Page 156 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

146 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS
so that
b

2

T D[ f (x)] =−λ p(x) f (x) n (x,λ)dx + N α [ f (a)]N [ n (a,λ)]r(a)
α
x=a
− N β [ f (b)]N [ n (b,λ)]r(b) (9.17)

β
where
b

2
λ 2 n p(x) f n (x) n (x,λ n )dx = λ F n (λ n ) (9.18)
n
a
9.3 THE INVERSE TRANSFORM
The great thing about Sturm–Liouville transforms is that the inversion is so easy. Recall that
the generalized Fourier series of a function f (x)is

b
∞ ∞
n (x,λ n ) n (ξ, λ n ) n (x)

f (x) = f n (ξ)p(ξ) dξ = F(λ n ) (9.19)
n n n 2
n=1 a n=1
where the functions n (x,λ n )form an orthogonal set with respect to the weight function p(x).
Example 9.2 (The cosine transform). Consider the diffusion equation

y t = y xx 0 ≤ x ≤ 1 t > 0
y x (0, t) = y(1, t) = 0

y(x, 0) = f (x)

To find the proper kernel function K(x,λ) we note that according to Eq. (9.16) n (x,λ n )
must satisfy the Sturm–Liouville equation

[ n (x,λ)] =−p(x) n (x,λ)

where for the current problem

d 2
[ n (x,λ)] = [ n (x,λ)] and p(x) = 1
dx 2
along with the boundary conditions (9.11)
N α [ (x,λ)] x=a = x (0,λ) = 0

N β [ (x,λ)] x=b = (1,λ) = 0