Page 52 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

42 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
Separating variables as u = R(r)T(t),

2
1 dT 1 d R 1 dR 2
= + =−λ 0 ≤ r ≤ 1, 0 ≤ t (3.60)
T dt R dr 2 rR dr
(Why the minus sign?)
The equation for R(r)is

2

(rR ) + λ rR = 0, (3.61)
which is a Sturm–Liouville equation with weighting function r. It is an eigenvalue problem with
an infinite number of eigenfunctions corresponding to the eigenvalues λ n . There will be two
solutions R 1 (λ n r)and R 2 (λ n r)for each λ n . The solutions are called Bessel functions, and they
will be discussed in Chapter 4.

R n (λ n r) = A n R 1 (λ n r) + B n R 2 (λ n r) (3.62)

The boundary conditions on r are used to determine a relation between the constants A and
B. For solutions R(λ n r)and R(λ m r)

1

rR(λ n r)R(λ m r)dr = 0, n = m (3.63)

0
is the orthogonality condition.
2
n for all n. Thus, the solution of (3.60),
The solution for T(t) is the exponential e −λ t
because of superposition, can be written as an infinite series in a form something like
∞
−λ 2
u = K n e n R(λ n r) (3.64)
n=0

and the orthogonality condition is used to find K n as

1 1

2
K n = f (r)R(λ n r)rdr/ f (r)R (λ n r)rdr (3.65)
r=0 r=0

Problems
1. For Example 2.1 in Chapter 2 with the new boundary conditions described in Example
3.2 above, find K n and write the infinite series solution to the revised problem.