1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

330 Chapter 5 Higher Dimensions and Other Coordinates

Several problems in which the Bessel functions play an important role follow.
The details of separation of variables, which should now be routine, are kept
to a minimum.

A. Potential Equation in a Cylinder

The steady-state temperature distribution in a circular cylinder with insulated
surface is determined by the problem

1

r

∂

∂r

(

r∂u

∂r

)

+∂

(^2) u
∂z^2
= 0 , 0 <r<a, 0 <z<b, (2)
∂u
∂r
(a,z)= 0 , 0 <z<b, (3)
u(r, 0 )=f(r), 0 <r<a, (4)
u(r,b)=g(r), 0 <r<a. (5)
Here we are considering the boundary conditions to be independent ofθ,sou
is independent ofθalso.
Assuming thatu=R(r)Z(z)we find that
(
rR′

)′

+λ^2 rR= 0 , 0 <r<a, (6)

R′(a)= 0 , (7)

∣∣

R( 0 )

∣∣

bounded, (8)

Z′′−λ^2 Z= 0. (9)

Condition(8)hasbeenaddedbecauser=0 is a singular point. The solution
of Eqs. (6)–(8) is

Rn(r)=J 0 (λnr), (10)

where the eigenvaluesλ^2 nare defined by the solutions of

R′(a)=λJ 0 ′(λa)= 0. (11)

BecauseJ′ 0 =−J 1 ,theλ’s are related to the zeros ofJ 1. The first three eigen-
values are 0,( 3. 832 /a)^2 ,and( 7. 016 /a)^2 .NotethatR( 0 )=J 0 ( 0 )=1.
The solution of the problem Eqs. (2)–(5) may be put in the form

u(r,z)=a 0 +b 0 z+

∑∞

n= 1

J 0 (λnr)

[

ansinh(λnz)

sinh(λnb)

+bnsinh(λn(b−z))

sinh(λnb)

]

. (12)