Page 75 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

SERIES SOLUTIONS OF ORDINARY DIFFERENTIAL EQUATIONS 65
Example 4.9. Derive a Fourier–Bessel series representation of 1 on the interval 0 < x < 1.
We note that with J 0 (λ j ) = 0

1
1
2
x[J 0 (λ j x)] dx = [J 1 (λ j )] 2 (4.103)
2
x=0
and

1

xJ 0 (λ j x)dx = J 1 (λ j ) (4.104)
x=0
Thus
∞
J 0 (λ j x)
1 = 2 (4.105)
λ j J 1 (λ j )
j=1
Example 4.10 (A problem in cylindrical coordinates). A cylinder of radius r 1 is initially at
a temperature u 0 when its surface temperature is increased to u 1 . It is sufficiently long that
variation in the z direction may be neglected and there is no variation in the θ direction. There
is no heat generation. From Chapter 1, Eq. (1.11)
α
u t = (ru r ) r (4.106)
r
u(0,r) = u 0
u(t,r 1 ) = u 1
u is bounded (4.107)

2
The length scale is r 1 and the time scale is r /α. A dimensionless dependent variable that
1
2
normalizes the problem is (u − u 1 )/(u 0 − u 1 ) = U. Setting η = r/r 1 and τ = tα/r ,
1
1
U τ = (ηU η ) η (4.108)
η
U(0,η) = 1
U(τ, 1) = 0 (4.109)
U is bounded

Separate variables as T(τ)R(η). Substitute into the differential equation and divide by TR.

T τ 1 2
= (ηR η ) η =±λ (4.110)
T Rη