expression byρJ 0 (krρ) and integrate fromρ=0toρ=a, and use the
orthogonality of the Bessel functionsJ 0 (knρ), then the coefficients are given by (18.91) as

An=

2 T 0

a^2 J 12 (kna)

∫a

0

J 0 (knρ)ρdρ. (21.36)

The integral on the RHS can be evaluated using the recurrence relation (18.92) of
chapter 16,

d

dz

[zJ 1 (z)] =zJ 0 (z),

whichonsettingz=knρyields

1

kn

d

dρ

[knρJ 1 (knρ)] =knρJ 0 (knρ).

Therefore the integral in (21.36) is given by
∫a

0

J 0 (knρ)ρdρ=

[

1

kn

ρJ 1 (knρ)

]a

0

=

1

kn

aJ 1 (kna),